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Sixth Edition February 2014

Satellite Platform Design Peter Berlin

Department of Computer Science Electrical and Space Engineering Luleå University of Technology Kiruna, Sweden

Satellite Platform Design

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Sixth Edition

Co p by yri Lu ght leå © Un 201 ive 4, rsi ty o

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Peter Berlin

Luleå University of Technology Department of Computer Science Electrical and Space Engineering Kiruna, Sweden

olo gy . hn ec fT Co p by yri Lu ght leå © Un 201 ive 4, rsi ty o Satellite Platform Design Sixth Edition

The author would like to thank all those who granted permission to reproduce their work. Every effort has been made to seek permission to reproduce copyright material in this book. The author would like to hear from any copyright holder not here acknowledged, and will be happy to correct any errors or omissions in future editions.

ISBN 978-91-637-5330-5

Copyright © 2014, by Luleå University of Technology. All rights reserved. Printed in the United Kingdom. No part of this publication may be reproduced or distributed in any form or by any means, or stored in a data base retrieval system, without prior written permission of Luleå University of Technology, Department of Computer Science, Electrical and Space Engineering, P.O. Box 812, SE-98128 Kiruna, Sweden.

To order additional copies, email the author on [email protected].

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Contents

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Table of Contents

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Foreword .............................................................................................................................. 1 Foreword to the 6th edition .................................................................................................. 3 Acknowledgments................................................................................................................ 5 1. Design Drivers ............................................................................................................. 7 2. Orbital Dynamics ....................................................................................................... 11 2.1. Introduction ........................................................................................................ 11 2.2. Planetary Orbits ................................................................................................. 11 2.3. Satellite Orbits ................................................................................................... 13 2.3.1. Classic Orbital Elements ............................................................................ 13 2.3.2. True Anomaly vs. Elapsed Time ............................................................... 20 2.3.3. Orbital Elements Summary ........................................................................ 23 2.3.4. Derived Orbital Parameters........................................................................ 25 2.3.5. Orbital Perturbations .................................................................................. 29 2.4. Mission Analysis ................................................................................................ 32 2.4.1. Coordinate Systems ................................................................................... 32 2.4.2. Subsatellite Track....................................................................................... 33 2.4.3. Ground Station Coverage ........................................................................... 40 2.4.4. Eclipse ........................................................................................................ 43 2.4.5. Sun Angle................................................................................................... 50 2.4.6. Launch Windows ....................................................................................... 53 2.4.7. Launch Date and Launch Time .................................................................. 56 2.5. Specialized Orbits .............................................................................................. 60 2.5.1. The Geostationary Orbit (GEO)................................................................. 60 2.5.2. Sun-Synchronous Orbits (SSO) ................................................................. 70 2.5.3. Highly Eccentric Orbits (HEO) ................................................................. 72 2.5.4. Low Earth Orbits (LEO) ............................................................................ 73 2.6. Solved Problems ................................................................................................ 76 3. Power Management ................................................................................................... 77 3.1. Introduction ........................................................................................................ 77 3.2. Primary Power Supply ....................................................................................... 78 3.2.1. Solar Cell Design ....................................................................................... 78 3.2.2. Solar Cell Performance .............................................................................. 80 3.2.3. Solar Arrays ............................................................................................... 86 3.3. Energy Storage ................................................................................................... 89 3.4. Power Conditioning and Distribution ................................................................ 92 3.4.1. Voltage Conversion ................................................................................... 92 3.4.2. Voltage Stabilization.................................................................................. 93 3.5. Fuel Cells ........................................................................................................... 97 3.6. Solved Problems ................................................................................................ 98 4. Attitude Management................................................................................................. 99 4.1. Introduction ........................................................................................................ 99 4.2. Attitude Stabilization ....................................................................................... 100 4.2.1. Gyroscopic Stabilization .......................................................................... 102 4.2.2. Gravity Gradient Stabilization ................................................................. 114 4.2.3. Geomagnetic Stabilization ....................................................................... 119

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4.3. Attitude Measurement ...................................................................................... 120 4.3.1. Definition of Axes.................................................................................... 121 4.3.2. Attitude Measurement Hardware ............................................................. 122 4.3.3. Sun Sensors .............................................................................................. 123 4.3.4. Earth Sensors ........................................................................................... 128 4.3.5. Star Trackers ............................................................................................ 134 4.3.6. Magnetometers ......................................................................................... 135 4.3.7. Gyroscopes ............................................................................................... 136 4.3.8. GPS .......................................................................................................... 141 4.3.9. Attitude Sensor Summary ........................................................................ 142 4.4. Attitude Determination .................................................................................... 143 4.4.1. Measurement Data Filtering .................................................................... 143 4.4.2. Resolving Attitude Ambiguities .............................................................. 144 4.4.3. The Attitude Matrix ................................................................................. 144 4.5. Attitude Control ............................................................................................... 148 4.5.1. Spin-Stabilized Satellites ......................................................................... 153 4.5.2. Body-Stabilized Satellites ........................................................................ 158 4.5.3. Alternatives to Thruster Control .............................................................. 163 4.6. Solved Problems .............................................................................................. 168 5. Orbit Management ................................................................................................... 169 5.1. Introduction ...................................................................................................... 169 5.2. Orbit Measurement .......................................................................................... 170 5.2.1. Angle Tracking ........................................................................................ 170 5.2.2. Slant Ranging ........................................................................................... 171 5.2.3. Laser Ranging .......................................................................................... 176 5.2.4. Range Rate ............................................................................................... 178 5.2.5. Using the Global Positioning System ...................................................... 181 5.3. Orbit Determination ......................................................................................... 184 5.3.1. State Vectors ............................................................................................ 184 5.3.2. Computing COE from Topocentric Data ................................................. 187 5.3.3. Refining the Orbit Determination Accuracy ............................................ 192 5.4. Orbit Control .................................................................................................... 193 5.4.1. Control Principles .................................................................................... 193 5.4.2. Choice of Intersecting Points ................................................................... 196 5.4.3. Determination of Velocity Vectors .......................................................... 198 5.4.4. Propellant Consumption........................................................................... 201 5.4.5. Orbit Control Strategies ........................................................................... 203 5.4.6. Sample Propellant Budget........................................................................ 218 5.5. Solved Problems .............................................................................................. 221 6. Propulsion ................................................................................................................ 223 6.1. Introduction ...................................................................................................... 223 6.2. Thermodynamic Propulsion ............................................................................. 224 6.2.1. Propellant Characteristics ........................................................................ 224 6.2.2. Subsystem Architecture ........................................................................... 227 6.2.3. Propellant Storage .................................................................................... 228 6.2.4. Propellant Distribution ............................................................................. 230 6.2.5. Thrusters .................................................................................................. 232 6.3. Electrodynamic Propulsion (EP)...................................................................... 234 6.3.1. Electrothermal Thrusters .......................................................................... 235 6.3.2. Electrostatic Thrusters ............................................................................. 236

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6.3.3. Electromagnetic Thrusters ....................................................................... 238 6.4. Thruster Layout ................................................................................................ 239 6.5. Performance Summary..................................................................................... 243 6.6. Solved Problems .............................................................................................. 244 7. Tracking, Telemetry and Command (TT&C) .......................................................... 244 7.1. Introduction ...................................................................................................... 245 7.2. Telemetry and Telecommand Structuring ....................................................... 247 7.2.1. Telemetry ................................................................................................. 247 7.2.2. Telecommand ........................................................................................... 253 7.2.3. Bit Rates and Error Coding ...................................................................... 255 7.3. RF Portion ........................................................................................................ 260 7.3.1. Tracking and Ranging .............................................................................. 260 7.3.2. Choice of Radio Frequency ..................................................................... 261 7.3.3. Signal Quality Requirements ................................................................... 262 7.3.4. Satisfying the Quality Requirements ....................................................... 272 7.3.5. Noise and Attenuation.............................................................................. 285 7.4. Solved Problems .............................................................................................. 296 8. Onboard Data Handling (OBDH) ............................................................................ 297 8.1. Introduction ...................................................................................................... 297 8.2. Onboard Computer........................................................................................... 298 8.3. Data Storage ..................................................................................................... 300 8.4. Data Distribution.............................................................................................. 301 8.5. Remote Terminal Unit ..................................................................................... 304 8.6. OBDH Design Aspects .................................................................................... 305 9. Structures and Mechanisms ..................................................................................... 307 9.1. Introduction ...................................................................................................... 307 9.2. Structures ......................................................................................................... 308 9.2.1. Mechanical Configuration ....................................................................... 308 9.2.2. Shapes and Materials ............................................................................... 310 9.2.3. Quasi-Static and Dynamic Loads............................................................. 313 9.2.4. Mathematical Modelling .......................................................................... 318 9.3. Mechanisms ..................................................................................................... 320 9.4. Solved Problems .............................................................................................. 323 10. Thermal Management .......................................................................................... 325 10.1. Introduction ...................................................................................................... 325 10.2. Thermodynamic Principles .............................................................................. 326 10.3. Radiation .......................................................................................................... 327 10.3.1. Thermal Equilibrium ................................................................................ 327 10.3.2. Thermal Gradients ................................................................................... 331 10.3.3. Thermal Interaction .................................................................................. 336 10.3.4. Local Spacecraft Thermal Modelling ...................................................... 338 10.4. Conduction ....................................................................................................... 347 10.5. Hardware Implementation ............................................................................... 351 10.5.1. Passive Thermal Control .......................................................................... 351 10.5.2. Active Thermal Control ........................................................................... 355 10.5.3. Layout Summary...................................................................................... 356 10.6. Solved Problems .............................................................................................. 357 11. Launch Vehicle Selection .................................................................................... 359 11.1. Introduction ...................................................................................................... 359 11.2. Rocket Engine Architecture ............................................................................. 361

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11.3. Ascent and Injection ........................................................................................ 362 11.4. Choice of Launch Sites .................................................................................... 364 11.5. Launch Vehicle Profiles ................................................................................... 366 11.5.1. Small Vehicles ......................................................................................... 366 11.5.2. Medium Vehicles ..................................................................................... 367 11.5.3. Heavy Vehicles ........................................................................................ 368 11.6. Launch Vehicle Performance ........................................................................... 371 11.6.1. Overview .................................................................................................. 371 11.6.2. GTO or Direct Injection into GEO .......................................................... 373 11.6.3. Computation of Performance ................................................................... 374 11.7. Designing a Launch Vehicle ............................................................................ 380 11.7.1. Vehicles Without Strap-On Boosters ....................................................... 380 11.7.2. Vehicles With Strap-On Boosters ............................................................ 383 11.8. Launch Vehicle Reliability .............................................................................. 385 11.9. Launch Vehicle Availabilisty .......................................................................... 388 11.10. Launch Costs ................................................................................................ 389 11.11. Making the Final Selection .......................................................................... 389 11.12. Solved Problems .......................................................................................... 391 12. Launch and Space Environment .......................................................................... 393 12.1. Introduction ...................................................................................................... 393 12.2. Launch Environment ........................................................................................ 393 12.2.1. Static Loads .............................................................................................. 393 12.2.2. Dynamic Loads ........................................................................................ 395 12.3. Space Environment .......................................................................................... 399 12.3.1. Weightlessness ......................................................................................... 399 12.3.2. Vacuum .................................................................................................... 399 12.3.3. Aerodynamic Drag ................................................................................... 401 12.3.4. Atomic Oxygen ........................................................................................ 404 12.3.5. Van Allen Belts ........................................................................................ 404 12.3.6. Meteorids and Man-Made Debris ............................................................ 406 12.3.7. Galactic Cosmic Radiation ...................................................................... 408 12.3.8. Solar Wind and Particle Showers ............................................................ 408 12.3.9. Solar Pressure........................................................................................... 408 12.3.10. Ultraviolet Radiation............................................................................ 411 12.4. Internal Environment ....................................................................................... 411 12.5. Solved Problems .............................................................................................. 413 13. Product Assurance ............................................................................................... 415 13.1. Introduction ...................................................................................................... 415 13.2. Parts Engineering ............................................................................................. 415 13.3. Materials and Processes ................................................................................... 416 13.4. Reliability......................................................................................................... 417 13.4.1. Mathematical Modelling .......................................................................... 417 13.4.2. Component Derating ................................................................................ 419 13.4.3. Redundancy.............................................................................................. 419 13.5. Quality Assurance ............................................................................................ 431 13.6. Configuration Management ............................................................................. 431 13.7. Safety and Risk ................................................................................................ 432 13.8. Solved Problems .............................................................................................. 433 14. Development and Test ......................................................................................... 435 14.1. Introduction ...................................................................................................... 435

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14.2. Satellite Development ...................................................................................... 436 14.3. Qualification and Acceptance .......................................................................... 437 14.4. Assembly, Integration and Test ....................................................................... 437 14.5. AIT Facilities ................................................................................................... 437 14.5.1. Static Loads .............................................................................................. 438 14.5.2. Dynamic Loads ........................................................................................ 439 14.5.3. Temperature and Vacuum ........................................................................ 440 14.5.4. Weightlessness ......................................................................................... 442 14.5.5. Antenna Pattern Tests .............................................................................. 443 14.6. AIT at the Launch Site ..................................................................................... 444 14.7. In-Orbit Testing ............................................................................................... 444 14.8. Overall AIT Sequence...................................................................................... 445 14.9. Solved Problems .............................................................................................. 447 Appendix A: Spherical Trigonometry ............................................................................ 451 Appendix B: Matrix Algebra .......................................................................................... 453 Appendix C: Solved Problems ........................................................................................ 465 Appendix D: Equation Summary .................................................................................... 484 Appendix E: Spacecraft Hardware Internet Links .......................................................... 503 Appendix F: Launch Vehicle User Manual Internet Links ............................................. 509 Abbreviations ................................................................................................................... 511 List of Symbols ................................................................................................................ 513 References ........................................................................................................................ 515 Index ................................................................................................................................ 517

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Foreword

Foreword

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This is a book about designing unmanned, earth-orbiting satellites. More specifically, it is about the satellite platform rather than the payload. The idea is to provide scientists, engineers and students with the knowledge and the basic mathematical tools to develop an overall satellite and mission design based on a predefined payload.

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Why yet another book on the subject of spacecraft design? The aim has been to combine completeness, accessibility and editorial consistency with utmost affordability. Readily solvable equations and over 350 original illustrations facilitate the reader’s comprehension of the theories involved. The price is kept low by minimizing the production, marketing and distribution costs.

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As for the level of required mathematical proficiency, the reader will manage with basic algebra (including vectors), trigonometry (including spherical triangles), second-order differential equations, Laplace transforms, simple integration, and 3 x 3 matrix manipulation.

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The book covers the following traditional subsystems, listed along with their analogies in the human physiology: • • • • • • • •

Structures and Mechanisms Thermal Management Power Management Attitude Management Orbit Management Propulsion Tracking, Telemetry and Control Onboard Data Handling and Storage

Skeleton Skin Digestive system Eyes and balance Sense of place Legs and feet Speech and hearing Brain and nervous system

In support of the above, the book includes the following programmatic subjects: • • • • • • •

Primary Design Drivers Launch Vehicle Overview Launch and Space Environment Orbital Dynamics Mission Analysis Product Assurance, including Reliability Development and Test

Where appropriate, the chapters conclude with suggestions for exam problems. The Appendices contain basic mathematical tools, a quick reference to all the equations used in the book, solutions to the exam problems, and linksto the websites of equipment manufacturers and launch servicee providers.

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Foreword to the 6th Edition

Foreword to the 6th edition

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Since its initial publication in 2004, Satellite Platform Design has found its way to universities and corporations in Europe, America, Australia, Africa and Asia. Judging by reader comments, the book has achieved its goal of conveying an intuitive understanding of the physics and engineering involved in spacecraft design.

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In addition to some technical updates and editorial improvements, the 6th Edition includes enhanced descriptions of launch windows, triple junction solar cells, magnetometers, reaction wheels, attitude feedback control, GPS, Hamming code, onboard computers, onboard data storage, heat pipes, launch vehicle reliability, and launch vehicle selection strategy. The Internet links to manufacturers’ data sheets in Appendix E have been updated, and a new Appendix F has been added containing links to launch vehicle user manuals that may be downloaded from the Internet. The Index has been completely reworked for improved relevance.

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As before, illustrations are used extensively to supplement the descriptive narrative. Many of these have been developed further, and several illustrations and diagrams have been added. For clarity, outline drawings are preferred over photographs in most cases. Buckfastleigh, UK, March 2014 Peter Berlin

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Acknowledgments

Acknowledgments

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Satellite Platform Design is the standard textbook for courses in spacecraft engineering at the Department of Space Science of Luleå University. A respectable number of copies have also found their way to universities and corporations around the world. I am indebted to my readers who have offered suggestions for improvement of the earlier editions.

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I would like to thank Dr Uwe Feucht of ESA, MM Torbjörn Hult and Lars Ljunge of Ruag Sweden, Mr Baard Eilertsen of Kongsberg, and MM Priya Fernando and Martin Bohm of the Department of Space Science, for taking the time to review the original manuscript and offer valuable advice.

About the Author:

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Last but not least, I would like to thank Cambridge University Press for allowing me to reproduce illustrations from my book The Geostationary Applications Satellite; and John Wiley & Sons for giving me permission to adapt a drawing from Principles of Communications Satellites by Gary D. Gordon and Walter L. Morgan.

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Peter Berlin holds a Master’s Degree in Aeronautical Engineering from the Royal Institute of Technology in Stockholm, Sweden. After 25 years in engineering and project management at ESA and Inmarsat, he took early retirement in 1993 to engage in consulting, teaching and writing. He continues to advise spacecraft manufacturers in matters of systems engineering. His writing credentials include four books and some 100 newspaper and magazine articles on subjects ranging from spacecraft engineering to project management and cross-cultural awareness. He is also the translator of Roads to Space, a collection of memoirs by 35 Soviet space pioneers published by McGraw-Hill (ISBN 0-07607095-6).

Copies of Satellite Platform Design may be ordered directly from the author on [email protected].

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1 / Design Drivers

1. Design Drivers

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Designing a spacecraft is an iterative process. The mission definition is the obvious starting point, and the design of the spacecraft’s payload is the logical next step. Beyond that, the project team has to go through several design cycles before it is ready to start cutting metal. The point of entry into the design cycle could be the structure, the power supply, the stabilization method, or almost any other subsystem onboard the spacecraft. A better approach is to structure the work along a logical path, as suggested in Figure 1-1 for a typical satellite, and we have followed this logic in the table of contents of the present book. . Mission design

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1

Choice of orbit

Design drivers

Development & test philosophy

Product assurance

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Power management

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Orbit management

Propulsion

TT&C

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Space & launch environment

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Choice of launch vehicle

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Iteration loop

Attitude management

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Payload definition

Chapter

Onboard data handling & storage

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8

Thermal management

Structure & mechanisms

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Figure 1-1 Iterative methodology for designing a satellite. The circled numbers refer to Chapters.

The logic in the above diagram is based on the thought that the payload has a direct influence on the primary design drivers (1) and on the choice of orbit (2), all of which determine the sizing of the power supply (3). The mission design and the approach to electric energy supply through sunlight conversion is a major factor in the choice of attitude stabilization method (4). Propulsion is needed for both attitude and orbit control (5)(6). Control from the ground is exercised remotely via the tracking, telemetry and command subsystem (7). The avionics side of the satellite is crowned by a “brain” in the guise of an onboard data handling system (8). We now have a clear outline of the satellite and are therefore in a position to draw a structure (9). The next question is how to keep the onboard equipment within acceptable temperatures; the answer is found in our chosen means of thermal control (10). Knowing the mass and volume of the spacecraft, we may suggest a suitable launch vehicle (11). The launch environment is severe, as is the cosmic environment, and both need to be taken into account (12). With the satellite defined at least in outline, the time is ripe to 7

1 / Design Drivers scope the quality and reliability provisions (13), and to establish a test philosophy for the overall spacecraft (14).

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The most common type of earth-orbiting spacecraft is the geostationary communications satellite, and we will use here it to illustrate the above logic on a practical level. The underlying requirements come in two parts:

Functional specifications, by which the satellite owner informs the manufacturer what his primary expectations are. In response to the functional specifications, the manufacturer issues –

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Implementation specifications, in which the manufacturer tells the owner in considerable detail how the satellite will look and perform.

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Notice the difference between telling someone what to build and how to build it. The theory is that the customer knows best what is needed by the future users of the satellite, while the manufacturer has many years of experience in how to design, develop, manufacture and test satellites. In order to produce a cost-effective mission, the two sides are well advised to keep their respective roles separate. However, in practice this role division is not always respected, especially when the owner is already operating multiple spacecraft procured from different manufactuers. Such an owner will have developed strong views on how satellites should be built and is not afraid to say so in his "functional" specifications. In the following illustration, the text in bold is representative of a functional specification, while the Consequence is a telegraphic version of the implementation specifications. a) Mission definition: The satellite is to relay radio signals between at least two points on the ground, day and night, without interruption. Consequence: The orbit must be geostationary, travelling around the earth at the same speed and in the same direction as the earth rotates around its North-South axis. The geocentric altitude of a geostationary orbit is 42,164 km; the height above the equator is 35,786 km. b) Mission lifetime: The satellite is required to function in orbit during X years with a probability of Y percent (typically, X = 15 years and Y = 70%). Consequence 1: The specified lifetime is to be met by adopting a rigorous quality and reliability programme, including the duplication of critical components and units. Consequence 2: The satellite’s consumables (notably the propellant, the batteries and the solar cells) are depleted over time and must therefore be oversized to last the mission lifetime. As we shall see in the following, the lifetime requirement has a major impact on the satellite’s launch mass.

c) Signal quality: For the received signal to be intelligible on the ground (whether voice or data), the received power has to be greater than the received noise by a factor of at least 10. Consequence: This puts requirements on various satellite design parameters, notably the power consumption of the transmitter and the size of the antennas.

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1 / Design Drivers

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d) Power supply: The functional units onboard the satellite – and the highpower transmitter in particular – shall be supplied with electric energy 24 hours a day, 365 days a year. Consequence 1: Sunlight is the most common energy source (average power density near the earth is 1367 W/m2), and photovoltaic cells are used to convert sunlight to electric current. The size of the solar panels is proportional to the expected peak power consumption at the mission end-of-life. Consequence 2: During two 6-week periods each year, the satellite will spend up to 72 minutes in the earth’s shadow, during which the solar cells produce no electric current. This energy deficit must be filled by onboard rechargeable batteries.

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e) Satellite orientation: The satellite’s attitude in space shall be such that the antennas are pointed at the earth and the solar cells receive continuous illumination. Consequence: For the satellite to track both celestial bodies accurately, it must be hinged in some way, and the hinge axis (the dash-dot line in the figure) should be perpendicular to the sun-earth-satellite plane (Figure 1-2).

Figure 1-2 Orientation of GEO satellites.

f) Physical volume: The satellite shall fit inside the heatshield of the launch vehicle (typically 10 m x 4 m for the heaviest launch vehicles, but constrained by the “useable volume” inside the dotted contour). Consequence: Because the heatshield is bullet-shaped, the satellite should have a similar shape if it is to make efficient use of the available volume. Appendages like solar panels and antennas must be stowed, necessitating the inclusion of deployment mechanisms. The spacecraft’s main body must be large enough for the appendages to be folded up against it.

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g) Physical mass: The satellite shall be light enough for the chosen launch vehicle to carry it to the intended orbit (max 6 tons). It shall also carry enough propellant to make the necessary orbit and attitude adjustments during the satellite’s mission lifetime. Consequence: If a chemical propellant is chosen, the satellite’s “wet” mass (i.e. launch mass) is almost twice its “dry” mass, and is a severe challenge to the first requirement (of lightness). To reduce the wet mass, it is possible to switch from chemical to electric propulsion;

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1 / Design Drivers however, electric thrusters consume large amounts of electric energy, forcing the size and weight of the solar panels and the batteries to be augmented.

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Let us stand back now and see how far we have come in defining the satellite. Knowing the mission definition and the signal quality requirements, we may estimate the power consumption of the communications payload and of the supporting platform subsystems. We are therefore in a position to calculate the size of the solar panels and the mass of the batteries. The peculiarities of the geostationary orbit and of candidate launch vehicles tell us how much propellant we must take onboard to last the mission lifetime. The size of the solar panels and of the propellant tanks give us a fairly accurate estimate of the satellite’s shape and volume, and the calculated mass of the propellant says a great deal about the likely weight of the overall satellite.

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After this brief back-of-the-envelope analysis, we may begin the iterative process of designing the satellite in detail, as per the diagram in Figure 1-1. We may also search more systematically for a suitable launch vehicle which, after all, is going to add between 30% and 50% to the cost of having the satellite designed and built.

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The example we chose was a geostationary communications satellite with a mission lifetime of 15 years. Obviously the shape, volume, mass and functionality of the satellite would have turned out quite differently if we had chosen another payload (e.g. in support of science, remote sensing, meteorology, navigation, or military intelligence gathering) – the more so if the satellite were to be deployed in a low-altitude orbit with only a 5-year lifetime. However, the methodology of identifying the design drivers would have been similar. As we make the first round through the cycle in Figure 1-1, we will discover a number of conceptual incompatibilities. The designers of the orbit management subsystem may favour electric propulsion, and the propulsion engineers may have identified a suitable thruster technology, but the power management people have not yet factored in the attendant high power consumption and its consequences on the size of the solar panels and the batteries. Doubling the size of of these items is easy on paper, but now the structures people are faced with having to increase the size of the satellite’s main body so that the panels can be stowed during launch. Those responsible for choosing a launch vehicle complain that the satellite is becoming too large for their preferred vehicle ... and so forth. An intensive dialogue takes place within the project team. In the process of resolving the incompatibilities, the team is in fact going around the iterative cycle one more time, and will keep doing so until the mission and the satellite design are a perfect match. The iterative process is complicated further by colleagues in procurement, contracts, finance and marketing getting into the fray. For example, for a commercial mission to make sense, the satellite along with its launcher and ground stations must be affordable in relation to the expected revenue. It is therefore natural that non-technical staff should have a big influence on the overall technical solution.

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2 / Orbital Dynamics

2.1.

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2. Orbital Dynamics Introduction

The orbit is the harness that makes a satellite perform useful tasks, rather than hurtle aimlessly into deep space. The laws of Newton and Kepler allow us to tailor the size, shape and orientation of the orbit to suit our needs. The mathematical modelling is straightforward, at least in the first approximation.

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Some common orbit types are illustrated in Figure 2-1.

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In common space parlance, an orbit is a closed contour, in contrast to trajectories which are open-ended. Thus, a satellite travels in an orbit, whereas a rocket or a deep-space probe flies along a trajectory.

Low orbits

High orbits

Equatorial orbits

Elliptic orbits

Polar orbits

Inclined orbits

Figure 2-1 Basic orbit terminology.

2.2.

Planetary Orbits

Mediaeval astronomers (notably Copernicus) assumed that the orbits of planets in the solar system were circular, and that the planets themselves moved around the sun at constant speed. On closer examination the astronomers discovered that the observed speed actually varied, however slightly. Ensconced in their belief that orbits were by necessity circular, and that therefore the speeds had to be constant, they deduced that the orbits themselves might not be concentric, i.e. that their centre points were offset from the sun. This would explain why planetary speeds appeared to vary when observed from the sun and, by extension, from the earth (Figure 2-2). But despite elaborate mathematical offset modelling, they never succeeded in fully explaining these variations in planetary velocity.

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∆φ1

∆φ2

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2/ Orbital Dynamics

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Figure 2-2 In an eccentric system, the angular velocity ∆φ1/∆ ∆φ1/∆t is lower than ∆φ2/∆t, ∆φ2/∆ creating the illusion that the orbital velocity of the planet varies.

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Johannes Kepler (1571 - 1630) was the first to suggest that the orbits of planets might not be truly circular, but were in fact slightly elliptical. He proceeded to formulate three laws that offered a satisfactory explanation to the irregularities in planetary motion

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Kepler’s First Law states that the generic orbit is an ellipse with two focal points, one of which is occupied by the sun (Figure 2-3). Planet

Focal points

r

Aphelion

•

Sun

Perihelion

•

Figure 2-3 Kepler’s First Law

The point of the orbit closest to the sun is called the perihelion. The point furthest away from the sun is known as the aphelion. For example, the earth passes through the perihelion around 3 January each year. Kepler’s Second Law states that the radius vector r sweeps across equal areas of the ellipse at equal times (Figure 2-4).

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2 / Orbital Dynamics

Aphelion ∆t

olo gy .

∆t Perihelion

Figure 2-4 Kepler’s Second Law

ec

hn

One fundamental consequence of the Second Law is that a planet moves more slowly through the aphelion than through the perihelion. Worded differently, a planet decelerates as it moves away from the sun, and accelerates as it moves closer.

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Kepler’s Third Law says that the orbital period τ squared is proportional to the semimajor axis a cubed, i.e. τ 2 = const. * a 3

(2.1)

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The orbital period is defined as the time it takes the planet to travel a complete turn around the sun.

2.3.

Satellite Orbits

The above discussion concerning planetary motion around the sun is equally valid for the motion of artificial satellites around the earth. The equivalent term for aphelion is apogee, and perihelion becomes perigee.

2.3.1. Classic Orbital Elements

Orbital analysts tailor the shape, size and position of orbits to suit the mission of individual satellites. For example, the optimum position for most telecommun-ications satellites is in the geostationary orbit (GEO) above the equator, where they travel around the earth with the same angular velocity that the earth spins around its axis. Seen from a terrestrial observer, these GEO satellites appear stationary over a fixed point in the sky, which makes them eminently suitable for relaying radio signals between points on earth, day and night. Satellites that take pictures of the earth may require global coverage, including the Arctic regions, and therefore the orbits are inclined to cover the poles. Mathematically speaking, all orbits are elliptic to a higher or lesser degree. A circular orbit is simply an elliptic orbit whose two focal points coincide. Traditionally, orbits are defined by five orbital elements denoted a, e, i, Ω and ω. The first two elements determine the shape of the orbital ellipse, while the remaining three establish the position of the orbit in space. A sixth element, denoted ν, gives the position

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2/ Orbital Dynamics of the satellite along the orbit. The definition of each of these elements will be described in the following.

b

olo gy .

Semimajor axis a: The ellipse is formed by its major axis = 2a and its minor axis = 2b (Figure 2-5).

r

a

a

ec

hn

b

Figure 2-5 Axes of the orbital ellipse.

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With a being half the distance of the major axis, it is referred to as the semimajor axis. Similarly b is called the semiminor axis. The satellite moves around the earth at a variable distance r, known as the radial distance, or altitude. The corresponding vector r is called the radius vector. The choice of semimajor axis a determines the height above the earth where the satellite will operate (e.g. to achieve optimum radio signal strength or camera resolution). Often the choice is a compromise between mission priorities and the need to protect the satellite from harmful influences, such as atmospheric drag or radiation in the Van Allen belts.

Note that the orbital height of a satellite is its altitude minus the earth’s radius R. For example, the height of the apogee Ap = ra – R, where R = 6371 km on average and 6378 km at the equator.

Eccentricity e: The eccentricity is a measure of the flatness of the ellipse. One such measure could have been the ratio b/a or, alternatively, 1-b/a. However, in order to simplify orbital calculations in a wider context, we introduce the distance 2c between the focal points, and define the eccentricity as e = c/a (Figure 2-6).

a

b

a

a

c

b

Figure 2-6 Definition of eccentricity e.

14

2 / Orbital Dynamics The position of the focal points is defined by c as shown in Figure 2-6. With c2 = a2 – b2, we have:

c b2 e = = 1− 2 a a

(2.2)

olo gy .

It follows that 0 ≤ e ≤ 1. In the case of a circular orbit, the two focal points coincide, such that c = e = 0. At the other extreme (e = 1), the orbit is flattened to a straight line between the two focal points.

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An orbital analyst may choose a value for the eccentricity e ≠ 0 to create variations in orbital height and velocity. For example, a scientific experiment may be designed to sample the earth’s atmosphere at different heights during each orbital revolution, so as to establish the physical or chemical composition of the atmosphere. But most satellites have no such requirement and therefore travel in circular orbits (e = 0).

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Inclination i: Since a satellite orbit is anchored to the centre of the earth, the question arises whether the satellite travels in an east-west direction along the equator, or in a north-south direction from Pole to Pole, or somewhere in between. The angle i that defines the slope of the orbital plane in relation to the equator is called the inclination angle (Figure 2-7).

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Figure 2-7 shows the projection of a typical orbit onto the earth’s surface – the so-called subsatellite track. The circular arrow around the North Pole indicates the sense of the earth’s rotation from west to east. The point where the satellite crosses the equator travelling northward is known as the ascending node. The opposite crossing is therefore called the descending node. N

N

Ascending node

i

i

Ascending node

Figure 2-7 Definition of the inclination angle i. The spacecraft movement in the left figure is prograde (i.e. in the same direction as the earth’s spin), while the movement in the right figure is retrograde (i.e. in the opposite direction).

Note that the inclination i is the angle to the right of the ascending node. The inclination is an angle of rotation around the line between the nodes. It follows that i = 0° denotes an equatorial orbit, and i = 90° is a polar orbit. When i < 90°, the satellite’s motion is said to be prograde, with an eastward component in the same direction as the earth’s own rotation. With i > 90°, the satellite’s movement becomes retrograde, with a westward component running counter to the earth’s rotation.

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2/ Orbital Dynamics

olo gy .

The inclination allows the orbital analyst to formulate the degree of global coverage. An equatorial satellite (i = 0°) only covers the equatorial zone, whose width is determined by the orbital height (i.e. by a). A polar satellite (i = 90°) covers the entire globe, over time, as the earth rotates underneath the subsatellite track. For example, during the 90 minutes it takes a low-flying satellite to complete an orbital revolution, the earth will turn 22.5° around its axis (90 / [24 · 60] · 360° = 22.5°). Intermediate values offer coverages which are better than equatorial but less than global.

hn

Why, then, don’t all satellites travel in polar orbits? Is global coverage not always preferable to a more limited coverage? No, not necessarily. For example, if the geostationary orbit were polar instead of equatorial, the satellite would no longer stay “fixed in the sky.”

N

fT

ec

Right ascension of the ascending node Ω (abbreviated RAAN): Recall that the inclination is an angle around the line between the nodes. Similarly, the RAAN is an angle of rotation around the Poles, i.e. it defines the position of the nodes in space. For this definition to be meaningful, we need a spatial reference, such as the line formed by the intersection between the equatorial plane and the ecliptic (Figure 2-8). The ecliptic is the orbital plane of the earth around the sun. The earth’s equator is inclined at an angle ε = 23.45° relative to the ecliptic.

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Equatorial plane

Ecliptic plane

ε

Vector to First Point of Aries

ϒ

Figure 2-8 Definition of reference vector in inertial space seen from the side of the ecliptic plane.

The desired reference vector passes along the line of intersection between the ecliptic and equatorial planes and points towards the first point in the star constellation Aries, traditionally represented by a ram’s horns (ϒ ϒ). Looking at the sun and the earth from the ecliptic north (Figure 2-9), the reference vector is the line from the sun to the first point of Aries, as seen from the earth during the vernal (spring) equinox.

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2 / Orbital Dynamics

olo gy .

21 March Vernal equinox

21 December Winter solstice

21 June Summer solstice

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21 September Autumn equinox

ec

ϒ

Figure 2-9 Inertial reference vector seen from the north.

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N

fT

Using the nodal vector as reference, the RAAN is defined as the geocentric angle of the ascending node in the prograde direction (Figure 2-10).

Ω

i

ϒ

Figure 2-10 Definition of RAAN.

The choice of the RAAN (Ω) determines the orientation of the orbital plane with respect to the sun, and hence the solar illumination of the satellite. For example, a judicious choice of Ω might allow the satellite to avoid passing through the earth’s shadow, such that the solar panels remain illuminated at all times. An orbit’s initial RAAN is determined by the time of launch on any given day. The orbital analyst can therefore achieve the desired value of Ω by prescribing the allowable dates and times of the day when the launch may take place (the so-called launch window – see Section 2.4.7). Argument of perigee: The inclination and the RAAN imply rotations of the orbit outside of its own plane. Stated differently, i and Ω represent rotation angles for the orbital plane around two degrees of freedom, namely the nodal line and the polar axis, respectively. It now remains to define the rotation angle of the orbital ellipse within its own plane (Figure 2-11). The angle ω between the line of apsides (pronounced 17

2/ Orbital Dynamics

Perigee Nodal line

ω

hn

Line of apsides

olo gy .

“ápsidees”) and the nodal line (i.e. the equator) is called the argument of perigee. It is equivalent to a rotation around the orbit plane normal, and is measured positive in the direction of satellite motion.

ec

Apogee

Figure 2-11 Definition of the argument of perigee ω.

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The argument of perigee ω, in combination with the inclination i, establishes the latitude of the apogee and the perigee, which may be of interest e.g. in certain scientific missions where the nature of samples is latitude dependent.

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True anomaly ν: Let us return to Figure 2-3 and introduce the true anomaly ν, i.e. the angle of the radius vector r with respect to the perigee (Figure 2-12).

r

ν

Apogee

Perigee

Figure 2-12 Definition of true anomaly ν.

Intuitively, one would expect the radial distance r to be a function of a, b and ν or, alternatively, of a, e and ν. It can indeed be shown that a (1 − e 2 ) r= 1 + e cos ν

(2.3)

µ r = 0 , where m is the mass of the r3 satellite, r is the radius vector, r is the radial distance (= |r|), and µ is the earth’s gravitational parameter = 398,601 km3/s2. This differential equation is of fundamental importance, in that it neatly describes the balance between the inertial, Newtonian force

by solving the differential vector equation mr

+ m

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2 / Orbital Dynamics

of a speeding satellite ( m
r
) and the harnessing gravitational pull from the earth ( m

µ r ). r3

A derivation of Eq 2.3 may be found in Appendix C.3 of Sellers [9].

Ap

R

R

hn

olo gy .

To illustrate Eq 2.3, take an orbit with a perigee located at a height Pe = 200 km and an apogee at Ap = 1000 km above the earth. The earth’s average radius R = 6371 km. Let ra = Ap + R be the geocentric altitude of the apogee, and rp = Pe + R the geocentric altitude of the perigee. From Figure 2-13:

Pe

Apogee

Perigee

rp

fT

ec

ra

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Figure 2-13 Calculation of the semimajor axis a.

a = ½(ra + rp )

(2.4)

Inserting the chosen values for Pe and Ap yields a = ½(6571 + 7371) = 6971 km. The eccentricity e is obtained from Eq 2.2: e=

rp r c = 1− = a −1 a a a

(2.5)

In our example: e = 1 – 6571/6971 = 7371/6971 – 1 = 0.057

Having calculated the values of a and e, we are now in a position to plot the radial distance r as a function of the true anomaly ν – see Figure 2-14. The satellite height h = r – R above the earth’s surface is shown in Figure 2-15. Note that, according to Eq 2.3 above, r = a for a circular orbit (e = 0).

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2/ Orbital Dynamics

7400 7200

6400 6200

300

280

260

240

220

200

180

160

140

120

100

80

60

40

20

0

6000

hn

True anomaly ν (deg)

360

6600

340

6800

olo gy .

7000

320

Radial distance r (km)

7600

Figure 2-14 Example of radial distance r as a function of true anomaly ν.

ec

1000

fT

800 600 400

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Satellite height h (km)

1200

200

360

340

320

300

280

260

240

220

200

180

160

140

120

100

80

60

40

20

0

0

True anomaly ν (deg)

Figure 2-15 Example of orbital height h as a function of true anomaly ν.

Having defined the size, shape and position of the orbit through a, e, i, Ω, ω, it remains to calculate the true anomaly (or angular position) ν of the satellite along the orbit at any given time. We must know where the satellite is if we are trying to point our ground station antennas to communicate with it. The true anomaly also determines the relative phasing of multiple satellites in similar orbits.

2.3.2. True Anomaly vs. Elapsed Time

From the definition of true anomaly ν (Figure 2-12) and from Kepler’s Second Law (Figure 2-4), it is evident that, for an elliptic orbit, ν does not grow linearly with time. Figure 2-16 shows the relationship between ν and the elapsed time t from perigee for a typical Ariane 5 geostationary transfer orbit (GTO), whereby Ap = 35,790 km and Pe = 200 km.

20

2 / Orbital Dynamics

10 8 6

olo gy .

Elapse time t (hrs)

12

4 2 0

hn

True anomaly ν (deg)

Figure 2-16 Relationship between ν and t for an Ariane 5 GTO.

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ec

As one would expect from Kepler’s Second Law, the satellite can be seen to cover a lot of ground during the first two hours from perigee (ν ≈ 155°). It then slows down around apogee (ν = 180°), before accelerating again after the apogee passage. The orbital period τ = 10.5 hours.

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In order to establish analytically where the satellite is to be found at any given time – i.e. to correlate ν and t – it is necessary to introduce two additional “anomaly” concepts, namely the eccentric anomaly E and the mean anomaly M. The definition of E is shown in Figure 2-17.

E

Apogee

Perigee

a

a

Figure 2-17 Definition of eccentric anomaly E.

The mean anomaly M is defined as: t M = 2π (radians) τ M or, conversely, t=τ 2π

(2.6) (2.7)

where τ is the orbital period, i.e. the time it takes for the satellite to complete an orbital turn (see Section 2.3.4.1 below). In other words, M is the angle-equivalent of time, and cannot be illustrated geometrically for an elliptic orbit. The eccentric anomaly E is obtained from:

21

2/ Orbital Dynamics e + cos ν 1 + e cos ν  1− e  E = 2 tan −1  tan(ν / 2)  1+ e 

cos E =

(2.8b)

olo gy .

or

(2.8a)

Eq 2.8b lends itself to finding E in the correct quadrant using the ATAN2 function in Excel or Matlab. Note also the following relationship between M and E: M = E − e sin E

(radians)

(2.9)

ec

hn

With the help of these equations, establishing t = f(ν) is straightforward, taking the route t = f(M), M = f(E), and E = f(ν), as per Eq 2.7, 2.9 and 2.8. The relationship is exemplified in Figure 2-16 above.

cos E − e 1 − e cos E

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cos ν =

fT

However, it is usually more interesting to know ν = f(t), i.e. the position of the satellite as a function of elapsed time. To establish this relationship it is necessary to take the route ν = f(E), E = f(M), and M = f(t). From Eq 2.8: (2.10)

Unfortunately, the next step is less straightforward, due to the irreversible nature of Eq 2.9. Given this complication, we may try a series expansion to obtain an explicit expression for E = f(M), and better still for ν = f(M). With some sleight of hand, we find that for small values of e (all angles in radians): 1 5 11 ν = M + (2e − e 3 ) sin M + ( e 2 − e 4 ) sin 2 M + ... (2.11a) 4 4 24 The following closed-form, curve-fit approximation is adequate for e < 0.5 and is easier to use: sin M ν = M + 2 .2 e (2.11b) 1.5 − cos M

The last step, i.e. M = f(t), is covered by Eq 2.6. above. Figure 2-18 shows ν, E and M as a function of ν for the sample transfer orbit. The elapsed time is also shown.

22

2 / Orbital Dynamics Elapsed Time from Perigee (hours) 0

1

2345678 9

10

10.5

360

olo gy .

Anomaly (deg)

300 240

M (deg) E (deg)

180

nν(deg)

120

0 0

60

120

180

240

300

360

ec

True anomaly ν (deg)

hn

60

fT

Figure 2-18 Relationship between various anomalies and time for a typical Ariane 5 geostationary transfer orbit.

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Note that, for circular orbits, the eccentricity e = 0, and therefore M = E = ν. Historical note: It may seem strange that the term “anomaly” (which means “deviation”) is used to label the angles ν, E and M. The reason is that early astronomers found the motion of the planets to be irregular, i.e. they deviated from linear motion; hence the argument of their radius vectors became known as “anomalies.”

2.3.3. Orbital Elements Summary

The semimajor axis a defines the size of the orbit:

while the eccentricity e describes the “flatness” of the orbit:

23

olo gy .

2/ Orbital Dynamics

hn

The inclination i determines the slope of the orbital plane with respect to the equator:

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i

ec

N

Co p by yri Lu ght leå © Un 201 ive 4, rsi ty o

The right ascension of the ascending node Ω is a measure of how the orbit is turned in the east-west direction: N

Ω

ϒ

while the argument of perigee ω measures how far the perigee is from the equator:

24

2 / Orbital Dynamics Lastly, the true anomaly ν indicates the position of the satellite along the orbit:

hn

olo gy .

ν

Alternatively, E or M is sometimes shown instead of ν as the sixth orbital element.

ec

Q: Why are these seemingly arbitrary elements used to describe orbits, rather than a more traditional xyz system?

Co p by yri Lu ght leå © Un 201 ive 4, rsi ty o

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A: r and ν represent polar coordinates, as opposed to xyz which are Cartesian coordinates. Polar coordinates offer more convenient mathematical modelling of physical phenomena involving circular motion, such as orbits. Given that the nodes and the argument of perigee drift fairly rapidly over time, a satellite’s orbit is actually a complex elliptic spiral, as observed in a geocentric Cartesian coordinate system. Using the elements a, e, i, Ω and ω has the benefit of treating the orbit as a “solid disc” at any moment in time, i.e. the disc can be re-shaped and rotated at will within the coordinate system. Because of this, the parameters a, e, i, Ω, ω and ν are sometimes referred to as osculating elements (“osculating” = having points in common).

2.3.4. Derived Orbital Parameters

Using the six classic orbital elements a, e, i, ω, Ω, ν, we are now in a position to calculate a set of parameters of critical importance in mission planning. 2.3.4.1.

Orbital Period

Kepler’s Third Law states that the square of the orbital period τ is equal to a constant multiplied by the cube of the semimajor axis a (Eq 2.1). The precise equation is:

τ2 =

or:

4π 2 3 a µ

τ = 2πa a / µ

(2.12)

where µ is the earth’s gravitational parameter = 398,601 km3/s2. (µ = GM, where the universal constant of gravitation G = 6.6742 x 10-11 m3/kg2/s2 and the earth’s mass M = 5,9736 x 1024 kg.) Given the dimension of µ, it follows that a needs to be entered in km, and that τ is measured in seconds. Note that the orbital period τ is only a function of the semimajor axis a, regardless of the eccentricity or the inclination of the orbit. 25

2/ Orbital Dynamics

30

olo gy .

Orbital period τ (hours)

Figure 2-19 shows τ for a range of a-values, while Figure 2-20 gives τ for various combinations of apogee and perigee heights.

25 20 15 10 5

hn

0

ec

Semimajor axis a (km)

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Figure 2-19 Orbital period τ as a function of circular orbit height h. 25

Perigee height (km)

Oribtal period (hours)

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20 15 10 5 0

0

10000

20000

30000

500 5000 10000 15000 20000 25000 30000 35000

40000

Apogee height (km)

Figure 2-20 Orbital period τ as a function of Ap and Pe.

The orbital period τ allows the mission planner to predict the cyclical nature of mission operations, e.g. the need for battery recharging in preparation for eclipse, the temperature variations of satellite equipment, or the time of satellite passages over a certain ground station. 2.3.4.2.

Orbital Velocity

Recall from Kepler’s Second Law that the satellite’s velocity V along the orbit varies as it travels from the perigee to the apogee and back.

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2 / Orbital Dynamics

2 1 µ −  r a

V=

(2.13)

Vcirc =

olo gy .

Evidently the velocity depends on both a and r, whereby r in itself is also a function of a, e and ν, according to Eq 2.3. In the case of a circular orbit, r = constant = a, and Eq 2.13 is simplified as follows:

µ a

(2.14)

hn

The orbital velocity of a satellite in the Ariane 5 sample transfer orbit is shown in Figure 2-21 as a function of the true anomaly ν.

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10

fT

8 6 4

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Orbital velocity V (km/s)

12

2

360

340

320

300

280

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200

180

160

140

120

100

80

60

40

20

0

0

True anomaly ν (deg)

Figure 2-21 Orbital velocity V as a function of true anomaly ν.

Knowing the speed of a satellite along its orbit is important for a variety of reasons, such as tracking it, predicting the Doppler effect in its radio transmissions, or planning an orbital change manoeuvre. Historical note: The early astronomers derived Eq 2.14 by studying the balance between the gravitational and centrifugal forces (Fg and Fc, respectively) acting on a planet. Given the radius a of the planet’s orbit (obtained from Kepler’s Third Law) and the sun’s gravitational parameter µs, they knew from Newton’s theories that

Fg = µ s

m a2

and

Fc =

mVcirc a

2

Eq 2.14 results from setting Fg = Fc.

2.3.4.3.

Flight Path Angle

The flight path angle is the angle η between a satellite’s velocity vector V and its radius vector r – see Figure 2-22. 27

2/ Orbital Dynamics

η

V

olo gy .

r

hn

ν

Figure 2-22 Definition of the flight path angle η.

1 + e cos ν e sin ν

(2.15)

fT

tan η =

ec

The flight path angle is obtained from the following identity:

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Figure 2-23 shows η as a function of ν for the Ariane 5 sample orbit.

Flight path angle η (deg)

160 140 120 100 80 60 40 20

360

340

320

300

280

260

240

220

200

180

160

140

120

100

80

60

40

20

0

0

True anomaly ν (deg)

Figure 2-23 Flight path angle η as a function of ν for an Ariane 5 GTO.

Knowing the flight path angle is of interest e.g. when planning orbital manoeuvres or taking pictures from space.

2.3.4.4.

Orbital Energy

The total energy of a satellite in orbit is the sum of its kinetic and potential energy:

E = E kin + E pot =

28

1 mµ mV 2 − 2 r

(2.16)

2 / Orbital Dynamics

olo gy .

It may come as a surprise that the potential energy is a negative quantity. This convention has come about because, even though Epot increases with the radial distance r, it reaches zero at r = ∞. For this to work, Epot must start out at the earth’s centre as an infinitely large negative quantity. Another useful expression for the total energy is:

E=−

mµ 2a

(2.17)

Together, Eq 2.16 and 2.17 yield Eq 2.13.

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2.3.5. Orbital Perturbations

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ec

Because the satellite moves around the orbit like a mass element, the orbital plane manifests gyroscopic properties, i.e. it remains fixed in inertial space unless it is perturbed by external torques. If perturbed, its nodes will begin to drift in such a way that the orbit plane normal undergoes a conical motion. Compare this with a spinning top whose axis starts “coning” when slowing down, because the force of gravity is no longer parallel with the spin axis – see Figure 2-24.

Perturbing force of gravitation

Figure 2-24 The coning motion of a spinning top.

In the case of orbits, the perturbing forces acting on a satellite consist of gravitational pull from the sun, the moon and the earth’s oblateness, as well as solar pressure.

2.3.5.1.

Nodal Drift

Node rotation is primarily the result of the earth’s oblateness, i.e. the fact that the earth is an ellipsoid rather than a sphere. The phenomenon is visualized in Figure 2-25. As the satellite approaches the equatorial bulge from the south, the gravitational pull grows and draws the spacecraft “upward,” resulting in an increase of the inclination angle. When the satellite is past the equator, the bulge pulls the satellite the other way, such that the inclination resumes its original value, but at that stage the node will already have drifted an amount ∆Ω degrees. The process is repeated as the satellite passes the descending node.

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2/ Orbital Dynamics N

Arrows show vertical component of gravitational pull

olo gy .

N

∆Ω ∆Ω

hn

Figure 2-25 Nodal drift, as seen from the ascending node (left) and the descending node (right).

(deg/day)

(2.18)

fT

dΩ − 10 cos i ≈ dt (a / R ) 7 / 2 (1 − e 2 ) 2

ec

The nodal drift is approximately:

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As before, we count Ω as positive in the direction of the earth’s rotation. Note that the nodal precession is retrograde (i.e. dΩ/dt is negative, such that Ω decreases) for i < 90°, and is prograde for i > 90°. Figure 2-26 shows dΩ/dt for various values of a and i in a circular orbit (e = 0). The nodal drift rate is quite significant at low inclinations and altitudes. Only the polar orbit has no nodal drift at all. The diagram is symmetrical for i > 90 deg, except that the nodal drift now turns positive.

Nodal drift dW/dt (deg/day)

0

-1

0

-2

15

-3

30

-4

45

-5

60

-6

75

-7

-8 7,000

9,000

11,000

13,000

15,000

Semimajor axis a (km)

Figure 2-26 Nodal drift as a function of a and i. For i > 90° the drift is prograde (i.e. the values along the y-axis become positive).

2.3.5.2.

Perigee Drift

Another consequence of external perturbations is that the line of apsides – i.e. the line between the apogee and the perigee – experiences a significant in-plane drift over time. Eq 2.19 approximates the drift rate:

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2 / Orbital Dynamics

dω 5(5 cos 2 i − 1) ≈ dt (a / R ) 7 / 2 (1 − e 2 ) 2

(deg/day)

(2.19)

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This equation is only meaningful for an elliptic orbit, since there is no apogee or perigee in a circular orbit. To illustrate the order of magnitude of perigee drift, dω/dt = 0.821 deg/day for a typical transfer orbit with Pe = 200 km and Ap = 35,793 km – an important consideration e.g. when planning the time available between launch and the firing of a satellite’s apogee engine.

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Inclination (deg)

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16 14 12 10 8 6 4 2 0 -2 -4 7,000

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Perigee drift dω ω /dt (deg/day)

Figure 2-27 illustrates the magnitude of perigee drift as a function of a and i for an orbit with moderate eccentricity (e = 0.1). Note that the horizontal axis shows semimajor axis rather than orbital height, since the orbit is elliptic.

12,000

17,000

0 15 30 45 60 75

22,000

Semimajor axis a (km)

Figure 2-27 Perigee drift dω ω/dt as a function of a and i for an elliptic orbit with e = 0.1.

As can be seen in Figure 2-27, the rate of perigee drift is quite dramatic for low values of a. (Note, however, that e = 0.1 and a < 7000 km puts the perigee below ground!) Inspection of Eq 2.19 leads to the conclusion that increasing the eccentricity e has the effect of increasing the drift rate further. From Eq 2.19 it is clear that the perigee drift rate is 0 when 5 cos2i – 1 = 0, i.e. when i = 63.4°. This phenomenon is exploited in some space missions – see Section 2.5.3. 2.3.5.3.

Drift of Other Orbital Elements

In addition to Ω and ω, the other orbital elements (a, e and i) are also subject to perturbations, but their magnitude is much smaller. Their effect is most noticeable in the geostationary orbit – see Section 2.5.1.

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2.4.

Mission Analysis

• • • • •

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Mission analysis examines the operational consequences of the choice of orbit. Here are some typical questions which need to be answered through mission analysis: Over what points of the earth will the satellite pass, and when? When, and for how long, will the satellite be visible over a given ground station? When will the satellite pass through eclipse (the earth’s shadow), and for how long? How will the satellite be illuminated, i.e. what will the solar incidence angle be? What is the launch window for the satellite?

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The following sections provide the mathematical tools for working out the answers to these questions.

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2.4.1. Coordinate Systems

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As we have already noticed, some orbital parameters (for example r, i and ω) are defined in relation to an earth-centered coordinate system with the N/S axis and the equator as the references. Other orbital parameters (notably Ω and τ) belong in an inertial coordinate system linked to the stars and the inertially fixed reference ϒ. In the following we will be moving back and forth between these basic systems and their variants, so this is a good place to define them and give them names. z

z

y

x=ϒ ϒ

y

x

Heliocentric inertial

Heliocentric rotating

z=N

x=ϒ ϒ

y

Geocentric-equatorial inertial

z=N

x = zenith

z =N

z=i

y =east

x=Greenwich meridian

y

Geocentric-equatorial rotating

Topocentric

x=ascending node Geocentric orbital

Figure 2-28 Nomenclature of coordinate systems.

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y=perigee

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2.4.2. Subsatellite Track

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The subsatellite track is the projection of the satellite’s orbital journey onto the earth’s surface. Stated differently, it is the track that the radius vector traces on the earth as it cuts through the earth’s surface (Figure 2-29).

x

Figure 2-29 Subsatellite track

Recall from Figure 2-12 that the radius vector r is defined in a geocentric coordinate system fixed to the orbital ellipse (“geocentric orbital”). The orbit, in turn, is anchored to an inertial coordinate system centered in the earth (“geocentric-equatorial intertial”). The earth, finally, is rotating with respect to the inertial system (“geocentric-equatorial rotating” – see Figure 2-30). To generate the subsatellite track mathematically, it is necessary to find the position of the radius vector over time in a coordinate system that is fixed to the rotating earth. This involves a series of coordinate transformations through ν, ω, i, Ω and φ.

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rs ν

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ω N N

ν+ω

r i

Ω

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ϒ

Greenwich meridian

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φ

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ϒ

Figure 2-30 Successive steps of transferring the radius vector r from the orbital ellipse coordinate system to a rotating earth system.

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Now let xs be the unit vector of the satellite’s radius vector rs, such that xs is the x-axis in a geocentric orbital coordinate system. The satellite is shown travelling anti-clockwise. According to the right-hand rule, the z-axis must be pointing out of the paper towards the reader, and the y-axis completes the orthogonal, right-handed system. Hence rs = 1 xs + 0 ys + 0 zs = (1, 0, 0). (It makes no difference for the geometry of the problem whether we use full-length vectors or unit vectors; think of the earth as having the radius “1”.)

ys

xs= rs

zs

Figure 2-31 Coordinate system anchored to the range vector.

As the next step, examine xs, ys, zs and rs in the context of a system xν, yν, zν where the xν-axis coincides with the line of apsides. In this scenario the radius vector coordinate

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2 / Orbital Dynamics system is rotated back to the perigee-fixed coordinate system. Note that the direction of rotation is “left-handed” – see the arrow around the zs-axis. yν

ν

xs= rs ν

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xν

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ys

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zs=zν

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Figure 2-32 Rotation of the range vector system to the apsides-based system.

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In this new coordinate system, rs would seem to become rν through the following transformation:

 r1   x s  xν   rν = r2  =  y s  xν  r3  ν  z s  xν

xs  y ν

y s  yν z s  yν

xs  z ν  1  cos ν sin ν 0 1 y s  z ν   0 = − sin ν cos ν 0  0 z s  z ν  0  0 0 1 0 s

However, because the rotation is left-handed in what is a right-handed coordinate system, the argument ν must be viewed as negative. Therefore:

 r1   cos(− ν ) sin (− ν ) 0 1 cos ν − sin ν 0 1 cos ν    rν = r2  = − sin (− ν ) cos(− ν ) 0  0 =  sin ν cos ν 0  0 =  sin ν   r3  ν  0 0 1 0  0 0 1 0  0  The [ν] system is rotated back to the line of nodes in the same manner, yielding rω as follows:

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 r1  cos ω − sin ω 0  r1    rω = r2  =  sin ω cos ω 0  r2   r3  ω  0 0 1  r3  ν

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Figure 2-33 Rotation of the apsides-based system to the nodal system.

The range vector is now brought into the geocentric equatorial inertial coordinate system by rotating the ω-system through i, i.e. by aligning the orbit plane normal with the earth’s North Pole vector.

zi = north

zω

yω

yi

i

i

xi=xω

Figure 2-34 Alignment of the orbit plane normal to the earth's spin axis.

0 0   r1   r1  1    ri = r2  = 0 cos i − sin i   r2   r3  i 0 sin i cos i   r3  ω

The RAAN is similarly accounted for:

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zΩ =zi

yi Ω

yΩ

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Ω

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xi

xΩ =ϒ ϒ

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 r1  cos Ω − sin Ω 0  r1  rΩ = r2  =  sin Ω cos Ω 0  r2   r3  Ω  0 0 1  r3  i

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Figure 2-35 Rotation of the pole-aligned system to the geocentric inertial system.

Lastly, we bring the rΩ vector into the geocentric-equatorial rotating coordinate system by making the earth is made to spin through an angle φ around the zΩ axis. Note that the rotation is clockwise, which leads to a sign change in the sinus term.

zφ=zΩ

φ

yΩ

yφ

φ

xφ

xΩ

Figure 2-36 Transformation of the earth's rotating system to the geocentric inertial system.

 r1   cos φ sin φ 0  r1    rφ = r2  = − sin φ cos φ 0  r2   r3  φ  0 0 1   r3  Ω

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Using shorthand, the full matrix transformation therefore becomes:  r1 

rφ = r2  = [φ] [Ω] [i] [ω] [ν] rs

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(2.20)

 r3  φ

Recalling that the components (r1, r2, r3)φ represent direction cosines of rφ to the three coordinate axes, we obtain the latitude λ and longitude Λ of the range vector from Figure 2-37:

cos-1r3 r λ

yφ

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Λ

cos-1r2

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cos-1r1

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zφ

xφ = ϒ

Figure 2-37 Derivation of latitude and longitude from direction cosines.

r3 = cos(90°-λ) = sinλ

According to the laws of spherical triangles (see Appendix A):

r1 = cosλ cosΛ

Therefore:

λ = sin-1(r3)

(2.21)

Λ = cos-1(r1/cosλ) = tan-1(r2/r1)

(2.22)

In Eq 2.22 it is better to use the tan-1 function rather than cos-1, because it allows us to track the angle through 360 deg using the ATAN2 function in Excel or Matlab.

Using equations 2.21 and 2.22, the subsatellite track of an Ariane-type transfer orbit is shown in Figure 2-38. The curious backward sweep of the track around the equator, as observed in the S-curve, arises because the satellite’s angular velocity at the apogee is slower than that of the earth.

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2 / Orbital Dynamics Apogee

Apogee

Perigee

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Latitude (deg)

X

50

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Launch site

10 9 8 7 6 5 4 3 2 Perigee 1 0 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 0

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Perigee

100

150

200

250

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East Longitude (deg)

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Figure 2-38 Ariane-type transfer orbit subsatellite track immediately following launch. (Ap = 35,793 km, Pe = 250 km, i = 6°, ω = 180°)

Figure 2-39 illustrates the subsatellite track of a geosynchronous satellite at i = 10°, an undesirable inclination value for this type of satellite, but one that is sometimes found where a GEO satellite approaches the end of its service life. 15

Latitude (deg)

10

5 0

-5

-10

-15 179.4

179.6

179.8

180.0

180.2

180.4

East longitude (deg)

Figure 2-39 Subsatellite track of a geosynchronous satellite (i = 10°).

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Q: Is it important in which order one makes the coordinate transformations? Could the above sequence s → ν → ω → i → Ω → φ just as well be reversed?

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A: The sequence as a whole may be reversed, but not individual groups of matrices within the sequence. Remember that matrices are not generally commutative, i.e. [A]·[B] ≠ [B]·[A]. Think of it this way: If we rotate an aircraft first along the roll axis 90° and then along the pitch axis 45°, we get a different attitude than if we first pitch the aircraft 45° and then roll it 90° (see Figure B-2 in Appendix B).

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In calculating the subsatellite track, we recognize that all the above arguments refer directly (Ω, φ) or indirectly (ω, i, ν) to the geocentric-equatorial inertial coordinate system xΩ , yΩ , zΩ, i.e. the frame where the x-axis coincides with the γ-vector. We must therefore rotate our local coordinate systems backwards one by one, until they arrive “home” to the reference frame. Equally important is to keep track of the sense of rotation (left-hand or right-hand), since it affects the sign of the argument (negative or positive, respectively) and hence the sign of the sinus terms.

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2.4.3. Ground Station Coverage

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One of the main uses of computing the subsatellite track is to establish when, and for how long, the satellite will pass over a given ground station. Here again, the use of vectors and coordinate transformations is necessary. The geometry of the problem is as shown in Figure 2-40. Let the geographical coordinates of the ground station be represented by the geocentric unit vector a, and the edge of coverage by the geocentric unit vector b. Vector a thus points from the earth’s centre to the ground station’s geographical location at λ deg latitude and Λ deg longitude.

a

b α

n

Figure 2-40 Ground station coverage zone.

To plot the coverage contour on the earth’s surface, we need to find a way of rotating vector b to form a cone around a, all the while keeping track of the b-vector’s coordinates in terms of latitude λ and longitude Λ. The simplest way is to form a normal vector n that

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2 / Orbital Dynamics is perpendicular to both a and b, and to let it make the 360° round. This normal vector is obtained from the cross product of a and b, i.e. n sin α = a x b. The coverage angle α is a function of the orbital height h (Figure 2-41): R R+h

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R

(2.23)

α R

h

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Figure 2-41 Satellite coverage.

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α = cos −1

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The earth’s average radius R = 6371 km. Let us begin by defining a geocentric-equatorial rotating coordinate system anchored to the earth’s centre. Here a forms the z-axis, while the x-axis cuts through the ground station meridian Λ, and the y-axis completes the righthanded system – see Figure 2-42. zβ =a

b

φ

b

α

n

β

yβ

β

xβ

Figure 2-42 Motion of vectors “n” and “b” around the “a”-vector.

As the n-vector begins to rotate around the a-vector through an angle β, it “drags” the bvector with it. Inspection of Figure 2-42 yields b = (sinα cosβ, sinα sinβ, cosα). The time is ripe to bring the xβ, yβ, zβ-system back to the geocentric-equatorial rotating system x, y, z, i.e. to establish b in terms of λ, Λ, with β as the variable. To do so, we go through the steps shown in Figure 2-43.

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Greenwich zβ meridian (Λ = 0°) λ

z

Greenwich meridian (Λ = 0°)

yβ

z

zλ yλ

y

y

x Equator

x Equator

xβ

Greenwich meridian (Λ = 0°)

Λ xλ

xΛ

x Equator

zΛ yΛ

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z

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Figure 2-43 Coordinate transformations for ground station coverage.

The corresponding transformations are as follows:

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zΛ zβ = a

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90° - λ

yΛ =yβ

90° - λ

x

xβ

Figure 2-44 Latitude rotation.

 b1   sin λ 0 − cos λ  sin α cos β    b λ = b2  =  0 1 0    sin α sin β  b3  λ cos λ 0 sin λ   cos α  z=zΛ

yΛ

Λ

y

Λ

x

xΛ

Figure 2-45 Longitude rotation.

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2 / Orbital Dynamics

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 b1  cos Λ − sin Λ 0  b1    b = b2  =  sin Λ cos Λ 0  b2  b3   0 0 1 b3  λ

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Figure 2-46 shows the coverage contour generated by having the b-vector sweep through 0°< β 6000 km. As stated earlier, the local time on the earth overflown by a satellite in SSO always stays the same, give or take 12 hours. Therefore, the illumination conditions on the ground also remain the same, which is the primary advantage of the SSO. Note, however, that the geographical areas themselves will vary from one day to the next unless the orbital period is a submultiple of a sidereal day of 1436 minutes. For example, if there are exactly 14 revolutions in a day, then t = 1436/14 = 102.6 minutes, corresponding to a = 884 km according to Eq 2.12.

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2.5.3. Highly Eccentric Orbits (HEO)

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Eccentric orbits are of interest e.g. in scientific space missions where sampling of physical data is to take place across a range of different altitudes. Another field of interest is space telecommunications.

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Note in Eq 2.19 that the perigee drift rate is zero when 5 cos2i – 1 = 0, i.e. when i = 63.4°. The Soviet Union exploited this phenomenon in their Molniya telecommunications satellite programme. In the early days of space exploration, the Soviets lacked rockets powerful enough to inject satellites into GEO from their relatively high-latitude launch sites (Baikonur being at latitude 46°N). Besides, satellites in equatorial GEO appeared too close to the horizon for comfort, as seen from Soviet territory. The Soviets therefore launched their Molniya satellites into highly elliptic, 12-hour orbits (Ap = 40,000 km, Pe = 500 km) and parked the apogee above their territory. Each time a satellite approached the apogee, it slowed down until it became practically “geostationary” for several hours. When it finally began to drop down towards the perigee, another satellite took its place at apogee – and so forth until the 24-hour day was completely covered. In this manner the Soviets were able to create a pseudo-geostationary service that fell within the lift capability of their launch vehicles (Figure 2-86). Apogee

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Apogee

Perigee

Perigee

Figure 2-86 Principle of the Molniya system. The square represents a ground station.

The only problem was how to prevent the apogee from drifting away from Soviet territory. By selecting i = 63.4°, the apogee was made to stay put. Since the national territory extended in latitude from 45°N to 75°N, this inclination had the added benefit of enabling Soviet ground stations to view the satellites near zenith, instead of low on the horizon where the radio signals would frequently have been corrupted by buildings, mountains and multipath effects. Figure 2-87 shows the subsatellite track of a typical Molniya satellite. Out of the 12-hour orbital period, the satellite spends no less than 10 hours within each closed loop. In practice, the requirement for “stationarity” is such that only the top portion of the loop is used, corresponding to about 4 hours of almost stationary ground coverage. Three 72

2 / Orbital Dynamics

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satellites are needed to populate the system for continuous 24-hour coverage (3 satellites x 4 hours = 12 hours = τ, after which the process repeats itself). The 12-hour orbital period creates synchronism with the earth’s 24-hour rotational period, thereby ensuring that the loops do not shift eastward or westward over time.

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Notice that the system could equally well have been exploited in Canada.

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Figure 2-87 Molniya subsatellite track (Ap = 40,000 km, Pe = 500 km, i = 63.4°, ω = 270°).

2.5.4. Low Earth Orbits (LEO)

The proximity to the earth makes LEO satellites obvious candidates for photography and telecommunications, and also for many scientific and military applications. When applied to circular obits, the term LEO usually brings to mind the mobile telecommunication satellite systems Iridium and Globalstar, which went into liquidation around the turn of the century after losing billions of dollars each for their investors (although they have since recovered). These systems provide global coverage from near-polar LEO, taking advantage of the favourable radio link budgets compared to GEO. Figure 2-88 outlines a typical LEO constellation.

Seamless coverage footprints

Figure 2-88 Outline of the Iridium system with its intersatellite links.

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2/ Orbital Dynamics Equations 2.54 and 2.55 spell out how many orbital planes are needed, and how many satellites per plane, in order to achieve global coverage from the orbital altitude h. Circular, near-polar orbits are assumed, with equally spaced Ω and ν.

 2π  P=   3α' 

(2.54)

No. of satellites per plane:

 2π  N =    3α' 

(2.55)

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No. of orbital planes:

No. of satellites in the constellation: S = P ∗ N

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(2.56)

The ground station coverage half-angle α’ is obtained from Eq 2.42 as:

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 R  α' = cos −1  cos δ  − δ R+h 

(2.57)

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Not surprisingly, the number of satellites S required for seamless global coverage increases with reduced orbital height h. Figure 2-89 shows this dependency for δ = 10°.

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Taking a specific example, the Iridium satellite constellation for mobile telecommunications has an orbital height h = 780 km. The mission analysts decided that the minimum elevation above the local horizon had to be δ = 8.2° to avoid signal obstruction. Eq 2.57 yields α’ = 20°. Inserting α’ in radians in Eq 2.54 – 2.56 gives us P = 6, N = 11, and S = 66, which is indeed how the Iridium system is configured.

When assessing the economics of a global constellation, system planners trade off the number of satellites required against the price per satellite, given that communications satellites become cheaper as the link budget improves with lower height.

No. of satellites

1000

100

Iridium

10

1

100

1,000

10,000

100,000

Orbital height h (km)

Figure 2-89 Number of satellites in polar orbit required to achieve global coverage.

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Commercial example: Each of the small Iridium satellites is rumoured to cost $5 million. A Delta II rocket can launch 8 Iridium satelllites at a time and costs $80 million, so the total cost of launching 66 satelllites for global coverage plus 8 in-orbit spares would be 74 x ($5 + $80/8) = $1,110 million.

To achieve global coverage with a hypothetical GEO satellite system, three large satellites plus one in-orbit spare are required. If each satellite costs $150 million, and if each Proton launch has a price tag of $100 million, the total bill is 4 x ($150 + $100) = $1,000 million.

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Thus, the difference in capital outlay to deploy the two systems is not significant. The choice should be made on other grounds, e.g. the cost of the ground terminals, the service lifetime of the system, etc.

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2.6.

Solved Problems

See Appendix C for solutions.

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2.6.1 A satellite in a circular orbit has an orbital period τ = 96 minutes. What is the orbital height h and the velocity V ? The earth’s average radius R = 6371 km and the gravitational parameter µ = 398,601 km3/s2.

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2.6.2 A satellite is travelling in an equatorial geostationary transfer orbit with the apogee height Ap = 35,786 km and the perigee height Pe = 300 km. What is the orbital velocity V when the true anomaly ν = 90°? The earth’s equatorial radius R = 6378 km and the gravitational parameter µ = 398,601 km3/s2

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2.6.3 The satellite in Problem 2.6.2 has a mass m = 1000 kg. Calculate its kinetic and potential energy when ν = 90°.

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2.6.4 Is it possible to design an orbit where the perigee drift equals the nodal drift? If so, what are the values of a, e and i?

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2.6.5 A satellite that never travels through the earth’s shadow needs no batteries. Design the lowest circular orbit that guarantees the satellite uninterrupted sun illumination. Assume the eclipse contour to be a cylinder with radius R = 6371 km. 2.6.6 A satellite is to be launched south on 1 March from Vandenberg (lat 34.7N, long 239.4E) into a sun-synchronous orbit with i = 98° and Ω = 70°. At what local time must the satellite be launched? Vandenberg is 8 time zones west of Greenwich.

2.6.7 A satellite constellation is being designed to provide global, uninterrupted telecommunications from an altitude h = 1000 km. A fundamental requirement is that at least one satellite shall be visible at an elevation δ > 20° above the local horizon anywhere on earth. How many satellites will be needed if the orbits are polar?

2.6.8 A satellite is launched in 2003 into the geostationary orbit with an initial inclination i0 = 3 deg and an initial RAAN Ω = 270 deg. How long will it take for the inclination to drift back to 3 deg, and what is the minimum inclination encountered en route? 2.6.9 A typical Molniya orbit has an apogee of 40,000 km and a perigee of 500 km. How many Molniya satellites are needed for round-the-clock coverage if each satellite is to be used from the ground while passing through the true anomalies ν = 169.8 deg and 190.2 deg? The Earth’s average radius R = 6371 km. The Earth’s gravitational parameter µ = 398,601 km3/s2.

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3 / Power Management

3.1.

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3. Power Management Introduction

The power subsystem is the satellite’s lifeline. Using the human analogy, it is the food intake and digestion needed to sustain the organism.

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Whatever the satellite payload – a telecommunications repeater, a camera, or a scientific sensor – the demand for electric energy is likely to be a major design driver for the rest of the spacecraft. This is so because the primary energy source for most earth-orbiting satellites consists of photovoltaic solar cells, whose solar radiation conversion efficiency is relatively poor. Moreover, the intensity of solar illumination in the earth’s vicinity is quite modest. Consequently, in order to satisfy a typical satellite’s appetite for electric energy, the size of the solar panels can be substantial – so large, in fact, that they have to be folded during launch and require a sufficiently big spacecraft body to be folded up against.

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The size of the satellite is further influenced by the need for rechargeable batteries to sustain the onboard subsystems whenever the satellite passes through the earth’s shadow (“eclipse”), i.e. when the solar panels cease to produce energy. The efficiency of batteries is limited, and there are also restrictions on how deeply a battery may be drained. Accommodating the weight and bulk of batteries remains a challenge for satellite designers. The payload is not the only subsystem to demand electric energy. Most platform subsystems also need access to electric power in order to function. For example, certain electric micropropulsion thrusters claim a significant proportion of a satellite’s power generation capacity. But payloads tend to dominate. Nearly all payloads use radio signals to relay their mission parameters to the ground, and radio transmitters are notoriously power-hungry. The present chapter deals with – • •

•

•

primary power supply, i.e. the direct conversion of solar energy to electric energy; energy storage in rechargeable batteries to permit continued satellite operation in eclipse; power conditioning and distribution, i.e. the stabilization and conversion of bus voltages within the satellite; and alternative means of generating and storing electric energy.

The concepts of power and energy are often used loosely in the literature. Keep in mind that power = energy produced or consumed per unit of time.

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3.2.

Primary Power Supply

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3.2.1. Solar Cell Design

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A typical solar cell is shown in Figure 3-1.

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Figure 3-1 Typical solar cell for satellites.

Photovoltaic solar cells generate electric energy by allowing photons from incoming sunlight to dislodge electrons in a semiconducting p-n junction. By interconnecting the player of one cell with the n-layer of an adjacent cell, a current is made to flow. Desired levels of current and voltage are achieved by stringing a number of cells together (Figure 3-2).

n p

Figure 3-2 Stringing of solar cells.

The n-layer, being less prone to radiation damage than the p-layer, is placed on top. Further radiation protection is provided by a glass cover slip, which is coated with a suitable UV filter, as well as with an electrically conductive film to prevent the build-up of static electricity across the cell array. The cover slip is attached to the n-layer using a transparent adhesive. More adhesive is employed to secure each cell to the solar panel substrate (Figure 3-3).

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p-layer

Substrate

Figure 3-3 Constituents of a solar cell.

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The active semiconductor element has traditionally been made of silicon, whereby the n and p-layers are created through a suitable level of doping. Silicon cells absorb sunlight in the visible spectrum and converts it to electric energy. Higher energy conversion efficiencies are being achieved with gallium arsenide ( GaAs). The larger and thinner the cell, the higher the efficiency, the practical limit being set by fragility considerations. Typical sizes are 4 cm by 6 cm, with thicknesses around 100 µm.

UV

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So-called triple junction GaAs cells are able to convert sunlight not only in the visible spectrum, but also in the ultraviolet (UV) and near-infrared (IR) spectra. The process is illustrated in Figure 3-4. VIS

IR

Solar Radiation (W/cm 2/µ µ m)

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0.25 0.20 0.15 0.10 0.05 0.00

0.1

1

10

Wavelength λ (µ µ m)

Current

Triple junction solar cell

Figure 3-4 Energy conversion using triple junction GaAs cells.

The upper diagram of the Figure shows the energy density of sunlight in near-earth space within each of the three spectral bands. The triple junction GaAs solar cell is outlined at the bottom of the Figure. We now see how the top layer of the solar cell absorbs the UV light but allows the VIS and IR light to pass through. The middle layer of the cell then absorbs the VIS light while letting the IR light pass through to the bottom layer. Hence the improved efficiency of the triple junction GaAs cell compared to the traditional silicon cell.

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The following curve-fit equation applies to Figure 3-5:  U  I = 130 cos 0.1    380 

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(3.1)

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A representative silicon solar cell performance curve is shown in Figure 3-5 for a cell measuring 2 cm x 2 cm. The open circuit voltage is in the region of 600 mV, while the short-circuit current might be 60 mA, the actual value depending on the cell size. The cell is exploited most efficiently if the working point is chosen at the “knee” where the output power reaches a maximum. This becomes evident if the y-axis is made to show power rather than current, as in Figure 3-6.

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The power output in Figure 3-6 is obtained by multiplying U and I along the curve in Figure 3-5.

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60 40 20 0

0

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Cell voltage U (mV)

Figure 3-5 Typical cell I/U profile.

Cell output power (mW)

70 60 50 40 30 20 10

0

0

100

200

300

400

500

600

700

Cell voltage U(mV)

Figure 3-6 Equivalent cell output power profile.

In our example, Figure 3-6 shows peak cell power output at ≈ 550 mV, so this is where the working point should be located for maximum efficiency. The corresponding cell

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Some modern satellites come equipped with automatic maximum power tracking circuitry (a so-called Maximum Power Point Tracker, or MPPT) which adjusts the voltage bias to keep the operating point at the knee of the I-U curve. Conversely, the operating point may be dynamically adjusted away from the knee, in order to adapt the array power output to a variable power requirement.

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The curve-fit: The advantage of a reasonably accurate curve-fit equation is that derived parameters can be obtained analytically. In the above example, U, I and P at the optimum working point are calculated as follows.

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The first step is to guesstimate which type of function (polynomial, trigonometric, ...) might provide the simplest and best fit. In this instance the curve-fit might take the form I = a c cos (bU). The optimum working point is where the power output is maximum, i.e. where dP/dU = 0. With P = U•I, we have

and therefore: c

c-1

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c

P = aU cos (bU)

tan( bU ) =

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dP / dU = a[cos (bU) – bcUcos (bU) sin(bU)] = 0;

1 bcU

With a = 130, b = 1/380 and c = 0.1, we find the optimum working point at U = 543 mV, I = 107 mA, P = 58 mW.

Obviously the accuracy of the analytical result is only as good as the accuracy of the curvefit.

The cell performance outlined in Figure 3-5 is temperature sensitive, as suggested in Figure 3-7. The sensitivity translates itself into a change in cell conversion efficiency, which will be discussed below.

Figure 3-7 Temperature-dependent cell conversion profile.

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The solar power flux in the earth’s vicinity varies between Smax = 1413 W/m2 in early January and Smin = 1321 W/m2 in mid-summer. The variation is due to the slight eccentricity of the earth’s orbit around the sun. For preliminary design purposes, an annual mean value S = 1367 W/m2 is often assumed. It is also common practice to assume that this value applies at all orbital altitudes up to GEO. Thus it is tempting to think that a solar panel measuring 1 m2 should produce 1367 W of electric power, on average. In practice, we are lucky to obtain even 15% of this output.

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Firstly, the efficiency of present-day solar cells is less than 30% using modern gallium arsenide (GaAs) cells – a value which drops to 20% in the case of the more conventional silicon (Si) cells. Even these modest values drop significantly when the solar cells become warm (Figure 3-8).

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Gallium Arsenide (GaAs)

20

Silicon (Si)

15 10

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Cell efficiency η (%)

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5 0

-50

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100

150

200

Cell temperature T (°C)

Figure 3-8 Solar cell efficiency η as a function of cell temperature T.

A reasonably good curve fit of Figure 3-8 is obtained using the following equations: GaAs: Si:

 T + 50    250   T + 50  η = 27 cos1.2    200 

η = 30 cos1.2 

(3.2) (3.3)

Secondly, the solar cell output depends on the solar incidence angle according to the cosine rule illustrated in Figure 3-9. For instance, if the sun angle φs is offset by 60° from the cell normal, the power output drops to 1/2 of the maximum value (cos 60° = 0.5).

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Figure 3-9 Cosine rule of solar cell output.

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Thirdly, there is a seasonal variation of solar cell output as the distance re between the earth and the sun varies during the year, and the satellite designer has to assume the “worst case” for dimensioning purposes. Given that the semimajor axis of the earth’s orbit around the sun is ae = 149,597,870 km and the eccentricity ee = 0.0167, we find from Eq 2.3 that: 2 a e (1 − ee ) re = 1 + ee cosν e

The solar power density is a function of the square of the distance, so we introduce a cyclic variation factor Fc = (ae/re)2

(3.4)

as the earth moves through νe from perihelion to aphelion and back – see Figure 3-10.

Figure 3-10 Cyclic variation factor Fc.

Lastly, the performance of a solar cell degrades over time. This secular degradation is due to damage caused by an influx of high-energy protons and electrons in the Van Allen belts, and of ultraviolet radiation from the sun. The corresponding degradation factor Fs is approximately 83

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Fs ≈ 0.7 + 0.3 e-t/1000

(3.5)

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where t is the number of days from launch (Figure 3-11).

Figure 3-11 Secular degradation factor Fs.

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P = Fc Fs cos φ s Pmax

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Combining the above degradations, we obtain the following approximate expression for solar cell power output: (3.6)

Figure 3-12 plots Eq 3.6 for a geostationary satellite whose solar panels are parallel with the earth’s N/S axis. Note that, though the panels are pointed towards the sun, the sun angle φs will vary between 0° and 23.45° due to the panels being aligned with the N/S axis (Figure 3-13), according to φs = sin-1[sinξ sin(23.45°)]

(3.7)

Hence the overlay of two different cyclic variations, one with a 12-month period caused by the annual variation of re, the other with a 6-month period as φs varies from +23.45° at summer solstice to –23.45° at winter solstice. The winter dip is shallower than the summer dip because the earth is then closer to the sun.

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1.00

Spring equinox

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0.85

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Autumn equinox

0.90

Worst case

0.80 0.75 0.70

Winter solstice

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Summer solstice

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Years from launch

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Figure 3-12 Combined variations in relative solar cell output.

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23.45°

23.45°

23.45°

ξ

γ

Figure 3-13 Seasonal ±23.45° variation of sun illumination of GEO solar panels.

Since the output is time-dependent, the solar panels have to be sized for the worst-case scenario, which usually occurs at the end-of-life (EOL) of the satellite. Take, for example, a GEO satellite designed for a 10-year lifetime whose GaAs solar cells rise to a temperature of 20°C at EOL. In order to yield P = 1367 W at worst-case EOL, the solar panel would require a surface area of: A=

P SηFs Fc cos φ s

In our example: A =

(3.8)

1367

= 5 .8 m 2 1367 ∗ 0.28 ∗ 0.7 ∗ 0.97 ∗ cos 23.45° i.e. nearly 6 times larger than in the ideal condition where η, Fs, Fc and cos φs are all = 1.

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3 / Power Management As mentioned earlier, cell conversion efficiency increases with increasing cell size. A typical size of modern solar cells is 4 cm x 6 cm. It follows that, for the above satellite, the number of 4 x 6 cm cells required is 5.7/0.0024 = 2375.

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So far we have only considered performance losses associated with the solar cells themselves. In addition, there are circuit and conversion losses within the satellite, and these must also be taken into account when sizing the solar arrays.

3.2.3. Solar Arrays

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Solar cells are connected in series in order to obtain the desired power bus voltage onboard the satellite (Figure 3-14). Multiple strings are connected in parallel to yield the required current. (Compare with an ordinary household appliance such as a flashlight, in which 1.5 V batteries are connected in series to achieve a higher voltage.)

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10A 50V

Figure 3-14 Cell string assembly on a solar panel (in this example the panel yields 500W)

The strings are mounted on structural members to form solar panels. Care must be taken to account for the possibility of shadowing from adjacent satellite members during certain times of the day and the year (Figure 3-15). To mitigate the energy loss due to shadowing e.g. from an antenna dish, the strings should be orientated as shown in the upper example, rather than as in the example below.

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Array shadowing

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Cell string orientation

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Figure 3-15 Solar panel shadowing and cell string orientation.

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Moreover, diodes are used in various configurations to protect inactive cells due to shadowing from the spacecraft body or failure of adjacent cells (Figure 3-16). In the absence of such protection, a cell or an entire string might experience damaging reverse bias voltage under the influence of active cells within the affected string.

Figure 3-16 Bypass diodes.

Solar arrays typically come as flat panels, as in the case of body-stabilized satellites, or as cylindrical shells in the case of spin-stabilized spacecraft (Figure 3-17). Other variants exist, e.g. flexible panels that are deployed and retracted like motorized roller blinds.

Solar arrays

Figure 3-17 Solar arrays for body (left) and spin stabilized (right) satellites.

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The solar panels on body-stabilized satellites are often mechanically steered to keep them pointed towards the sun while the spacecraft body stays pointed towards the earth. In the case of a satellite in GEO, the panels are aligned with the earth’s N/S axis and are made to rotate 360° per day so as to track the apparent motion of the sun.

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d

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d

Deployable telescopic skirt

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On spin stabilized satellites the arrays are not steerable, so the attitude of the entire satellite must be maintained such that the array faces the sun. The illuminated portion of a cylindrical array equals d·h, while the total cylindrical surface area = πdh. The dh 1 “efficiency” of a cylindrical array is therefore = ≈ 30% . The possibility exists to πdh π double the array area by introducing a concentric cylindrical “skirt” which is telescoped out after launch, at minimal cost in launch mass and volume (Figure 3-18).

h

Figure 3-18 Illumination of single and double cylindrical arrays.

How much of an array is illuminated?

If a satellite is covered with solar cells, some of the cells may be illuminated head on, others at an angle, and others again not at all. To calculate the equivalent illuminated cell area, imagine a screen placed directly behind the satellite, as seen from the sun. The illuminated 2 area Α is simply the area of the shadow on the screen. In the case of a sphere, A = πd /4, where d is the sphere’s diameter. For a cylinder facing the sun as shown in the figure, A = dh, where h is the cylinder’s height. For an arbitrary body shape, it suffices to calculate the projection of the body onto the screen – i.e. the shadow.

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3.3.

Energy Storage

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Rechargeable batteries are used onboard satellites as a secondary power supply to allow satellite equipment to function in eclipse, when the solar arrays are no longer useful. Sometimes batteries are also called upon to supplement the solar arrays during peak load situations in sunlight.

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Traditionally, spacecraft batteries have employed nickel-cadmium (NiCd) or nickelhydrogen (NiH2) technology. In NiCd cells, the nickel and cadmium electrodes are interleaved to form a rectangular box. NiH2 cells experience hydrogen gas pressure up to 30 atm and are therefore shaped like cylindrical pressure vessels with spherical tops and bottoms. In both technologies the cell voltage is typically 1.2 V during the level part of the discharge cycle. The cells are connected in series to yield the required bus voltage (Figure 3-19).

Figure 3-19 NiCd and NiH2 battery cells connected in series.

Allowable depth of discharge %

Batteries require careful charge/discharge management in order to guarantee maximum lifetime in orbit. Their capacity, measured in ampère-hours (Ah) must not be depleted entirely. For a GEO spacecraft, the permissible depth of discharge (DOD) of NiCd batteries is around 60% of the rated capacity, while 80% DOD is often acceptable for NiH2. The allowable DOD for satellites in low orbits is significantly lower due to the increased charge/discharge frequency (Figure 3-20). 90 80 70 60 50

NiCd

40

NiH2

30 20 10

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Ratio te/ts

Figure 3-20 Permissible DOD as a function of eclipse/sunlight duration ratio.

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The weight of batteries is of concern to spacecraft designers. The energy yield for NiCd is around 40 Wh/kg, and for NiH2 it is in the region of 60 Wh/kg. Hence the mass of a NiCd battery needed to supply a medium-sized GEO satellite drawing 3200 W in maximum eclipse (1.2 h) would be 3200 · 1.2/(40 · DOD) = 160 kg. A NiH2 battery on the same satellite would weigh 120 kg.

Forced discharge to ≈ 100% DOD

100

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Forced discharge to ≈ 100% DOD

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Capacity (% of C)

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Both battery technologies are said to possess a “memory,” i.e. their capacity deteriorates over time at a rate, which is dependent on how well earlier charge-discharge cycles were managed (Figure 3-21). The deterioration has two components: one that is reversible through reconditioning, and another that is permanent. Reconditioning consists in fully discharging the battery at a high rate across a purpose-made load. This is a delicate procedure, as it risks running the battery into short circuit and destroying it.

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Reversible loss

Reversible loss

0

0

1

2

Reversible loss

Irreversible loss 3 Time (years)

Figure 3-21 Reversible and irreversible deterioration of battery capacity. (Source: European Space Agency)

The choice between NiCd and NiH2 technologies is influenced by the type of orbit and bus regulation technique (see Section 3.4.2 below). NiCd is favoured in low earth orbit applications where eclipse transits are long and frequent ( ≤ 40% of every circular orbit), since this technology performs best when the DOD is shallow. NiCd also manifests a relatively constant output voltage during the discharge cycle, thereby reducing the need for voltage stabilization in eclipse. NiH2 batteries, with their preference for deeper DOD and fewer discharge cycles, are sometimes found onboard geostationary satellites where eclipse transits are limited to max 72 minutes during a 45-day window around spring and autumn equinox. The voltage drop is more prominent during the discharge cycle than in the case of NiCd, making full bus voltage regulation desirable. New battery technologies have been developed for satellites to reduce the mass and the rate of deterioration through the memory effect. Two of these technologies are familiar from cellular telephones, namely Ni-MH and Li-Ion, whose specific energy amounts to ≈ 70 and 120 Wh/kg, respectively. Li-Ion is becoming the norm onboard modern satellites. Despite its advantages, careful voltage balancing among the battery cells is necessary to maintain the original capacity and avoid overheating.

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Numerical Example

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A three-axis stabilized geostationary satellite is equipped with two solar wings that are aligned with the earth’s N/S axis. The satellite’s power consumption is constant at Psat = 2000 W, its bus voltage Ubus = 100 V and its operational lifetime is 10 years. Calculate the charge capacity and mass of its NiH2 battery, as well as the size of its GaAs solar panels operating at a worst-case temperature of 80°C.

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Solution: Let us begin by sizing the battery. The maximum eclipse duration in GEO, te = 72 minutes (1.2 hours), occurs around the two annual equinoxes. The battery must be capable of yielding a current I = 2000/100 = 20 A during 1.2 hours; therefore the battery’s useable capacity Cused = 20 · 1.2 = 24 Ah. Figure 3-20 suggests a permissible DOD of 80%. The rated battery capacity must therefore be Crated = 24/0.8 = 30 Ah.

Cused = 24 Ah Crated = 30 Ah. mbatt = 50 kg

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In summary:

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The mass of the battery is obtained from the knowledge that the specific energy µ of a NiH2 battery is around 60 Wh/kg, so the battery weighs mbatt = Crated Ubus/µ = 30 x 100/60 = 50 kg.

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Turning now to the solar arrays, we know that they must produce Psat = 2000 W in sunlight to sustain the satellite, plus the necessary power to recharge the battery Pbatt. The total array power output is therefore Parray = Psat + Pbatt, where Pbatt is an unknown. To assess the magnitude of Parray, it is useful to analyse the energy balance. The solar array must produce enough energy during the time ts in sunlight (ts = 24.0 – 1.2 = 22.8 h) to sustain the satellite in sunlight, and also to replenish the energy taken from the battery during the time te in eclipse. We have: Parray ts = Psat ts+ Pbatt te;

i.e. Parray = Psat + Pbatt te/ ts

where Pbatt = 1.2 Ubus Cused / te. The factor 1.2 comes about because of battery efficiency considerations, i.e. the recharge voltage needs to be approximately 20% higher than the discharge (bus) voltage. Thus: Parray = Psat + 1.2 Ubus Cused / ts = 2000 + 1.2 · 100 · 24/22.8 = 2126 W The size of the array is calculated considering the solar influx energy density S = 1367 2 W/m , the degradation over 10 years = 0.7, the efficiency of GaAs cells at 80°C (25%), and the worst-case sun vector deviation from the solar panel normal = 23.45° around the solstices. We have:

A array =

In summary:

2126

o = 9.7 1367 ⋅ 0.7 ⋅ 0.25 ⋅ cos(23.45 )

m

2

Parray = 2126 W 2 Aarray = 9.7 m

Immediately after reconditioning, it is of course necesssary to recharge 100% of Crated, since in that scenario Cused = Crated.

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3.4.

Power Conditioning and Distribution

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Onboard users receive electric energy from the solar array or from the battery via a power bus. The bus is equipped with fuses or other current-limiting devices to prevent a short circuit in one unit from bringing down all the other units. These devices are placed at strategic points along the bus to minimize the functional damage to equipment in order of priority. For example, isolating a shorted propellant tank heater is more affordable than losing the entire telecommand function, since a satellite becomes crippled the moment it is unreachable by telecommand. The reliability of energy distribution can, in many cases, be increased by employing redundant buses.

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The voltage of the power bus is typically chosen in the range of 28V – 100V DC, and is achieved by connecting a corresponding number of solar cells in series, as well as battery cells in series (recalling that a solar cell delivers approximately 0.5V and a battery cell 1.2V). The lower the voltage, the higher the current drawn by a user for a given power requirement (since P = U·I), and the thicker and heavier the wire harness has to be to avoid excessive resistive power losses (P = R·I 2) - see Figure 3-22. The harness can therefore be thinner and lighter if a higher bus voltage is chosen, but then it becomes more fragile and prone to breakage during installation. Higher voltages also challenge the state of the art in semiconductor switching technology.

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Mass (kg/m) 1.0 0.8 0.6 0.4 0.2 0.0

0

50

100 150 Current (A)

200

Figure 3-22 Ability of copper wire to conduct current depending on gauge.

Many electronic units onboard a satellite require several input voltages. Moreover, these voltages usually have to be stable. On the other hand, the supply voltage from the solar array and the battery can vary considerably over time, as illustrated in Figure 3-25 below. A need therefore exists to manipulate the supply power by performing DC/DC voltage conversion as well as voltage stabilization. Conversion and stabilization are known collectively as conditioning. 3.4.1. Voltage Conversion

Traditional induction transformers are not usable in DC circuits, since AC is needed to achieve induction. One solution is to chop up the direct current using a pair of transistors 92

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°

°

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A more efficient method of performing DC/DC conversion is called pulse width modulation (PWM). Here the direct current is made to pass through an electronic switch that opens and closes the circuit with variable ON/OFF duty cycles. A capacitor serves as a charge storage device, which is filled during the ON period and is drained during the OFF period, thereby ensuring continuity of current flow. An inductor helps smooth the inevitable voltage ripple (Figure 3-23). The ratio between the ON and the OFF periods determines the output voltage (Figure 3-24).

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Uout

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Figure 3-23 PWM circuit.

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U

Uin

ON

OFF ON OFF ON OFF ON OFF

Uout time

U

Uin ON

OFF ON OFF ON OFF ON OFF

Uout time

U

Uin ON

OFF ON OFF ON OFF ON OFF

Uout time

Figure 3-24 Principle of PWM voltage conversion.

The feedback loop in the above circuit diagram controls the switching period needed to achieve Uout. 3.4.2. Voltage Stabilization

If left unregulated, the bus voltage will vary over time to a degree that is unacceptable for most electronic equipment. In sunlight, the power output of the solar array is a function of the array temperature, as shown in Figure 3-7. If the satellite passes through the earth’s shadow, the array cools down during the eclipse transit and warms up during the active sunlit part of the orbit; hence the variation of the unregulated bus voltage in sunlight. The battery output voltage is also temperature dependent and changes during discharge in 93

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eclipse. The unregulated bus voltage may therefore behave as in Figure 3-25 during an orbital period.

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Bus voltage

Desired voltage Eclipse

Sunlight

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Time

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Figure 3-25 Unregulated bus voltage profile.

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Voltage regulation may be performed centrally at source, or at the level of individual units. The former is preferred when new equipment is being designed, while the latter solution may be appropriate when using off-the-shelf equipment with its own built-in voltage regulators.

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At bus level, there are essentially three levels of regulation (Figure 3-26). The unregulated bus is as described above. A sun-regulated bus employs voltage regulators during the sun-lit phase of the orbit, while the battery’s relatively constant output voltage is deemed to provide adequate regulation during eclipse. A fully regulated bus offers voltage stabilization throughout. Fully regulated

Unregulated

Array

Charge switch

Discharge switch

Array

Charge regulator.

Discharge regulator

Load

Battery

Battery

Shunt

Array

Charge regulator.

Load

Battery

Shunt

Sun-regulated

Figure 3-26 Power bus regulation methods.

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Series

Shunt

“Variable resistor”

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Voltage regulators come in two basic designs: serial and parallel (Figure 3-27). The latter is also known as a shunt regulator. In a series regulator, a zener diode is connected to the base of a transistor to provide the voltage reference. The transistor acts as a variable resistor connected in series with the load. A shunt regulator, on the other hand, has the transistor connected parallel with the load, and is controlled in such a way that excess current is made to bypass the load.

Shunt Array

Controller

Load

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Figure 3-27 Basic principle of series and shunt regulators.

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An example of an unregulated bus is the use of automatic maximum power tracking (MPPT) mentioned in Section 3.2.2. The aim is to extract as much electric power as possible from the solar arrays at all times, regardless of cell temperature or aging. The fact remains, however, that solar cells do age, so the maximum obtainable power will inevitably decrease over time. Most satellites have a constant power consumption profile over their entire lifetime. Since the array must be sized to satisfy a satellite’s needs at the end of its life, it follows that it is oversized at the beginning of life. If MPPT is adopted, there will be a great deal of excess power from the outset that needs to be dissipated through a shunt in the form of heat. This may be undesirable from a thermal balance point of view. MPPT is therefore better suited for satellite missions where the power consumption in greater at the beginning of life than towards the end. For example, a satellite using power-hungry electric propulsion to work its way from GTO to GEO as quickly as possible may fire several thrusters during the transition, whereas only one or two thrusters will be needed later for station-keeping in GEO. Fully regulated bus voltage is nowadays achieved using a technique called Sequential Switching Shunt Regulator (S3R). A bus voltage sensor activates a set of electronic switches that disconnect superfluous array strings by placing them in open circuit. All the remaining strings but one yield the bulk of the required power. Fine-tuning of the bus voltage is achieved by varying the output of that last string through shunting (Figure 3-28).

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Solar array

Satellite body

Array string

° ° ° ° ° ° ° °

olo gy .

° ° ° ° ° ° ° °

Shunt

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Loads

Figure 3-28 Array switching regulator.

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Figure 3-29 shows a very basic block diagram of a typical satellite power regulation and distribution architecture.

Array

Charge regulator.

Discharge regulator

50 V

Battery

Shunt

Figure 3-29 Power regulation and distribution.

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Payload

Attitude & Orbit Control (AOCS) Tracking, Telemetry & Command (TT&C) Onboard Data Handling (OBDH) Propulsion

Mechanisms Thermal Control (TCS)

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3.5.

Fuel Cells

olo gy .

Solar panels have the disadvantages of being idle in eclipse and, for low-orbiting satellites, causing aerodynamic drag. Partly to overcome these problems, so-called fuel cells have been developed (Figure 3-30).

H2 O

O2

Cathode

hn

H+

Electrolyte

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ec

Load

e–

Anode

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H2

Figure 3-30 Principle of fuel cell operation.

In principle, a fuel cell operates like a battery but does not run down or require recharging. It will produce energy in the form of electricity and heat as long as fuel is supplied. A fuel cell consists of an electrolyte sandwiched between two electrodes. Oxygen passes over one electrode and hydrogen over the other, generating electricity, water and heat. The hydrogen is fed into the anode of the fuel cell. The oxygen enters the fuel cell through the cathode. A catalyst helps split the hydrogen atom into a proton and an electron, which take different paths to the cathode. The proton passes through the electrolyte, while the electrons create a separate current that can be utilized before they return to the cathode, to be reunited with the hydrogen and oxygen in a molecule of water. The main disadvantage of fuel cells is the need to carry onboard fuel in the form of hydrogen and oxygen. Today’s fuel cells are also bulky, and their space use is currently restricted to manned missions where the water byproduct is used by the crew for consumption and household chores. As smaller sizes become available, they will likely be incorporated in unmanned spacecraft as well. Future miniature fuel cells might allow consumers to talk for up to a month on a cellular phone without recharging. They also promise to change the telecommuting world by powering laptops and palm pilots substantially longer than present-day batteries.

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3.6.

Solved Problems

olo gy .

See Appendix C for solutions.

3.6.1 A spin-stabilized cylindrical satellite is deployed in geostationary orbit. Its spin axis is parallel with the earth’s N/S axis, and its cylindrical surface is covered with solar cells which operate at a constant temperature. The cells yield a total of PBOL = 1000 W at the beginning of life (BOL), which occurs at spring equinox. How much do they produce at summer solstice 7¼ years later?

hn

3.6.2 What storage capacity should a NiCd battery have, in order to sustain a satellite in a 600 km circular orbit? The satellite’s bus voltage is Ubus = 50 V, and its maximum power requirement Psat = 1000 W.

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3.6.3 How many 4 cm x 6 cm GaAs solar cells are needed to supply a GEO satellite with 4 kW of power after 10 years in orbit? The satellite’s solar panels are aligned with the earth’s N/S axis. Assume that the worst-case cell temperature is 70°C. The annual average solar power density S = 1367 W/m2.

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3.6.4 Solar cells on older satellites are prone to permanent damage in case of local shadowing of the panels. The damage is due to reverse voltage breakdown as adjacent sunlit cells connected in series continue to produce power. Modern satellites (from around 2000 onwards) avoid the problem by installing bypass diodes around each cell. Older satellites do not always provide this protection. A rescue satellite (“chaser”) is launched with the aim of intercepting an old, ailing GEO communications satellite (“target”) and docking with it. Both satellites have the classic GEO construction with a central body and two solar wings. The chaser must not shadow the target’s solar panels under any circumstances, i.e. neither during the final approach nor after docking, because of the risk of permanent cell damage. How is partial shadowing of the target to be avoided?

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4.1.

olo gy .

4. Attitude Management Introduction

“Attitude” is aerospace lingo for “orientation”. The very first manmade satellite, Sputnik, fulfilled its task brilliantly without any attitude stabilization or control whatsoever; it just tumbled randomly in orbit. So what has changed?

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In fact, all that Sputnik managed to do was to tansmit a couple of beeps of data per second from a few hundred kilometers above the earth’s surface. The satellite had no solar cells, and the onboard battery ran flat after only 3 weeks.

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Since then, the demands on satellites have grown. Modern satellites transmit and receive speech, television and camera images at data rates amounting to several million bits per second from altitudes as high as 100,000 km above our planet. High-gain antennas are needed to focus the transmission energy. These antennas, along with cameras, telescopes and other sensors have to be aimed at their respective targets. All the while, the solar arrays must stay pointed at the sun continuously for up to 15 years. This is why nowadays we launch satellites with an attitude. Attitude management involves four primary tasks: • • • •

to stabilize the satellite, to measure its attitude, to reconstitute the attitude from the measurements, and to re-orientate the satellite.

We will divide the present chapter into four main sections: Attitude Stabilization, Attitude Measurement, Attitude Determination, and Attitude Control. The attitude management subsystem has different names in different satellite projects. The most common variants are Attitude Control Subsystem (ACS), Attitude Determination & Control Subsystem (ADCS), Attitude & Orbit Control Subsystem (AOCS), and Guidance, Navigation & Control Subsystem (GN&C). The latter two lump attitude management (Chapter 4) and orbit management (Chapter 5) into a single subsystem, which can be confusing since the two tasks are separate from the point of view of physics, engineering and operations. That said, attitude and orbit control share some of the same resources, such as the computer and the thrusters - but just to confuse matters further, the thrusters belong to a different subsystem, namely Propulsion (Chapter 6). Figure 4-1 attempts to clarify the scope of the AOCS.

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4 / Attitude Management OBDH (Chapter 8)

Data bus

Attitude Management (Chapter 4) Reaction Wheels

Thrusters

Ranging Transponder

Propulsion (Chapter 6)

TT&C (Chapter 7)

GPS

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Attitude Sensors

Orbit Management (Chapter 5)

olo gy .

AOCS Computer

AOCS

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Figure 4-1 Scope of the AOCS.

Attitude Stabilization

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4.2.

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An important quantity in stability analysis is the mass moment of inertia I, so this is a good time to recapitulate its meaning. Just as the mass m of a body resists linear acceleration by mobilising a reaction force F = m·a, so the moment of inertia I resists rotational acceleration by countering with a reaction torque T = Iω
. (Or, conversely, it takes a force F to impart a linear acceleration on a mass m, and a torque T to accelerate the rotation of a body around a spin axis.) The inertia is directly proportional to the magnitude of m and I, respectively.

If we imagine a spinning satellite as being made up of n infinitely small mass elements mi, we obtain the satellite’s moment of inertia by adding up the products of each mass mi times its distance ri squared from the spin axis. n

I sat = ∑ mi ri

2

(4.1)

i =1

or, in the case of a homogeneous body: Isat = ∫r2 dm

(4.2)

In practice, of course, the mass elements onboard a spacecraft are not infinitely small but consist of substantial units i with their own moments of inertia Ii around an axis parallel to the satellite’s spin axis. With the help of Steiner’s theorem we are able to reformulate Eq 4.1 as follows: n

I sat = ∑ ( I i + mi ri ) i =1

100

2

(4.3)

4 / Attitude Management z

I1 m1

m5 r1

I2

olo gy .

z

r2 x

y

r3

m3

m2 m4

Figure 4-2 Moments of insertia.

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hn

In the above illustration, m1 and m2 represent the mass of two pieces of equipment, while m3 is the mass of the cylindrical wall, and m4 and m5 refer to the circular top and bottom plates. Instead of using the subscript sat, it is customary to indicate the chosen spin axis, in our example the z-axis.

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The axis around which a body has its maximum moment of inertia is called the major axis. The axis showing the smallest moment of inertia is the minor axis. Miraculously, these two axes are always perpendicular in all rigid bodies, regardless of shape or content. The major and minor axes, along with the third axis that forms a right-handed orthogonal triad, are called principal axes of inertia, and we denote the corresponding moments of inertia Ix, Iy, Iz. Formally speaking, the principal axes are those around which the products of inertia Ixy, Ixz, Iyz = 0. In practice, the principal axes are usually those which define relative body symmetry, as illustrated below. The analysis of the dynamic behaviour of a satellite and its constituent parts is greatly simplified if the body coordinate system xyz is anchored to the centre of mass and is chosen to coincide with the principal axes. y

x

z

Figure 4-3 Principal body axes.

In the case of a body-stabilized satellite, the x-axis usually coincides with the orbital velocity vector; the y-axis points south; and the z-axis points towards the earth or some other celestial target. The angular momentum vector H is defined as:

H=Iω

(4.4) 101

4 / Attitude Management

olo gy .

where ω is the spin axis vector, the direction of which is determined by the right-handed “screw” convention. The momentum vector is therefore aligned with the spin axis vector – at least in the case of a clean spin. Angular momentum is analogous with linear momentum P = mV, where m is the mass of the body in motion and V is its velocity vector. Here are some additional analogies which we will need later in this chapter:

Linear

Rotational Mass moment of I inertia Angular H = Iω momentum

m

Mass Linear momentum

P = mV

T =dH/dt = I dω/dt

Linear impulse

Ilin = F∆t

Irot = T∆t

Kinetic energy

E = ½mV2 = ½PV

E = ½Iω2 = ½Hω

hn

F = ma = dP/dt = m dV/dt

Torque

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Rotational impulse Rotational energy

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Force

Table 4-1 Physical analogies in linear and rotating space.

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The law of conservation of angular momentum states that the magnitude and direction of the satellite’s overall H will not change due to torques originating within the spacecraft (e.g. propellant sloshing, wheel acceleration, or the turning of solar panels). External forces are needed to change H (e.g. expelling propellant through thruster operation, interaction with the earth’s magnetic field, or solar pressure).

4.2.1. Gyroscopic Stabilization 4.2.1.1.

Gyro Dynamics

One of Nature’s most useful gifts to Man is gyroscopic stiffness, the mysterious phenomenon whereby the spin axis of a freely rotating body keeps its orientation fixed in inertial space and resists any attempt to change it. Figure 4-4 illustrates how an undisturbed spinning satellite maintains its inertial attitude as it orbits the earth.

(a)

(b)

(c)

Figure 4-4 Attitude stability of a spinning body in inertial space.

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olo gy .

Looking at Figure 4-4, orientating the spin axis perpendicular to the orbit plane, as in (c), is usually preferred over (a) and (b), since it is the only attitude where one side of the spacecraft always faces the earth, for the benefit of telecommunications and earth observation. The gyroscope thus provides not only stability, but also an inertial reference for a spinning body’s changing orientation in space. The latter thought will be developed further in Section 4.3. Though there are several ways of stabilizing a satellite in space, the gyroscope is the device most commonly used. There are basically three implementations:

hn

a) Spinning the entire spacecraft;

ec

b) Using a dual-spin concept whereby the satellite’s main platform is made to spin while part of the payload (e.g. a camera or an antenna) is “despun” to stay pointed at the earth;

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c) Installing a spinning momentum wheel inside a “despun” body.

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In addition there exists a stabilization method which does not, à priori, rely on gyroscopic stiffness, namely by d) Installing a reaction wheel along each of the satellite’s roll, yaw and pitch axes.

These four implementations are illustrated in Figure 4-5. There are variants of these methods, but they are too rare on earth-orbiting satellites to merit analysis in this book.

(a)

(b)

(c)

(d)

Figure 4-5 Stabilization methods. (a) Spinning spacecraft, (b) Dual spin, (c) Bias momentum wheel, (d) Zero momentum reaction wheels.

The implementations shown in (a) and (b) are called spin-stabilized satellites, while those in (c) and (d) are known as body-stabilized or three-axis stabilized satellites. In principle, there is not a big difference between (b) and (c), since in both cases one part of the satellite is spun and the other is despun; but because of their physical appearances, they fall into the stated categories.

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olo gy .

Solution (a) is the simplest and most reliable method of stabilizing a spacecraft. The method also facilitates feeding propellants from tanks (with the aid of centrifugal forces), and provides a benign thermal environment due to the “barbecue” effect. The major disadvantage is the difficulty of pointing a payload continuously to a target such as the earth. Other drawbacks include the relatively high propellant consumption for attitude manoeuvres (to overcome the gyroscopic stiffness), and the fact that less than one third (1/π) of the solar cells on the cylindrical shell are illuminated at any one time.

hn

If the member to be pointed is an antenna, then one solution is to install a cylindrical drum on top of the main body, and to mount radiating elements on its cylindrical wall, as shown in (a). The elements are scanned electronically like a despun phased array. This is the solution adopted e.g. for the early Meteosat and GOES weather satellites.

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In solution (b), the unit to be pointed is mounted on the shaft of an electric motor. As we shall see later, the despun platform has a calming effect on the satellite’s unstable tendencies and allows it to be spun around a minor moment of inertia axis. This in turn enables designers to build a taller, more slender spacecraft that make better use of the available volume inside the launch vehicle’s heatshield.

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In (c), the spinning flywheel inside the satellite provides gyroscopic stiffness around the axes orthogonal to the spin axis, but not around the spin axis itself. Rotation around this axis is achieved by varying the speed of the wheel. Because the wheel maintains angular momentum by virtue of its spinning motion, the method is known as momentum bias. This design overcomes the disadvantages of (a) and (b). In contrast to (c), the three wheels in (d) are normally at a standstill, i.e. they have no momentum bias. If the satellite experiences a disturbance torque, the appropriate wheel(s) will accelerate so as to correct the resulting attitude error through action and reaction; hence the label reaction wheels. It follows that method (d) is a zero momentum approach, and therefore does not employ gyroscopic stiffness as a means of stabilization, although a degree of stiffness appears as soon as a wheel begins to turn. This design is more complicated and therefore less reliable than (c), but has the advantage of greater agility and pointing accuracy around all three axes. Spin-stabilized satellites are a vanishing species, whether in pure spin as in (a) or dual spin as in (b). Even so, we will devote the next few Sections to spinning bodies, since a good comprehension of gyroscopic principles facilitates the understanding of gyros and momentum wheels. Moreover, a few nominally body-stabilized GEO satellites are spinstabilized in GTO to improve directional stability during apogee motor burn.

The gyroscopic stiffness is proportional to the moment of inertia I of the body around its spin axis, as well as to the spin rate ω. If we want to force the gyro to change its orientation, we must apply a torque T. A torque is created by a force pair, as when two hands turn a steering wheel. If we turn the wheel to the right, the torque vector T points into the steering column, in accordance with the right-handed “screw” convention. By applying the torque vector perpendicular to the angular momentum vector H for a short duration ∆t, we create a small angular displacement ∆θ of the momentum vector (Figure 4-6), such that H ∆θ = T ∆t, or:

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4 / Attitude Management T ∆θ T = = ∆t H Iω

(4.5)

T ∆t ∆θ

hn

H

olo gy .

The fact that the displacement velocity ∆θ /∆t decreases with increasing ω and/or I suggests resistance, or “stiffness.”

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Figure 4-6 Illustration of gyroscopic stiffness.

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Q: Is there a more intuitive explanation to why a gyroscope feels “stiff” when we try to turn its axis?

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A: Yes. Newton’s 1st Law states that a particle in motion wants to move along a straight line, and it takes a force to make it deviate. The greater the mass and/or the velocity of the particle, the more it will resist any diversion. This is why firearm bullets are made of lead and travel fast. If we spin a flywheel in the weightlessness of space fast enough, the centrifugal forces will eventually overwhelm the molecular cohesion among the mass elements. The wheel disintegrates, and the mass elements fly off along straight lines in all directions. When we attempt to turn the spin axis of a flywheel, we are in fact making an out-of-plane change in the “orbit” of the mass elements, and they will resist that, too. Hence the stiffness feel of a gyroscope.

Mass element

As we saw in Figure 4-4, an undisturbed spinning spacecraft will maintain a constant spin axis direction in inertial space, no matter where the satellite is along the orbit. In real life, a spin-stabilized satellite is disturbed all the time, either intentionally to achieve a reorientation of the spin axis, or involuntarily by internal or external forces. Table 4-1 above contains an entry which states that the spin rate ω will change if a torque is applied along the spin axis: T = dH/dt = I dω/dt

(4.6)

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Let us now apply a torque vector T at some arbitrary angle (Figure 4-7). z ωz

hn

T

ec

y x

olo gy .

In this case the orientation of the spin axis does not change. Conversely, Figure 4-6 showed us that a torque applied perpendicular to the spin axis will change its orientation but not its magnitude.

Using vector notation:

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T = dH/dt = I dω/dt

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Figure 4-7 Satellite with rotating coordinate system xyz, as seen from inertial space.

(4.7)

Note that, just as the torque T has three components Tx, Ty, Tz, then so does the resulting spin vector ω, namely ωx, ωy, ωz. This leads us to the intuitively obvious but nonetheless important conclusion that an arbitrary torque will cause a rotational movement not just around the spin (z) axis, but around all three axes. To illustrate the point, take a spinning top and give it a slight push (i.e. an instantaneous torque around its centre of mass). What happens? The spin axis is now subject to precession (a coning motion), which is to say that there are spin components around all three axes. To better understand the dynamic process, we need to climb out of the satellite-fixed, rotating xyz-system and move to a fixed point in inertial space some distance away from the spacecraft. Eq (4.7) then takes the form

T = dH/dt + ω x H

(4.8)

Eq 4.8 is known as Euler’s equation of motion and is a close cousin of the Coriolis theorem.

Given that H = Iω, and assuming that we have chosen the coordinate system axes xyz to coincide with the satellite’s principal axes (usually the axes of relative symmetry), Eq 4.8 may be broken down into its component parts as follows:

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yˆ

xˆ

ωy I yω y

ωz I xω z

olo gy .

xˆ  Tx   I xω
x       Ty  =  I yω
y  + ω x  T   I ω
 I ω x x  z  z z which translates to:

Tx = I xω
x + ω yω z ( I z − I y ) T y = I yω
y + ω xω z ( I x − I z )

(4.9)

Tz = I z ω
z + ω xω y ( I y − I x )

Stability Criteria

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4.2.1.2.

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Recall that there are two kinds of torques: those caused by external forces, and those generated internally to the satellite. Firing a torque-generating thruster pair gives rise to an external torque, since the thrusts are in reaction to propellant being ejected. Accelerating a momentum wheel generates an internal torque. Both kinds may upset the balance of a spinning satellite, but there is an important difference: External torques change the magnitude and/or the orientation of the satellite momentum vector H, whereas internal torques do not.

Eq 4.9 gives us a complete mathematical model of a disturbed spinning satellite’s complex motion, including the cross-coupling between axis rotations. This set of equations may also be used to analyse the stability of the satellite motion, i.e. if it will keep spinning around the intended axis or degenerate into some other spin mode. Unfortunately, the three equations cannot be solved in a closed form as they stand, but solutions may be found if we make some simplifying assumptions. For example, if we remove the torque in order to study the satellite at rest, the first two identities in Eq 4.9 reduce to: 0 = I xω
x + ω y ω z ( I z − I y )

0 = I yω
y + ω xω z ( I x − I z )

Let us eliminate ωy by differentiating the first equation with respect to time:

0 = I xω

x + ω
y ω z ( I z − I y ) 0 = I yω
y + ω xω z ( I x − I z )

Here we have assumed that ωz is constant, which is true at least during the onset of a disturbance. Elimination of ω
y gives us: 

2 ω

x + ω z 1 −



Iz Ix

 I  1 − z ω x = 0    I y 

(4.10)

Taking the Laplace transform of Eq 4.10, we arrive at the characteristic equation: 107

4 / Attitude Management  2 I  I  2  s + ω z 1 − z 1 − z ω x ( s ) = 0  I x  I y  

Iz Ix

 I  1 − z     I y 

Let

ω nut = ω z 1 −

Then

s 2 + ω nut = 0; s 2 = −ω nut



2

olo gy .



(4.11)

(4.12)

2

hn

The subscript nut stands for “nutation” which we will deal with later. Recall from the theory of feedback control systems that ωnut must be real if there is to be dynamic stability. According to Eq 4.12, this is the case if Iz > Ix and Iy, or else if Iz < Ix and Iy, but not if Iz lies somewhere in between Ix and Iy.

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What does all this signify in practice? It means that a satellite will spin cleanly around the z-axis if the corresponding moment of inertia Iz is either the greatest or the smallest of the three principal moments of inertia. Otherwise the satellite will tumble in some complicated fashion.

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However, this is not to say that the spin motion is stable in both cases. The laws of physics dictate that dynamic systems always try to shed kinetic energy to reach a certain minimum energy threshold. The same is true for a spinning satellite. Energy may be dissipated by processes internal to the spacecraft, usually some form of friction caused by structural flexing or liquid sloshing. Recall from Table 4-1 that the energy E = ½Hω. Given that the angular momentum H stays constant also during the energy dissipation process, ω must decrease as the energy E decreases. But with H = Iω, and with H being constant, I must increase as ω decreases. This means that the satellite will endeavour to spin around the major axis, i.e. the axis representing maximum moment of inertia I. If that happens to be Iz – our intended spin axis – then all is well; but if Iz is smaller than either Ix or Iy, then the satellite is no longer stable and will choose the largest of the latter two as its new spin axis. We then have a condition known as flat spin, which often spells the end of the mission. It is worth repeating that the degeneration from unstable spin around a minor axis to stable spin around the major axis only occurs if there exists some internal energy dissipation device. Otherwise the satellite will happily continue spinning around the minor axis. In conclusion, a satellite is spin-stable if Iz/Ix > 1 and Iz/Iy > 1 at all times, with Iz being the major principal axis around which the satellite is spinning. There is a more compact way of expressing these criteria:

  Iz  I − 1 z − 1 > 1    Ix  I y 

λ = 1 + 

where λ is the stability margin.

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(4.13)

4 / Attitude Management If the satellite is axisymmetric (Ix = Iy), then Eq 4.13 reduces to:

Iz >1 Ix

(4.14)

olo gy .

λ=

hn

As a rule, rigid satellites are designed with a small stability margin, e.g. λ ≈ 1.1. A bigger margin, while desirable in principle, would mean a flatter and wider satellite, making poor use of the available volume inside the launch vehicle’s heatshield. That said, if a satellite contains substantial volumes of propellant that are prone to sloshing, the margin needs to be increased accordingly.

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Historical note: Spacecraft designers’ inadequate understanding of spin dynamics has caused several mission failures. Flat spin is a case in point.

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The first mission to fail in this manner was America’s very first satellite, Explorer 1, launched on January 31, 1958. A more recent example is NASA’s Lewis spacecraft, built under the Agency’s new “better, faster, cheaper” motto and launched on August 23, 1997. As a result of flat spin, the satellite’s solar panels ceased to be adequately illuminated – a predicament which also prevented the onboard battery from being recharged.

4.2.1.3.

Nutation

Spin-stabilized satellites

When a spinning satellite experiences an internal disturbance, the angular momentum vector H maintains its original orientation. If the disturbance originates outside the spacecraft, H may change in amplitude or direction. Either way, the disturbed satellite body goes into a coning motion called nutation (Figure 4-8).

H

Figure 4-8 Nutation.

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Q: Can we observe nutation here on earth?

z

olo gy .

A: Yes. A spinning top starts out spinning “cleanly,” but as the angular velocity ωz diminishes due to friction and air drag, gravity gets hold of the top’s centre of mass. The influence of gravity constitutes a torque T around the tip of the shaft, which causes the top to nutate. The nutation amplitude θ as well as the torque T increase as ωz decelerates further and the gyroscopic stiffness weakens. Eventually the top falls over.

H

ωz ωnut

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g

hn

θ

T

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The earth is a gyro that nutates under the influence of gravitational pull from the sun and the moon. Because of the earth’s equatorial bulge, the pull is not symmetrical, thus giving rise to a torque around the earth’s centre of gravity. Consequently the earth’s spin axis precesses one turn every 26,000 years. A satellite orbit may also be seen as a gyro containing a single mass element, namely the satellite itself. Without orbital corrections, the nodes of a geosynchronous orbit (i ≠ 0°) will complete a turn in 54 years.

Let us return to Euler’s equation of motion to try to understand this phenomenon. By setting ωz = constant, we arrived at the differential equation in Eq 4.10: 

2 ω

x + ω z 1 −



Iz Ix

 I  1 − z ω x = 0    I y 

(4.15)

Note the similarity between this equation and the linear force equation m
x
+ kx = 0 , or k
x
+ x = 0 , which has a natural oscillation frequency ω = k m . We should therefore m expect our spinning satellite to exhibit an oscillation when disturbed, and that is precisely what the nutation is. Thus:



ω nut = ω z 1 − 

I z  I z  1− I x  I y

   

(4.16)

Most spinning satellites are mass-symmetric with regard to the x and y-axes (“axisymmetric”) such that Ix = Iy. Eq. 4.16 then reduces to: 2

ω nut = ω z

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 Iz  1 −  = ω z  Ix 

2

I   Iz   − 1 = ω z  z −1  Ix   Ix 

(4.17)

4 / Attitude Management The third equation in Eq 4.9 now reads (in equilibrium): 0 = I z ω
z + ω xω y ( I x − I x )

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i.e. I zω
z = 0 , and therefore ωz = constant, so there is no change or variation of the satellite’s original spin rate due to nutation.

  

(4.18)

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 Iz  I − 1 + ω z = ω z  z  Ix   Ix

ω nut = ω z 

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Eq 4.17 gives the nutation rate as perceived by an observer locked inside the spacecraft (or, more likely, by a piece of equipment). The difference between an observer inside the spacecraft and outside in inertial space is that the former does not perceive the body spin ωz, whereas the outside observer does. Therefore, seen from a point in inertial space, Eq 4.17 becomes:

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There are in fact two rotations involved: the nutation ωnut of the z-axis around the H-axis, and the rotation ωz of the body itself around its z-axis. The latter may go either way depending on whether Iz/Ix >1 (stable) or Ix (a short, chubby, stable spacecraft). The drawings feature two invisible cones, namely a “space cone” which remains fixed in inertial space, and a slewing “body cone” containing the zaxis at its centre. The two cones facilitate the understanding of the dynamics of nutation. The picture to the left is the easiest to visualize, in that ωz and ωnut rotate in the same sense as the body cone rolls on the outside of the space cone. In the right-hand picture, ωz rotates in the opposite sense to ωnut. One can convince oneself that this is so by pressing a

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4 / Attitude Management pencil against the inside wall of a glass. If the pencil is rolled around the glass clockwise, the pencil itself will rotate counter-clockwise.

tan γ =

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As for the cone sizes, it can be shown that

Iz tan θ Ix

(4.19)

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where γ is the angle between the z-axis and the line of cone contact, while θ is the nutation amplitude. Therefore, for a stable spacecraft, γ > θ, which is confirmed by Figure 4-9.

Iz 1), it is the scenario to the right that interests us. The most stable spacecraft of all would have the shape of a flat disc. We know from classical mechanics that, for a flat disc, Iz = Ix + Iy. In our axisymmetric case, Iz = 2Ix , and therefore Iz / Ix = 2. Consequently, for stability, our moment of inertia ratio Iz / Ix must fall in the following range:

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Inserting this inequality in Eq 4.17 gives us the nutation rate interval in the body-fixed coordinate system: 0 ≤ ω nut < ω z

(4.20)

Similarly, from Eq 4.18 in the inertial reference frame:

ω z ≤ ω nut < 2ω z

(4.21)

Numerical example: Let us take a satellite in the form of a homogeneous aluminium cylinder spinning at 60 rpm. Physical measurements:

Diameter d = 1 m, radius r = 0.5 m, height h = 0.5 m 2 3 Volume V = πr h = 0.4 m 3 Density of aluminium ρ = 2,700 kg/m Mass m = ρV ≈ 1,060 kg

Moments of inertia:

Iz = ½mr = 133 kg m 2 2 2 Ix = Iy = 1/12 m(h + 3r ) = 88 kg m Iz/Ix = 1.5 → satellite spin-stable around the z-axis.

Spin rate around z-axis:

ωz = 60 x 2π / 60 = 6.28 rad/s = 360 deg/s

2

2

Nutation rate: ωnut = 360° x (1.5 – 1) = 180 deg/s ≡ 30 rpm in the body reference frame. ωnut = 360° x 1.5 = 540 deg/s = 90 rpm in the inertial reference frame.

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Dual-spin satellites

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A mechanically despun member on a spin-stabilized satellite (Figure 4-5 b) has a stabilizing effect. We will not present the rather complex mathematical analysis here. Suffice it to say that the spin motion is stable even around the minor axis as long as the energy dissipation in the despun part is greater than in the spinning body (see nutation damping below). The nutation rate, measured in the inertial frame, is given by: Iz *

Ix Iy

(4.22)

*

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ω nut = ω z

Iz Ix

*

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ω nut = ω z

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In the axisymmetric case, with Ix* = Iy*:

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where Ix* and Iy* are the transverse inertia axes of the overall satellite, including the despun member.

(4.23)

The despun member is usually much lighter than the spacecraft body, such that Ix* and Iy* are only marginally greater than Ix and Iy of the spinning body alone. As a result, the nutation rate obtained from Eq 4.23 is slower than in Eq 4.18, but not drastically so. Body-stabilized satellites

A satellite equipped with an internal momentum bias wheel (Figure 4-5 c) has similar dynamic characteristics to those of dual-spin satellites, in that the wheel has now become the spinning body, and the actual body corresponds to the despun member The main difference is that the magnitudes of moments of inertia are now reversed, and therefore the nutation rate is much slower.

2

Numerical example: Take a momentum wheel spinning at ωz = 600 rpm with Iz = 1 kg m . 2 The satellite body is axisymmetric with Ix = Iy = 1200 kg m . Then Hz = 600 x 2π/60 x 1 = 2 62.8 kg m /s = 62.8 Nms, and ωnut = 0.052 rad/s = 3.0 deg/s = 0.5 rpm.

Satellites employing zero-momentum reaction wheels for stabilization (Figure 4-5 d) are insensitive to the stability and nutation phenomena discussed above, unless the disturbance torques are of such a magnitude that one or more of the wheels attain a significant spin rate. The zero momentum can always be restored through a procedure known as momentum dumping (Section 4.5) – not to be confused with nutation damping (see below).

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4 / Attitude Management Nutation damping

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Nutation is nearly always undesirable, primarily because it upsets the pointing accuracy of onboard instruments and antennas. It may also give rise to elastic flexing of flimsy appendages (such as solar panels) and to sloshing in propellant tanks. Flexing and sloshing, while undesirable in their own right, may also alter the dynamic behaviour of the spacecraft, with sometimes dire consequences for the mission.

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Historical note: The original, spin-stabilized Meteosat satellite had a telescope that took pictures of the earth every 30 minutes. The telescope was pivoted from south to north in the course of 25 minutes, after which it rapidly retraced to its starting position for another 2½ minutes. The brutality of the retrace motion induced nutation, which would have seriously degraded the quality of subsequent images. The remaining 2½ minutes of each 30-minute slot were therefore allotted to nutation damping.

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As mentioned earlier, dynamic systems tend to shed excess kinetic energy and operate at the minimum achievable energy level. Nutation represents excessive energy, since it has been introduced artificially through internal or external torques, so if we install a suitable energy dissipator, we are able to remove nutation.

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The simplest solution is to install a passive nutation damper, which usually consists of a closed tube with a viscous liquid or a steel ball inside. The tube is mounted in a way to achieve maximum sloshing or ball movement apace with the nutation frequency. Energy is dissipated through heat generated by viscous friction in the liquid, or by a ball moving through a gas and slamming into the ends of the tube. Active nutation damping may be required if a faster elimination time is required, or if the inertial properties of the satellite are such that the nutation might degenerate into flat spin (e.g. due to propellant sloshing). In this procedure, an attitude thruster is fired in pulsed mode at the spin frequency, such that the pulses are phased in a way to counteract the nutation movement. Another method consists in using an accelerometer to sense the nutation frequency and amplitude around a particular body axis, and then feed this information to the control electronics of the satellite’s momentum wheel. By modulating the spin speed of the wheel, the nutation may be damped. (An accelerometer behaves like a damped mass on a spring, whereby the displacement of the internal mass relative to the casing is monitored and delivered as an electrical output.)

4.2.2. Gravity Gradient Stabilization

Gravity gradient stabilization exploits Newton’s law of general gravitation which, in the earth-satellite context, states that the gravitational pull F on a stationary mass m is proportional to the square of the range distance r from the earth’s centre: F =µ

m r2

(4.24)

Gravity gradient stabilization is a much simpler way of maintaining a satellite’s attitude than using gyroscopic techniques. Being simpler, it is also more reliable. While the

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4 / Attitude Management gyroscopic effect is inertial in nature, the gravity gradient phenomenon has the added advantage of always keeping the spacecraft earth-pointed – a state of affairs which suits most satellites.

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A satellite stabilized by gravity gradient techniques consists of two distinct masses, which are located some distance apart (5 – 10 m) and are interconnected by a mast (Figure 4-10).

Figure 4-10 Earth pointing by means of gravity-gradient. (not to scale)

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As the name implies, we stabilize the satellite by exploiting the difference in gravitational pull on the main body (the cube in Figure 4-10) and the auxiliary mass (the sphere) due to the difference in their distance from the earth. As we shall see, that minute difference is sufficient to keep the configuration aligned with the radius vector at all times. It does not matter which of the two bodies is closest to the earth; the configuration is stable either way (Figure 4-11).

Figure 4-11 Reversed satellite attitude.

The configurational ambiguity may create a problem when deploying the satellite after launch. To illustrate this point, we will pretend that the spacecraft in Figure 4-11 is the Russian Gonyets telecommunications satellite. The satellite is launched with the telescopic mast compressed to fit inside the heatshield of the launch vehicle. After injection into orbit, the mast is deployed by telecommand, and the satellite may adopt either one of the attitudes in Figure 4-10 and Figure 4-11. The attitude in Figure 4-11 is 115

4 / Attitude Management

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not desired, because the antenna is pointing straight out into space instead of at the earth. The only way to rectify the situation is to retract the mast using the onboard electric motor, wait for the satellite to tumble out of sync, and redeploy the mast. With luck, the satellite will now align itself as shown in Figure 4-10; if not, the procedure has to be repeated until it does. Let us analyze the physics of gravity gradient. Assume that the satellite immediately after injection and deployment finds itself in the attitude shown in Figure 4-12.

CoM r

F2

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F1

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M

Figure 4-12 Differential gravitational pull.

If the auxiliary mass is closest to the earth, then the gravitational pull F1 will be greater than F2. Consequently a moment M will appear around the satellite’s centre of mass (CoM) which will endeavour to align the mast with the gravitation vector (parallel with the range vector r). The torque Tgg occasioned by the two forces F1 and F2 is obtained as follows (Figure 4-13): 3µ Tgg = 3 (I z − I x )sin(2θ) (4.25) 2r with

and

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µ = the gravitational parameter = 398,601 km3/s2 r = range distance from the earth’s centre (km) θ = angle between the satellite's "long" axis of symmetry and the radius vector Iz = the satellite’s moment of inertia around the z-axis Ix = the satellite’s moment of inertia around the x-axis Iz >> Ix

4 / Attitude Management z Iz

Ix x

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Tgg r

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Figure 4-13 Moments of inertia.

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θ

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Eq 4.25 tells us that the torque Tgg is maximum when θ = 45 deg, that it is proportional to the magnitude of the difference between Iz and Ix, and that it becomes rapidly weaker with altitude. It follows that gravity gradient stabilization is most effective for low-orbiting satellites that can afford carrying a long, deployable boom with a heavy tip mass. The torque itself is weak. As an example, a satellite in a circular orbit at 600 km altitude with Iz = 40 kg m2 and Ix = 10 kg m2 will experience a maximum stabilizing torque Tgg = 5.3 · 10-5 Nm.

The fact that the main body has more mass than the auxiliary body does not mean that F2 > F1. Think of it this way: the CoM is weightless in space, meaning that its gravitational and centrifugal forces are in balance. The auxiliary mass in Figure 4-12 is travelling around the earth at the same speed as the CoM, but it is closer to the earth. Therefore it experiences a greater gravitational pull and a smaller centrifugal force, which means that the gravitational pull predominates. For the main body the opposite is true, i.e. the centrifugal force predominates over the gravitational. So while the auxiliary body is attracted to the earth, the main body wants to head out into space. The net result is a moment about the CoM as shown in Figure 4-12. Note that, when θ = 90 deg, the two satellite bodies travel along exactly the same orbit trajectory; hence there is no gravity gradient torque. However, this attitude is unstable, since the slightest change in θ due to drift will activate the gravity gradient phenomenon. Despite these major advantages, gravity gradient is not commonly used as a stabilization method. There are three primary reasons for this:

•

The pointing accuracy is poor (approximately 5° around the nadir axis, as compared to 0.001 - 1° using gyroscopic properties);

•

Only the nadir axis is stabilized, not the other body axes, to the detriment of earthimaging instruments and antenna coverage;

•

The technique only works well in low earth orbit where the gravitational pull is strong. 117

4 / Attitude Management That said, the method remains attractive for a select number of low-budget, low-orbit missions where the cost of developing, building and launching the satellite(s) is of greater concern than the pointing accuracy.

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There are two reasons for the poor pointing accuracy:

a) Any pendulum oscillation around the stable direction following deployment will continue indefinitely unless it is damped.

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b) The earth’s gravity field is not homogeneous, i.e. the gravity vector deviates from the range vector r as the satellite moves along the orbit due to varying mass concentrations in and on the earth (mountains, ocean trenches, etc.).

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Even if we manage to damp out the oscillation in (a) with a suitable energy dissipator, we are left with the problem in (b).

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Point (b) may be appreciated if we use the analogy with the nut part of a coconut. The nut is basically round, but its surface has many wrinkles. The same is true for the earth’s gravitational field potential at any particular altitude. Eq 4.26 is a fairly accurate mathematical model of the wrinkled field. The model is based on spherical harmonics, which is a three-dimensional form of Fourier series.

µ

n ∞ n  R 1 +  ∑ ∑   (C nm cos mΛ + S nm sin mΛ ) Pnm (sin λ ) r  n = 2 m =0  r  

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(4.26)

where U(r,λ,Λ) = is a potential function describing the gravitational surface at each point r, λ, Λ; r = radial distance to the satellite; λ = latitude; Λ = east longitude; R = earth's average radius = 6371 km; Pnm = associated Legendre polynomials; Cnm and Snm = harmonic coefficients.

The gravitational force acting on a satellite at point r, λ, Λ above the earth is: T

1 ∂U   ∂U 1 ∂U T F = ∇U =  , ,  = (Fr , Fλ , FΛ )  ∂r r ∂λ r cos λ ∂Λ 

(4.27)

Eq 4.26 is not as difficult to use as it looks, since the various polynomials and coefficients are readily available in the literature. That same literature also offers suggestions how far to expand the series for a given accuracy requirement (e.g. stopping at n = 10). The purpose of showing Eq 4.26 and 4.27 here is only to illustrate the methodology, which is also used for modelling the earth’s magnetic field (Section 4.2.3). We mentioned earlier that the satellite is only stabilized along the nadir axis, meaning that the other two axes are free to drift uncontrollably. A degree of control may be introduced by mounting reaction wheels along the two free axes – but in so doing, we have defeated

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4 / Attitude Management the very purpose of having gravity gradient stabilization, namely simplicity and reliability.

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4.2.3. Geomagnetic Stabilization The earth’s magnetic field may be approximated by a dipole whose axis forms an angle β ≈ 11° with the earth’s spin axis (Figure 4-14). To be more precise, the geomagnetic pole is currently located at latitude 79°N, longitude 109°E, and is slowly drifting northwestward.

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Geographic Geomagnetic North North β

S

Figure 4-14 Geomagnetic stabilization of satellites in equatorial and polar orbits.

The strength of the geomagnetic field is given by the magnetic flux density B, measured in gauss, tesla or Wb/m2. Wb stands for weber. 1 Wb = 1 V-s; 1 tesla = 1 Wb/m2 = 1 kg/A-s2 = 1 N/A-m = 104 gauss. The magnitude of B in the simplified dipole model may be computed from the following expression, assuming β = 0 deg: B = 2.6 ⋅ 1011

B0 1 + 3 sin 2 λ [tesla] 3 r

(4.28)

where r is the radial distance to the satellite in km, λ is the geomagnetic latitude, and B0 is the magnetic flux density at the geomagnetic equator on the earth’s surface = 0.30 gauss = 3 · 10-5 tesla. For example, the value of B at 300 km altitude above the equator (r = 6,671 km) is 2.6 · 10-5 tesla, while at geostationary altitude it is found by setting r = 42,164 km, which gives BGEO ≈ 10-7 tesla. Aligning a satellite along the geomagnetic field lines is straightforward, since all it requires is a magnet onboard. The method has some validity in the equatorial orbit, while its usefulness in polar orbit – or in any inclined orbit for that matter – is dubious due to the changing orientation of the field lines encountered by the satellite.

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As in the case of gravity gradient stabilization, this technique is only of interest in low orbits, since the geomagnetic field strength decreases with r3. Also, as with the gravitational field, the geomagnetic field pattern is far from homogeneous, to the detriment of the satellite’s pointing accuracy. However elegant Eq 4.28 may seem, it is not very useful, partly because of the significant tilt β, and partly due to the fact that the actual field is far from regular. If the earth’s gravity field is the shell part of a coconut, then the magnetic field may be likened to a green pepper vegetable, surface irregularities and all. Here again, spherical harmonics provide a more precise model of the field in a geographically linked coordinate system: n +1

(Gnm cos mΛ + H nm sin mΛ ) Pnm (cos λ* )

(4.29)

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∞ n R U (r , λ* , Λ) = R ∑ ∑   n =1 m = 0  r 

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where U(r,λ∗,Λ) is a potential function describing the gravitational surface at each point r, λ∗, Λ; λ∗ = 90° - λ is the colatitude; R = 6371 km; Pnm = associated Legendre polynomials; Gnm and Hnm = harmonic coefficients.

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Note the strong similarity with Eq 4.26, which models the earth’s gravitational field, the main difference being the use of the colatitude λ∗ rather than the latitude λ, whereby λ∗ = 90° - λ . The geomagnetic field density acting on a satellite at point r, λ∗, Λ above the earth is: 1 ∂U 1 ∂U   ∂U B = −∇U =  − ,− ,−  = ( B r , Bλ , B Λ ) * r ∂λ r sin λ* ∂Λ   ∂r

(4.30)

While the geomagnetic field is of only marginal interest for stabilization purposes, it is nevertheless an important element for attitude measurement (Section 4.3) and control (Section 4.5).

4.3.

Attitude Measurement

Attitude measurement means using sensors to establish the orientation of the satellite in space, and to relay the sensor output to the ground via telemetry. Attitude determination is the next step; it involves processing the sensor data to improve measurement accuracy and place the attitude in an operationally convenient coordinate system (“datum”). Operational convenience means facilitating the association of attitude with payload functions (e.g. taking images or pointing antennas) and housekeeping tasks (e.g. the preparation of attitude and orbit manoeuvres). For example, many geostationary communications satellites use an orbit-referenced coordinate system whose x-axis is aligned with the velocity vector, whose y-axis points south along the negative orbit normal, and whose z-axis coincides with the nadir vector. 120

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Other GEO satellites prefer a similar system but earth-referenced, such that the y-axis is aligned with the earth’s spin axis rather than the orbit’s normal. An astronomy spacecraft, on the other hand, will usually require an inertially fixed coordinate system, with selected stars as reference points. The attitude sensors are chosen to make the correlation with the preferred datum as direct as possible.

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In most satellite missions, attitude measurement takes place onboard the satellite, while attitude determination is carried out on the ground. But there are exceptions. For example, the attitude of a satellite may be measured on the ground by interferometric methods, i.e. measuring the phase difference between incoming radio carrier signals. Conversely, attitude determination may be carried out onboard the satellite, in addition to attitude measurement, by equipping it with GPS receivers (Section 4.3.8). 4.3.1. Definition of Axes

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We will introduce this Section by recalling the definition of roll, pitch and yaw, since this terminology is important in the context of body-stabilized satellites. In the case of a ship or an aircraft, roll is the “rocking” movement around the axis of symmetry (Figure 4-15). Pitch is the up-and-down movement of the symmetry axis (Figure 4-16), and yaw is the sideways movement of the symmetry axis (Figure 4-17). An aircraft uses the ailerons to achieve roll, the elevators to create pitch, and the rudder to induce yaw. Elevator

Aileron

Rudder

Aileron

φ

Figure 4-15 Definition of roll.

θ

Figure 4-16 Definition of pitch.

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ψ

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Figure 4-17 Definition of yaw (also known as “heading”).

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The three rotation angles φ, θ and ψ are called Euler angles. They are measured with reference to the equilibrium axes – i.e. level, horizontal flight with the fuselage aligned with the velocity vector in the case of an aircraft.

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To the extent that a satellite resembles an aircraft, it is easy and logical to use the same geometrical convention. Let us equate the solar panels with the wings, the spacecraft’s body with the aircraft’s fuselage, and the earth-facing side of the spacecraft’s body with the belly of the fuselage (Figure 4-18).

x Roll axis (= velocity vector in GEO)

z Yaw axis points towards the earth

Earth

y Pitch axis completes the right-handed coordinate system (= south in case of a GEO satellite)

Figure 4-18 Definition of roll, pitch and yaw for a satellite.

If the satellite’s geometry offers no analogy with an aircraft, it is up to the designer to name the axes. Usually one side of the satellite faces the earth; if so, it is common practice to let z denote the satellite-earth (nadir) axis. A right-handed xyz-convention is the norm as regards axis sequence and rotational direction. 4.3.2. Attitude Measurement Hardware

Note that there is a difference between a sensor and a detector. The detector registers the presence and magnitude of incoming radiation (mostly visible or infrared) and converts it to voltages or currents. The sensor is the structural box that houses one or more detectors.

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4 / Attitude Management It may also include optics and processors for performing power and signal conversion, as needed.

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The sensors most commonly used are sun sensors, earth sensors, star trackers, magnetometers, gyros and GPS receivers. The satellite designer’s choice among these sensors depends mainly on the payload’s preferred datum, the stabilisation method (spin or body), the required measurement accuracy, and the cost.

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The first three sensors measure the angle between the satellite’s axes and the vector from the satellite towards a celestial target. The position of the targets is either fixed in inertial space (i.e. the stars), or moves in space in a highly predictable fashion (the sun and the earth). Because the latter two sensors only resolve the attitude with respect to one or two of the satellite’s three axes, it is necessary to embark at least two of the sensors and to combine their measurements. This will be explained below.

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The following sensors are used on body-stabilised satellites. The magnetometer is relatively simple in construction, and is also less accurate. It measures the magnitude and direction of the instantaneous geomagnetic field vector in the body coordinate system. Since the geomagnetic field has been fairly well mapped, it is possible to translate the measurements to a more convenient terrestrial datum.

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The gyro is used during attitude manoeuvres and measures the inertial attitude of all three satellite axes with respect to the starting position. GPS receivers are flown on a trial basis to combine attitude measurement and determination directly onboard the satellite. Links to detailed hardware data sheets are found in Appendix E. 4.3.3. Sun Sensors

The sun is an ideal attitude reference for satellites in earth orbit because of its high luminosity and small apparent size (0.5° as seen from the earth). In its most basic form, a sun sensor is simply a solar cell, whereby the cell output current is proportional to the solar incidence angle φs according to the cosine rule. As illustrated in Figure 4-19, the measurement is ambiguous, since the incoming sunlight may lie anywhere along the mantle of the cone.

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Ambiguity cone

φs

Iout = I0 cos φs

Figure 4-19 Solar cell as basic sun sensor.

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Solar cell

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This type of sun sensor is sometimes mounted on the solar panels of GEO satellites to facilitate sun tracking. The ambiguity is of no consequence in this case if the panels only have one degree of freedom of rotation.

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A more sophisticated sun sensor is shown in Figure 4-20. Sunlight enters through a square window on the sensor wall and is projected onto the four solar cells below. The matrix of cells yields four different output currents, whose ratios determine the solar incidence angle with respect to the sensor’s three body axes.

φz

φx

φy

I1

I2

I3

I4

Figure 4-20 Static sun sensor with four solar cells.

In the above example, cell 1 is illuminated 70%, cell 2 – 30%, cell 3 – 60% and cell 4 – 100%. Note that this sensor only measures the direction of the sun vector relative to the 124

4 / Attitude Management three body axes. It still does not tell us how the satellite is rotated around the sun vector. Therefore it does not by itself allow us to determine the satellite’s attitude unambiguously, so we are left with three “ambiguity cones,” one for each body axis.

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The accuracy of the cell-based sensors is limited to a few degrees, mainly because of the flatness of the cosine curve near its peak. The FOV is similarly limited to ≈ 170°, since sunlight is not absorbed by the cell at grazing incidence angles due to total reflexion.

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Slit openings

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Several sun sensor designs exist for greater accuracy, with 0.1° being readily achievable. The one presented in Figure 4-21 is called a V-slit sun sensor.

Figure 4-21 V-slit sun sensor.

The working of the sensor is easiest to understand if we imagine it mounted on the cylindrical wall of a spin-stabilized satellite, and if we project the sunlight entrance slits onto a surrounding sphere (Figure 4-22). Using a sphere is justified on the grounds that the satellite changes its attitude by “nodding” around its centre of mass. ω

φs

φs

ω∆t

90°-φs

α 90°-φs

90°-α

ω∆t

ω∆t

Figure 4-22 V-slits projected onto sphere whose spin-axis coincides with that of the satellite.

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Note that the satellite is inside the sphere looking out, and that we are viewing the same sphere from the outside.

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The quantity we wish to reconstitute is the solar incidence angle φs, i.e. the angle between the sun vector and the satellite’s spin axis. With the satellite – and hence the sphere – spinning at an angular velocity ω, the sun will trace a circle on the sphere as per the dotted contour. The trace first crosses the tilted slit, and ∆t seconds later the vertical slit. A detector inside the sensor records the slit crossings. If the slits form an angle α with each other, we arrive at the spherical triangle shown enlarged on the right. All the sides are great circles, so we may use spherical trigonometry (see Appendix A) to derive φs. Napier’s rule gives us:  tan α  φ s = tan −1    sin(ω ∆t ) 

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(4.31)

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For example, take a satellite spinning at ω = 60 rpm = 2π rad/s. The spacecraft comes equipped with a V-slit sun sensor whose slits are inclined at an angle α = 30°. Figure 4-23 plots the sun angle φs versus the measured transition time ∆t. From the diagram it is evident that the lower limit of measurable sun angle equals the angle between the slits. 90

Sun angle φ s (deg)

80 70 60 50 40 30 20 10

0

0

50

100

150

200

Slit transition time ∆ t (msec)

Figure 4-23 Sun angle versus slit transition time for V-slit sun sensor, with the angle between slits = 30° and spin rate = 60 rpm.

The V-slit sun sensor measures the sun angle with respect to only one body axis – in this example the spin axis. The orientation of the spin axis in inertial space remains as ambiguous as in Figure 4-19, i.e. the spin axis may lie anywhere on the mantle of the ambiguity cone whose apex half-angle = φs. To resolve this ambiguity, we need a second, independent sensor (see Section 4.4.2). It is possible to use a V-slit sun sensor on a body-stabilized spacecraft. In the absence of a spinning motion, the sun is made to sweep across the entrance slits by placing an oscillating mirror in front of them. Since the sensor measures the sun angle with respect to only one body axis, a second sensor may be mounted perpendicular to the first to obtain the sun angle relative to a second body axis (Figure 4-24). Placing the second

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4 / Attitude Management

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olo gy .

sensor on a different body face is not a good idea, since both sensors might not see the sun at the same time, and the attitude may drift in the meantime.

Figure 4-24 V-slit sun sensors on a body-stabilized satellite.

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The sun sensors described so far are analogue by nature, and the sensor output must therefore be digitized before transmission to the ground via telemetry. Digital sun sensors eliminate this step in the data relay chain. A sensor developed by Adcole allows the sunlight to enter through an entry slit and project onto a floor, which has rectangular holes (reticle slits) laid out in a Gray-code pattern. The sun beam thus coded falls onto underlying photo detectors, which produce electrical 1’s, and 0’s (Figure 4-25).

φs

0

0

1

1

1

0

1

Figure 4-25 7-bit digital sun sensor. Credit: Adcole Corporation

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4 / Attitude Management The field of view (FOV) of this sensor is in the order of 60° x 60°. In Gray-code, only one bit changes from one transition to the next, whereas in binary code there can be several bit changes. The Gray-code is therefore more resilient to bit errors.2

olo gy .

4.3.4. Earth Sensors

With most satellites being focussed on the earth, our planet is an obvious target for attitude referencing. The reference vector of interest is the one from the satellite to the earth’s centre. However, finding the centre is not trivial, given the size of the earth’s disc as seen from a satellite. The earth’s disc subtends an angle 2ρ = 2 sin-1[R/(R+h)]

hn

(4.32)

R ρ ρ

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R

h

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R

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at the satellite according to Figure 4-26.

Figure 4-26 Earth’s disc as seen from a satellite.

For example, seen from a GEO satellite, the earth subtends 2ρ = 2 sin[R/(R+h] = 2 sin(6,378 / 42,164) = 17.4°. From a 600 km LEO, 2ρ = 132°. To make matters worse, only the daylight portion of the globe is illuminated in the visible spectrum, and the edge of the disc (the limb) is rather fuzzy due to the atmosphere. To overcome the illumination problem, infrared detectors are favoured over detectors operating in the visible spectrum, since the earth radiates heat day and night. The CO2 portion of the IR band (14 – 16 µm) is preferred due to its relatively flat emission profile, and is bandpass-filtered by the sensor optics. A simple method of finding the centre of the earth’s disc is shown in Figure 4-27.

2

Gray-codes are useful in analogue-to-digital (A/D) converters, since a slight change in location only affects one bit. Using a typical binary code, several bits could change, and slight misalignments between reading elements might cause wildly incorrect readings. To illustrate this point, decimal 63 = binary 0111111, while decimal 64 = binary 1000000, i.e. a change of 6 bits.

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4 / Attitude Management I1

I3

olo gy .

I2

I4

(a)

(b)

(c)

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Figure 4-27 Static infrared earth sensor.

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Four IR detectors in a star configuration view the earth. In (a) the FOV of the detectors are all covered at around 50%, so the centre of the earth must lie in the middle between them. In (b) one detector is covered 100%, another 0%, and the remaining two 20%. In analogy with the static sun sensor in Figure 4-20, the ratios of the IR detector outputs I1, I2, I3, I4 give us an idea where the centre of the earth has moved relative to the body axis. The same applies to (c).

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This type of sensor is suitable for body-stabilized satellites. It permits simultaneous attitude angle determination around two body axes (i.e. pitch and roll in the picture, but not yaw). However, because of the poor definition of the earth’s limb and the relative weakness of the IR radiation, the accuracy of this earth sensor is limited to a few degrees. From the accuracy viewpoint, a better approach to finding the centre is to somehow measure the length of a scan chord from limb to limb, and then divide that distance in half. This can be done by letting an infrared sensor scan the earth’s disc and recording the limb transitions (Figure 4-28).

γ

ω

2α

.

Figure 4-28 IR earth sensor type Dual-Beam Horizon Crossing Indicator.

The earth sensor works on the principle that crossing the earth’s limb is detectable, since the IR radiation goes abruptly from around 4 K to 293 K in the transition from space to earth – and vice versa. By allowing the IR sensor to scan the earth’s disc, we record two such limb crossings in each spin period, as well as the time difference ∆t between them. The chord length is therefore γ = ω ∆t. 129

4 / Attitude Management

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The optics of each telescope have a FOV of approximately 1°. The germanium lenses contain a filter for letting the CO2 spectral band through. The radiation is detected by a bolometer. A bolometer measures changes in the heat input from the surroundings and converts this into a measurable quantity such as a voltage or current. The earth sensor described above is known as a horizon crossing indicator (HCI) or, alternatively, a pencil-beam IR sensor.

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If the sensor has two detectors which are mutually inclined (Figure 4-28), we obtain two chords having arc lengths γ1 = ω ∆t1 and γ2 = ω ∆t2, as suggested in the right-hand illustration. The two chord lengths will be equal only if the satellite’s spin axis is truly perpendicular to the earth vector. Intuitively, therefore, we may conclude that the ratio of unequal chord lengths is proportional to the spin axis offset from that reference.

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As in Figure 4-22, the principle is easiest to understand if we choose a spin-stabilized satellite and circumscribe it with a sphere (Figure 4-29). We seek to calculate the earth angle φe between the satellite’s spin axis and the nadir vector.

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ω

γ1

φe

γ2

½ω∆t2

φe−α φe+α

φe

Nadir

ρ

α

ρ

α

½ω∆t1

φe-α

φe

φe+α

ρ

ρ

Figure 4-29 Calculation of the earth angle θe.

In Figure 4-29, we are again looking at the circumscribed sphere from the outside, recalling that the satellite is located at the centre of the sphere. The shaded circle is the earth’s disc projected onto the sphere, and the scanned chords are shown as thick black lines. The angle ρ is the angular equivalent of the earth’s radius R as per Eq 4.28. The two IR telescopes are inclined ±α degrees to each other (Figure 4-28). Knowing ρ, α, ω, ∆t1 and ∆t2, we may now compute φe from the magnified spherical triangles to the right in Figure 4-29. Using the cosine rule: cos ρ = cos φe sin α + sin φe cos α cos (½ω ∆t2)

(4.33)

cos ρ = - cos φe sin α + sin φe cos α cos (½ω ∆t1)

(4.34)

Eliminating cos ρ, we arrive at the following equation:

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4 / Attitude Management

  2 tan α φ e = tan −1    cos(½ω ∆t1 ) − cos(½ω ∆t 2 ) 

olo gy .

(4.35)

Figure 4-30 shows φe for various chord length ratios ω∆t1 / ω∆t2 pertaining to a GEO satellite spinning at ω = 60 rpm ≡ 2π rad/s, and having the two IR telescopes canted at α = ±10°. 93 92

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91 90 89 88

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Earth angle φ e (deg)

94

87 0.1

1.0

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86

10.0

Chord ratio γ 1 /γ2 = ∆t ∆t2 ∆ 1/∆

Co p by yri Lu ght leå © Un 201 ive 4, rsi ty o

Figure 4-30 Earth angle φe as a function of scan chord length ratio.

The corresponding geometry is shown in Figure 4-31.

∆t2

∆t2

φe

φe

∆t1

∆t1

Figure 4-31 Scan chords in perspective.

To be able to discriminate between the two chord lengths, α should be as large as possible. But note that α must be 100, we may replace 2N +1 with 2N. Thus:

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4 / Attitude Management d1 ≈ 2πRN + ωb R t1

We also know that d1 = c t1. If we solve for t1, we have: 2πNR c − ωb R

olo gy .

t1 =

Similarly, the travel time t2 through branch 2 is defined by: 2πNR c + ωb R

hn

t2 =

The difference in travel distance is therefore:

2

c2 − ωb R 2

Since ω2R2 3000 km, corresponding to a frequency f = c/λ = 100 Hz.

λ ∆φ ⋅ 2 360

(5.4)

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∆d =

ec

The phase measurement error ∆φ at the ground station is typically 3 deg, which translates to a slant range error of

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In the chosen example: ∆d ≥

3000 3° ⋅ = 12.5 km 2 360

This magnitude of error is usually not acceptable. If the orbit had been geostationary (d ~ 40,000 km), the error would have grown to ∆d > 330 km, which again is unacceptable due to the risk of interference with neighbouring satellites. The solution lies in adding a tone with a higher frequency since, according to Eq 5.4, the error diminishes with smaller wavelengths. The only problem with this approach is that we now introduce a phase ambiguity, i.e. we do not know within which period the phase difference is measured.

φ/360° · λ

d

d

λ

Figure 5-7 Short wavelength ranging tone.

Eq 5.3 now reads:

d=

174

λ φ  n −  2 360° 

(5.5)

5 / Orbit Management where n is an integer 1, 2, 3, ... etc., indicating the number of periods within the two-way range distance.

olo gy .

The temptation, then, would be to minimize the range error ∆d by choosing a super-high frequency for the second tone. Problems arise if the second wavelength is shorter than the error in the first, because we now move into a different kind of ambiguity. Even without this new ambiguity, Eq 5.5 might be satisfied by more than one value of n. It is therefore customary to add several intermediate tones. The European (ESA) ranging standard includes 7 tones at 8, 32, 160, 800 Hz, along with 4, 20 and 100 kHz. The U.S. (Goddard) norm adds an 8th tone at 500 kHz.

λ2 

ec

φ   n2 − 2  ; 2  360°  λ  φ  d = 4  n4 − 4  ; 2 360°  λ  φ  d = 6  n6 − 6  ; 2 360°  d=

(5.6)

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λ1 

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φ   n1 − 1  ; 2 360°  λ  φ  d = 3  n3 − 3  ; 2 360°  λ  φ  d = 5  n5 − 5  ; 2 360°  λ  φ  d = 7  n7 − 7  . 2 360° 

d=

hn

The equations needed to solve a set of ESA ranging measurements φ1, φ2, ... , φ7 will therefore look like this:

where the wavelength λi = c/fi, and fi = 8, 32, 160, 800 Hz, 4, 20 and 100 kHz. The correct value for d is obtained when n1, n2, n3 ... are all integers.

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Phase Tone (Hz) difference φ (deg.) 8 19.2 32 76.9 160 24.3 800 121.3 4000 246.6 20000 153.1 100000 45.7

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Numerical example: When tracking a LEO satellite, the following phase measurements were recorded at an ESA ground station:

d

hn

d

ec

Determine the orbital height d, as well as the height error ∆d, assuming that the accuracy of the phase meter on the ground is limited to ∆φ = 3 deg.

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Solution: We know that the satellite is in LEO (i.e. below 10,000 km); hence n1 = 0 for the first tone f1 = 8 Hz, since the wavelength λ1 ~ 37,000 km >> 2 · 10,000 km. Insertion in Eq 5.5 yields d = 1000 km. If this is the right answer, then n2 = 0 also for the second tone, since λ2 ~ 9400 km >> 2000 km. Solving Eq 5.5 one more time confirms that d = 1000 km. We therefore have reason to believe that d = 1000 km is the correct value, but just to be sure, let us look at the third tone with λ3 ~ 1900 km. This value is less than 2000 km, and therefore n3 must be at least 1. If we try this value in Eq 5.5, we again arrive at d = 1000 km. Continuing along the same line, we find that d = 1000 km for each of the subsequent integer values of ni shown in the table below. f (Hz) 8 32 160 800 4,000 20,000 100,000

λ (m) 37,474,057 9,368,514 1,873,703 374,741 74,948 14,990 2,998

n 0 0 1 5 26 133 667

φ (deg) 19.2 76.9 24.3 121.3 246.6 153.1 45.7

d (km) 1000 1000 1000 1000 1000 1000 1000

∆φ (deg) 3.0 3.0 3.0 3.0 3.0 3.0 3.0

∆d (km) 156.1 39.0 7.81 1.56 0.31 0.06 0.013

Independently of the orbital height, we may compute the measurement error ∆d from Eq 5.4. According to the last column of the table, the highest tone provides a theoretical accuracy of 13 m. In practice, such a high accuracy may not be achievable through ranging due to somewhat unpredictable signal delays in the satellite transponder and phase distortions in the atmosphere.

5.2.3. Laser Ranging

In theory, the laser is the ideal ranging tool with its ultra-short wavelength and associated high measurement accuracy. The principle is simple: a short laser pulse is transmitted at a time t1 towards the satellite equipped with a cluster of retroreflectors (Figure 5-8). Optical retroreflectors consist of open corner cubes whose orthogonal walls are covered with mirrors. A light beam arriving at the retroreflector from whichever direction is 176

5 / Orbit Management always returned to the source, and nowhere else. The laser pulse is thus returned to the originating laser station, where it is clocked in at time t2. The range distance from the ground station to the satellite is calculated as: d = ½ c (t2 – t1) = ½ c ∆t

olo gy .

(5.7)

ec

hn

using the same notation as in PRN ranging.

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Figure 5-8 Working principle of the concave optical retroreflector (shown in two dimensions on the left and three dimensions on the right).

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Retroreflectors are also used in the tail lights of road vehicles to ensure that the headlights from a following vehicle are reflected back to the driver of that vehicle, and nowhere else (Figure 5-9).

Figure 5-9 Retroreflectors in vehicle tail lights.

The greatest challenge using laser ranging is for the ground station to aim the laser telescope accurately enough to hit the satellite's retroreflectors.

Suppose, for example, that we are employing this technique on a GEO satellite whose cluster of retroreflectors is 1 m in diameter. The required pointing accuracy is then θ = 2 tan-1(½ · 1 / 35,786,000) = 1.4 · 10-6 degrees – and that is in the ideal situation with the satellite in zenith and the retroreflector perpendicular to the ground station. As for LEO satellites, the lower pointing accuracy requirement is outweighed by the need to track the satellite across the sky.

Another difficulty with laser ranging is that the weather has to be clear, because laser light is absorbed by clouds. Despite these challenges, laser ranging is sometimes used, and accuracies down to a few centimetres are claimed.

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5 / Orbit Management 5.2.4. Range Rate

2 1 V = µ −  ; r a

r=

olo gy .

In addition to measuring the slant range d itself, it is possible and useful to measure the rate at which the slant range changes, i.e. dd/dt. As we shall see in the following, rangerate measurements are particularly helpful in determining the satellite’s orbital velocity vector V. Knowing V is helpful for refining our knowledge of key orbital elements. Recall from Eq 2.13 and 2.3 that a (1 − e 2 ) 1 + e cos ν

hn

With these two equations, we are able to corroborate our ranging measurements in terms of a, e and ν at each instant.

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Range-rate measurements are based on the Doppler shift of the satellite’s signal carrier as the spacecraft approaches and moves away from the ground station. We are all familiar with the phenomenon where the horn of a car or a train appears to have a higher pitch on approach than on retreat. The reason is that a component of the vehicle’s velocity is added to the speed of sound, such that the wavelength entering our ears seems compressed. As a result, the observer perceives a higher tone than that of the horn itself. The reverse is of course true as the vehicle passes and draws away.

A

V

B

C

θ

V cosθ D

Figure 5-10 Principle of Doppler shift.

The Doppler effect is illustrated in Figure 5-10. The vehicle A is moving to the left with a velocity V while emitting a sound. Because of the limited propagation velocity of sound, the sound waves are bundled up in front of the vehicle, such that the listener B faces a shorter sound wavelength than the original. The shorter wavelength is perceived as a higher frequency according to f = λ/v, with v being the speed of sound. Conversely, the listener C encounters a longer wavelength, and therefore a lower frequency. And the listener in D perceives a frequency proportional to V cosθ.

In the space context, A is the satellite. D is a typical ground station, whose recording of the Doppler effect in the telemetry signal carrier varies over time apace with the variation of θ as the satellite passes overhead. The arithmetic is fairly complex, but we can accept 178

5 / Orbit Management intuitively that the Doppler effect is proportional to V cosθ, and hence to the range rate dd/dt = d
.

olo gy .

The slant range vector d can be expressed as follows in the defined topocentric coordinate system uvw:

 sin δ   d u    duvw = d  cos δ sin α  =  d v  cos δ cos α   d w 

(5.8)

hn

where α is the azimuth and δ is the elevation of the tracking antenna at each instant as it follows the satellite across the sky (Figure 5-11). The satellite’s velocity relative to the ground station is the time derivative of the range vector:

ec (5.9)

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d
uvw

  d
sin δ + δ
d cos δ  
=  d cos δ sin α − δ
d sin δ sin α + α
d cos δ cos α   d
cos δ cos α − δ
d sin δ cos α − α
d cos δ sin α    uvw

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If the ground station is equipped to measure the range rate d
, it is probably also capable of recording the rate of change of azimuth and elevation, i.e. α
and δ
. In that case the relative velocity vector d(duvw)/dt is obtained directly from Eq 5.9. z

z

δ

d

r

z w

d

v

α

u

s λ

Λ+φ

Λ+φ

y

x

x

y x

y

Figure 5-11 Definitions of α, δδ, r, d, s, λ, φ and the uvw-system.

Converting the range vector from the topocentric uvw-system to the geocentric-equatorial inertial xyz-system is straightforward (Figure 5-11):

 cos(Λ + φ) − sin(Λ + φ) 0   cos λ 0 − sin λ   sin δ        dxyz = d  sin( Λ + φ) cos(Λ + φ) 0    0 1 0    cos δ sin α  (5.10)  0 0 1   sin λ 0 cos λ   cos δ cos α   or, in shorthand: dxyz = [A(Λ, λ, φ)] duvw with Λ being the ground station’s longitude, λ its latitude, and φ the hour-angle of the Greenwich meridian relative to the inertial axis ϒ. The ground station provides the sidereal time t (in decimal hours) when the ranging measurement was made. This gives us: 179

5 / Orbit Management φ = 360° t/24 (deg)

(5.11)

olo gy .

It follows that d
xyz = [A(Λ, λ, φ)] d
uvw . However, d
xyz ≠ Vxyz , because we have yet to take into account the fact that the ground station itself is moving within the inertial xyz-system due to the rotation of the earth. If the earth’s angular velocity vector in the xyz-frame is ωe and the station location vector is s, then

Vxyz = d
xyz + ω e × s

(5.12)

ωe

hn

ωe ωe x s

d’xyz + ω e x s = Vxyz

d

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s

ec

ωe x s

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d’xyz

Figure 5-12 Determination of the satellite velocity vector V from range-rate measurements.

We already know that the earth turns around its axis 360° in 23 hours and 56 minutes (sidereal time), i.e. ωe ≈ 0.25°/s = 0.0044 rad/s. We also know that the rotation vector is aligned with the inertial z-axis; therefore ωe = ωe (0, 0, 1)T. From Figure 5-11 (middle), the station vector s is found to be: sxyz = R [(cosλ cos(Λ+φ), cosλ sin(Λ+φ), sinλ)]xyzT

(5.13)

Insertion of d
, ωe and s in Eq 5.12 yields the desired velocity vector Vxyz.

The range-rate approach is not the only method of determining V. An alternative is to determine successive slant range vectors d1, d2, d3 ... etc., while recording the precise timing for each (Figure 5-13). If the time difference ∆t is sufficiently small, then Eq 5.14 can be used to obtain V: V ≈ (r2 – r1)/∆t

(5.14)

However, real-life ranging measurements take rather longer, and a more sophisticated algorithm is therefore necessary. For more detail, consult Bate et al. [1].

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d3

olo gy .

d1 d2

hn

Figure 5-13 Determination of a satellite’s velocity vector V from successive range measurements.

5.2.5. Using the Global Positioning System

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The Global Positioning System, also known as Navstar, is a constellation of 24 operational satellites which travel in circular orbits at a height of 20,200 km and an inclination of 55°. There are 6 orbital planes, with 4 satellites in each. This configuration allows users anywhere on earth to be in radio contact with at least four satellites at any one time. The orbital height yields an orbital period of 12 hours, which means that each satellite passes over the same geographical points every day, the advantage being that the timing, azimuth and elevation of passes are readily predictable. The GPS antenna coverage and signal characteristics were originally designed for terrestrial applications. Satellite designers have since discovered that GPS is also suitable for position measurement of spacecraft, especially in LEO. The claimed measurement accuracy varies wildly. The “raw” real-time accuracy is probably in the order of 15 m, but much greater accuracies are achieved through mathematical filtering of large data sets. Velocity is readily computed from the position measurements. While satellites in orbits below that of GPS acquire GPS signals continuously the same way as terrestrial users, satellites in orbits above that of GPS (e.g. GEO) receive the signals more sporadically, and the signals are also much weaker. The reason becomes apparent when studying Figure 5-14.

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5 / Orbit Management GEO

olo gy .

GPS orbit

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Earth

Main lobe

26,600km

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42,164 km

Figure 5-14 GPS signal coverage of GEO.

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Because the GPS main beam and its sidelobes are directed towards the earth, a satellite in GEO will receive the signals only if it finds itself on the opposite side of the earth, i.e. at a distance of approximately 20,200 + 6371 + 42,164 ≈ 69,000 km. At that distance, the GPS signal received by the GEO satellite receiver is very weak and therefore unreliable. Moreover, the GEO satellite will only receive the GPS signal while travelling though the spill-over from the GPS main antenna lobe and its sidelobes. The net result is that the GPS coverage of GEO is patchy, such that a GPS receiver onboard the GEO satellite repeatedly loses GPS signal and must re-acquire it. For these reasons, the deployment of GPS receivers onboard satellites in GEO and higher is still in its infancy. Many position measurement systems employ a two-way signal path, whereby the user terminal emits a radio signal and listens for the echo (cf. ranging and radar). GPS differs from these systems by requiring only a one-way signal path. This simplification is achieved by having central ground facilities insert intelligence in the signal, such as time markers and orbital ephemerides. Stated differently, the GPS downlink signal includes accurate information about that satellite’s orbital position at any given moment, thereby allowing the GPS receiver to directly calculate its own position. A GPS receiver is therefore a relatively simple, reliable and inexpensive piece of radio equipment. It lends itself to integration onboard LEO satellites and permits autonomous orbit determination. To understand how GPS performs position measurement, refer to Figure 5-15. The configuration is analogous to that shown in Figure 5-3, except that the known reference points are GPS satellites instead of ground stations. The user’s satellite must receive at least three GPS signals simultaneously to achieve a position fix (cf. trilateration in Section 5.2.2). As in the case of trilateration, Pythagoras’ theorem in three dimensions gives us:

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(xs − x1 )2 + ( ys − y1 )2 + (zs − z1 )2 = d12 (xs − x2 )2 + ( ys − y2 )2 + (zs − z2 )2 = d 2 2 (xs − x3 )2 + ( ys − y3 )2 + (zs − z3 )2 = d32 GPS1

z d1 d2 d3

olo gy .

(5.15)

GPS2 GPS3

hn

r

y

ec

x

fT

Figure 5-15 Orbit determination of LEO satellites using GPS.

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The instantaneous position (x1, y1, z1) of the first GPS satellite is accurately determined by the GPS control centre and is relayed via the GPS satellite to the GPS receiver onboard the user satellite. The same is true for the second (x2, y2, z2) and third (x3, y3, z3) GPS satellites. Similarly, the distance d1 is determined by the GPS receiver by measuring the difference ∆t between the time t1 when the clock signal is transmitted by the GPS satellite, and the time t2 when it is received (t1 is also embedded in the GPS signal). The distance d1 = c/∆t = c/(t2 – t1), where c is the speed of light. The same procedure is followed to calculate d2 and d3. This leaves us with three unknowns, namely the three components of the satellite position r = (xs, ys, zs), as well as with three equations; hence it is possible to calculate the satellite position.

While the GPS satellites are equipped with extremely accurate onboard atomic clocks, the GPS receiver’s clock is far less sophisticated, the receiver being a commercial off-theshelf item. Consequently a timing error is introduced in ∆t, which translates to a distance error ∆d in d. Because ∆t is common to all three measurements, the error is the same for all three distances at any one moment in time. With ∆d being a fourth unknown, we now need simultaneous access to four GPS satellites. The complete set of simultaneous equations is then:

(xs − x1 )2 + ( ys − y1 )2 + (z s − z1 )2 = (d1 + ∆d )2 (xs − x2 )2 + ( ys − y2 )2 + (z s − z2 )2 = (d 2 + ∆d )2 (xs − x3 )2 + ( ys − y3 )2 + (z s − z3 )2 = (d 3 + ∆d )2 (xs − x4 )2 + ( ys − y4 )2 + (z s − z4 )2 = (d 4 + ∆d )2

(5.16)

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Unfortunately, solving these equations for (xs, ys, zs) is not trivial. Step-by-step methods can be found on the Internet by searching for GPS.

5.3.1. State Vectors

Orbit Determination

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5.3.

ec

hn

olo gy .

At this juncture, it is tempting to return to Section 2.5.4 to assess whether the GPS satellite constellation is correctly configured for global coverage. Assume that an elevation δ = 10 deg above the local horizon is required for adequate contact between the satellite and the GPS receiver on the ground. With h = 20,200 km, Eq 2.57 gives us α’ = 66.34 deg = 1.158 rad. Eq 2.54 yields P = 1.81 ≈ 2 orbital planes, while Eq 2.55 suggests N = 3.13 ≈ 3 satellites per plane. The total number of satellites needed for continuous contact with at least one GPS satellite is given by Eq 2.56 as S = 2 x 3 = 6. If we now require simultaneous coverage of 4 satellites, we need a total of 4 S = 24 satellites in total, as is indeed the case. Calculating the optimum P and N for the 24-satellite constellation is a little more complicated and must account for the inclination being 55 deg instead of 90 deg.

Co p by yri Lu ght leå © Un 201 ive 4, rsi ty o

The input for orbit determination is tracking data received from one or more satellite ground stations. The output is a state vector. A state vector is a set of data describing exactly where the spacecraft is in its orbit at a given point in time. State vectors can come in many forms: Cartesian, Spherical Polar, Classic Orbital Elements, and Two-Line Elements, to name a few. Regardless of how they are formatted, all state vectors have one parameter in common, namely time. It is necessary to relate all the parameters of the state vector to a particular epoch (i.e. a point in time). Without this, planning future mission events is impossible. Most of the formats use seven parameters (e.g. the six COE plus time) to define the orbit. Over the centuries, the position in the sky of the vernal equinox vector (ϒ) slowly changes. Since some of the orbital elements are measured with respect to this vector, it is important to refer them to the year for which they were derived. Table 5-1 contains a sample of a two-line element state vector printout produced by the North American Aerospace Defense Command (NORAD) which tracks all objects in earth orbit larger than a baseball (http://celestrak.com/NORAD/elements/). The printout refers to two Iridium mobile communications satellites.

184

5 / Orbit Management IRIDIUM 8 [+] 1 24792U 97020A

.00000090

00000-0

5858

86.3961 144.0586 0002080 105.7097 254.4328 14.34217417866198

IRIDIUM 7 [+] 1 24793U 97020B 2 24793

24944-4 0

13324.49480675

.00000556

00000-0

olo gy .

2 24792

13324.49555837

19161-3 0

5789

86.3966 144.2700 0002033 105.7407 283.1264 14.34216937866191

hn

Table 5-1 Example of two-line elements.

Co p by yri Lu ght leå © Un 201 ive 4, rsi ty o

fT

ec

The following legend interprets this rather telegraphic NORAD message.

Name of Satellite (NOAA 6) This is simply the name associated with the satellite. In the sample NORAD printout (Table 5-1) we read IRIDIUM 8, etc. International Designator (84 123A) The 84 indicates launch year was in 1984, while the 123 tallies the 124th launch of the year, and "A" shows it was the first object resulting from this launch. In the Iridium case, the letters A, B indicate that the two satellites were launched on a single rocket. Epoch Date and Day of Year Fraction The day of year fraction is the number of days (in GMT) passed in the particular year. For example, the date above shows "86" as the epoch year (1986) and 50.28438588 as the number of days after January 1, 1986. The resulting time of the vector would be 1986/050:06:49:30.94, computed as follows: Days: Hours:

Minutes:

50.28438588 days = 50 days 50.28438588 – 50 = 0.28438588 hours; 0.28438588 hours x 24 hours/day = 6.8253 = 6 hours UTC 6.8253 - 6 = 0.8253 hours; 185

5 / Orbit Management

Seconds:

0.8253 hours x 60 minutes/hour = 49.5157 = 49 minutes 49.5157 - 49 = 0.5157 minutes; 0.5157 minutes x 60 seconds/minute = 30.94 seconds

olo gy .

First Derivative of Mean Motion, or Ballistic Coefficient (0.00000140) The daily rate of change in the number of revolutions the object completes each day, divided by 2. Units are revs/day. This is a "catch all" drag term used in some orbital decay prediction models.

ec

hn

Second Derivative of Mean Motion (00000-0 = 0.00000) The second derivative of mean motion is a second order drag term used to model final orbit decay. It measures the second time derivative in daily mean motion, divided by 6. Units are revs/day3. A leading decimal must be applied to this value. The last two characters define an applicable power of 10. (12345-5 = 0.0000012345).

fT

Drag Term (67960-4 = 0.000067960) Also called the radiation pressure coefficient, the parameter is another drag term used in decay predictors. Units are earth radii-1. The last two characters define an applicable power of 10.

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Element Set Number and Check Sum (5293) The element set number is a running count of all two-line element sets generated for this object (in this example, 529). Since multiple agencies perform this function, numbers are skipped on occasion to avoid ambiguities. The counter should always increase with time until it exceeds 999, when it reverts to 1. The last number of the line is the check sum of line 1. Satellite Number (11416) This is the catalogue number for this object. The letter "U" indicates an unclassified object. Inclination (degrees) The angle i between the equator and the orbit plane.

Right Ascension of the Ascending Node (degrees) The angle Ω between vernal equinox and the point where the orbit crosses the equatorial plane (going north). Eccentricity (0012788) The constant e defining the shape of the orbit. A leading decimal must be applied to this value. Argument of Perigee (degrees) The angle ω between the ascending node and the orbit's point of closest approach to the earth (perigee). Mean Anomaly (degrees) The angle M, measured from perigee, of the satellite location in the orbit referenced to a circular orbit with radius equal to the semi-major axis.

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5 / Orbit Management

olo gy .

Mean Motion (14.24899292) The value N is the mean number of orbits per day the object completes. (The more common definition of mean motion is n = 2π/τ, which comes to the same thing as N after some unit conversions). Revolution Number and Check Sum (346978) The orbit number at Epoch Time. This time is chosen very near the time of true ascending node passage as a matter of routine. The last digit is the check sum for line 2. 5.3.2. Computing COE from Topocentric Data

fT

ec

hn

In the above two-line element message, NORAD has already preprocessed the raw ground station measurement data to provide us with four of the six classic orbital elements (COE = a, e, i, Ω, ω and ν), namely e, i, Ω, ω. The missing elements are a and ν. The semimajor axis a may be calculated from the mean motion N, since N = 86164/τ and τ = 2πa(a/µ)½ according to Eq 2.12 (the constant 86164 is the number of seconds in a sidereal day). The true anomaly ν is found from the mean anomaly M using Eq 2.11.*

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As explained in Section 5.2, the raw data from the ground station usually arrives in the form of a slant range distance d to the satellite, along with two antenna pointing angles: azimuth α and elevation δ. If we are lucky, the station might also provide us with rangerate data in the form of time derivatives of d, α and δ.

On the basis of the raw data, we are able to compute a state vector combining a geocentric range vector r and the satellite’s tangential velocity vector V. These vectors are usually expressed in a topocentric, Cartesian uvw coordinate system (see Section 2.4.1), and it is left to the satellite operator to convert these into COE. Mathematically, it stands to reason that the six components ru, rv, rw, Vu, Vv, Vw can be transformed to the six COE, and we shall now proceed to prove that this is so. We will be adding and multiplying vectors defined in different coordinate systems, so we must first agree on a common target frame. Since i, Ω, ω and ν are all defined in the geocentric-equatorial inertial system (Figure 5-16), let us choose this as our reference. The x-axis coincides with the intersection between the equator and the ecliptic, the z-axis with the earth’s spin axis pointing north, and the y-axis completes the right-handed orthogonal system.

 86164 µ   In the Iridium case, N = 14.34, as shown. Therefore, a =   2πN    h = a – R = 775 km. *

2/3

=7146 km;

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5 / Orbit Management

olo gy .

z= N

ν ω Ω

i

y

hn

x=ϒ

ec

Figure 5-16 Inertial geocentric-equatorial coordinate system.

Figure 5-16 recapitulates the definitions of i, Ω, ω and ν.

fT

Before proceeding, it is useful to calculate the angular momentum of the orbit. h=r ×V

Co p by yri Lu ght leå © Un 201 ive 4, rsi ty o

(5.17)

Next we define the eccentricity vector e:

e = (1/µ)[(V2 – µ/r)r – (r⋅⋅V)V]

(5.18)

This vector points from the earth’s centre to the perigee, regardless of r and V. The magnitude of e equals the eccentricity e. (For a derivation of the eccentricity vector, refer to Sellers [9]) We also require a vector n which points from the earth’s centre to the ascending node, and which we obtain from (Figure 5-17): n=z×h

z

h

V

r

n

Figure 5-17 Definition of vectors h and n.

188

(5.19)

5 / Orbit Management Lastly, Eq 2.16 provides the specific energy of the orbit: ε = V2/2 – µ/r where ε = E/m is the total energy divided by the satellite mass.

olo gy .

(5.20)

Notice that the parameters h, n and ε are derived solely from the two measured vectors r and V. We are now ready to proceed with the calculation of the six COE. µ 2ε

From Eq 2.17 and 5.20:

a=−

From Eq 5.18:

e = ex + e y + ez

From Eq 5.17:

h  h  i = cos −1  z  = cos −1  z  h  h 2

hn

2

2

h = hx + h y + hz

ec

(5.22)

n  n  Ω = cos −1  x  = cos −1  x  n  n

From Eq 5.19:

with

From Eq 5.18 and 5.19:

2

2

n = nx + n y + nz

with

2

2

r = rx + ry + rz

(5.24)

2

 n⋅e  ω = cos −1    n⋅e 

 r ⋅e  ν = cos −1    r ⋅e

From Eq 5.18:

(5.23)

2

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with

2

fT

2

(5.21)

(5.25)

(5.26)

2

It remains to determine which is the correct quadrant for the four angles i, Ω, ω, ν, given that the cos-1 function is ambiguous. These are the rules: Eq 5.23:

i must always be < 180°

Eq 5.24:

Use 360° - Ω if ny < 0

Eq 5.25:

Use 360° - ω if ez < 0

Eq 5.26:

Use 360° - ν if r⋅⋅V < 0

So far so good. Unfortunately, ground stations do not measure r and V directly, but rather the slant range d from the station to the satellite, as well as the antenna pointing angles in

189

5 / Orbit Management terms of azimuth α and elevation δ. Section 5.2.4 described how to convert these measurements to r and V.

Co p by yri Lu ght leå © Un 201 ive 4, rsi ty o

fT

ec

hn

olo gy .

Let us proceed with a numerical example. In the example, a value for r has been chosen which is the average between r2 and r1, since it offers the best match for V.

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5 / Orbit Management

d (km) α (deg) δ (deg)

Meas. 1 1000 0 90

Ground station coordinates Meas. 2 2100 5 17

λ (deg) Λ+φ (deg) R (km)

Magn.

Meas. 2 50 1.17 6371

s2 (km)

d1 (km)

d2 (km)

r1 (km)

r2 (km)

V (km/s)

4095.2 0 4880.5

4094.4 83.4 4880.5

642.8 0 766

-1141.2 151.8 1756.3

4738 0 5646.5

2953.1 235.2 6636.8

-6.37 0.84 3.54

7371

7267.9

7.34

fT

s1 (km)

Co p by yri Lu ght leå © Un 201 ive 4, rsi ty o

x y z

ec

Calculation of s, d and r

Meas. 1 50 0 6371

hn

Ground station measurements

olo gy .

Numerical example: A satellite ground station in Europe (Λ = 0°, λ = 50°N) delivers the following tracking measurements at midnight sidereal time: slant range d = 1000 km, azimuth α = 0°, elevation δ = 90°. 280 seconds later the station produces a new set of measurements: d = 2100 km, α = 5°, δ = 17°. Calculate the classical orbital elements, as well as the apogee, the perigee and the orbital period.

6371

6371

1000

2100

ε=

-28.08

km2/s2

r (km)

V (km/s)

h (km2/s)

n (km2/s)

e

3846 118 6142

-6.37 0.84 3.54

-4743 -52750 3980

52750 -4743 0

-0.054 0.005 0.006

Orbital energy:

Calculation of r, V, h, n, e

x y z

Magn.

7247

7.34

53112

52963

0.055

e 0.055

i (deg) 85.7

Ω (deg) 5.1

ω (deg) 173.6

ν (deg) 244.6

V◦r = n◦e =

-2694 -2885

Calculation of COE a (km) 7098

Calculation of Apogee, Perigee and Orbital Period Ap (km) 1116

Pe (km) 338

τ (min) 99

Let us analyse the above numerical example. In the first set of measurements, the antenna is pointing to zenith (δ = 90°), so the satellite is passing straight overhead at a height of d 191

5 / Orbit Management = 1000 km. Since the measurement was taken at 0 hours sidereal time, the Greenwich meridian must cross the ϒ-vector; hence Ω = 0°.

hn

olo gy .

By the time of the second measurement 280 seconds later, the elevation has dropped significantly, and a small azimuth angle is provided. This tells us that the satellite is approaching the northern horizon, so the orbit must be near polar. If i = 90°, the azimuth would have adopted a small negative value, since the station’s meridian has moved 1.17° eastwards from the ϒ-vector due to the earth’s rotation, such that the orbit would be passing to the west of the station. But the azimuth given is in fact positive, so i must be < 90° (namely 85.7° according to the table). Consequently the range vector r is for the moment in the first quadrant of the inertial xyz-system, as corroborated by the fact that rx, ry and rz are all ≥ 0.

ec

With r2 < r1 the orbit must be elliptic (e ≠ 0), and the satellite is on its way from the apogee to the perigee. We would therefore expect ω to be < 180° + λ, i.e. ω < 230°, which is confirmed by the calculations (ω = 173.6°).

z

Co p by yri Lu ght leå © Un 201 ive 4, rsi ty o

2

fT

The above kind of reasoning, as well as the sketch below, is useful for doing a sanity check on the results.

1

y

x=ϒ

Figure 5-18 Illustration of numerical example.

5.3.3. Refining the Orbit Determination Accuracy

Having determined the COE from tracking measurements at a particular ground station, it is possible to predict the timing, slant range, azimuth and elevation of the next station pass. Any errors in the initial determination will carry over to the next pass, where new measurements are taken. The difference between predicted and actual pass parameters constitutes a residual, which can be used to improve the predictions for subsequent passes. The magnitude of these residuals shrinks as new measurements are made. Various mathematical filtering techniques (r.m.s., Kalman filtering, etc.) are put to good use to keep the orbit determination errors to a minimum.

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5 / Orbit Management

5.4.

Orbit Control

olo gy .

5.4.1. Control Principles Conceptually, orbital manoeuvring is simple: Give the satellite an impulse somewhere along the orbit, and a new orbit will form. Depending on the location, magnitude and direction of the impulse, it is possible to tailor the new orbit to new requirements. The illustration sequence below shows how each of the six COE can be changed separately or in combination. The dotted line represents the original orbit, while the solid contour is the new orbit resulting from the impulse.

Hohmann transfer. The dash-dot line shows an intermediate orbit.

V2

∆ V21

V1

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∆ V21 = V2 – V1 ∆ V43 = V4 – V3

ec

Changing a, while keeping e = 0

fT

5.4.1.1.

hn

The black dot indicates where the impulse should occur. Note that the impulse location is always at a point where the old and new orbits intersect.

V3 ∆ V43 V4

∆V21 = V2 – V1 ∆V43 = V4 – V3

5.4.1.2.

Changing e, but not a

∆ V = V 2 – V1

V2

∆V = (V22 + V12 – 2V1V2 cos θ)½

∆V

The impulse may be entered at either one of the two orbital intersections.

θ

V1

193

5 / Orbit Management 5.4.1.3.

Changing a and e V2 ∆ V = V2 – V 1

olo gy .

∆V V1

∆V = V2 – V1

5.4.1.4.

hn

Usually it is possible to change more than one COE at a time.

Changing i, but not Ω

ec

N

To accomplish a pure inclination change, the impulse must be given at the ascending or descending node.

V2

fT

∆V

V1

∆i

i2

i1

Co p by yri Lu ght leå © Un 201 ive 4, rsi ty o

∆ V = V2 – V1

∆V = (V22 + V12 – 2V1V2 cos ∆i)½ with ∆i = i2 – i1

5.4.1.5.

Changing Ω, but not i

The manoeuvre must occur at the intersection closest to the poles.

N

V1

∆ V = V2 – V1

∆V = (V22 + V12 – 2V1V2 cos θ)½

θ

θ = cos-1(cos2i + sin2i cos ∆Ω)

θ

∆Ω

∆Ω

194

∆V θ V2

5 / Orbit Management 5.4.1.6.

Changing ω

∆ V = V2 – V1

olo gy .

θ

∆ω

V1

V2

∆V = (V22 + V12 – 2V1V2 cos θ)½

5.4.1.7.

Changing ν

ec fT

Alternatively, the impulse may be given at the intersection between the two perigees.

hn

θ = (ν1 + η1) – (ν2 + η2)

∆V

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V2

∆ V21 = V2 – V1 ∆ V43 = V4 – V3

∆V21 = V2 – V1 ∆V43 = |V4 – V3|

}

∆ V21

}

∆ V43

V1

V3

V4

The last example deserves elaboration. Assume that the satellite is to be re-phased in the circular orbit through a change in ν(t). The solution is to temporarily extend or shorten the orbital period τ by altering a, according to Eq 2.12: τ = 2πa(a/µ)½. In the illustration, a is extended in the first step, leading to a longer τ. Therefore, after the second step, the satellite will be trailing behind its original ν(t) position.

If instead the objective is to accelerate the satellite, a must be shortened for the intermediate orbit (i.e. the ellipse will lie inside the circle). From the above examples one may draw several conclusions: a) The old and the new orbits always have one or two points of intersection. This is the point where the manoeuvre will take place. 195

5 / Orbit Management

b) ∆V is the vector difference between V2 and V1 which represent the tangential velocities of the initial and final orbits at the chosen point of manoeuvre.

olo gy .

c) If one or both orbits are elliptic, the magnitude of the two ∆V vectors is likely to differ; the intersection with the smallest ∆V gives the lowest propellant consumption.

ec

hn

For the statement in (b) to be strictly true, the impulse must be instantaneous, i.e. of zero duration. In practice, the rocket motor responsible for the impulse will inevitably burn during a finite length of time. As a consequence, the magnitude and/or direction of V2 and V1 will change in the course of the manoeuvre, whereas ∆V will not (assuming the satellite’s attitude remains constant). To avoid missing the target, the burn should be kept very short (a couple of minutes at most), or else a modified ∆V must be calculated using numerical integration methods. Often the manoeuvre is completed through a sequence of brief motor burns at each passage through the chosen orbital crossover point.

2

2

fT

The magnitude of ∆V is found from simple vector algebra∗ to be: ∆V = V2 − V1 = V1 + V2 − 2V1V2 cos θ

(5.27)

Co p by yri Lu ght leå © Un 201 ive 4, rsi ty o

where θ is the angle between V2 and V1. In the case of a pure inclination change, we find that θ = ∆i. In all other cases, θ is obtained from spherical triangles involving i, ω and ν. When V2 and V1 are parallel (i.e. θ = 0°), we have a so-called Hohmann transfer with ∆V = V2 − V1 . The manoeuvres illustrated in Sections 5.4.1.1, 5.4.1.3 and 5.4.1.7 are Hohmann transfers, while the others are not. 5.4.2. Choice of Intersecting Points

The purpose of an orbital manoeuvre is to generate a new set of a, e, i, Ω and ω. Knowing a1, e1, i1, Ω1 and ω1 of the old orbit, as well as a2, e2, i2, Ω2 and ω2 of the new orbit, the mission planner must calculate V1 and V2 at the chosen point of intersection, in order to find ∆V. The required quantities are the true anomalies ν1 and ν2, which are readily obtained from spherical trigonometry (Figure 5-19).

∗

∆V2 = ∆V2 = (V2 – V1)2 = V22 + V12 – 2V1V2 = V22 + V12 – 2V1V2 cosθ

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5 / Orbit Management

θ i1

i2

ω1+ν1

ω2+ν2

i1

ec

∆Ω

hn

180°-i2

∆Ω

olo gy .

N

fT

Figure 5-19 Calculating ν1 and ν2 at the intersection.

To begin with, we determine the angle θ between the two velocity vectors using the Law of Cosines (see Appendix A):

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θ = cos-1[-cos i1 cos(180°-i2) + sin i1 sin(180°-i2) cos ∆Ω]

= cos-1[cos i1 cos i2 + sin i1 sin i2 cos ∆Ω]

(5.28)

The Law of Sines gives us:

sin(ω1 + ν1 ) sin ∆Ω = sin(180° − i2 ) sin θ

sin ∆Ω   ν1 = sin −1  sin i2  − ω1 sin θ  

(5.29)

sin ∆Ω   and therefore: ν 2 = sin −1  sin i1  − ω2 sin θ  

(5.30)

or:

We are now in a position to calculate the radius r (= r1 = r2), the velocities V1 and V2, the velocity increment ∆V and the flight path angles η1 and η2 at the chosen point of orbital intersection. 2

From Eq 2.3:

a (1 − e1 ) r= 1 1 + e1 cos ν1

(5.31)

From Eq 2.13:

V1 = µ(2 / r − 1 / a1 )

(5.32)

V2 = µ(2 / r − 1 / a2 )

(5.33)

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5 / Orbit Management

From Eq 2.15:

2

2

∆V = V1 + V2 − 2V1V2 cos θ 1 + e1 cos ν1 tan η1 = e1 sin ν1 1 + e2 cos ν 2 tan η2 = e2 sin ν 2

(5.34) (5.35)

olo gy .

From Eq 5.30:

ec

hn

Numerical example: A scientific satellite travelling in a polar, circular orbit (i1 = 90°) at a height of h = 600 km has completed its original mission. Because the satellite is expected to live longer, it has been decided to undertake a supplementary mission in a sun-synchronous orbit. The new orbit is also circular at the same altitude but, according to Eq 2.18, an inclination i2 = 97.8° is needed for sun-synchronism. A node rotation of ∆Ω = 10° is also desirable. Calculate the true anomalies ν1 and ν2 of the orbital intersections, as well as the ∆V required to perform the manoeuvre.

V1 = V2 = 7.56 km/s ∆V = 1.67 km/s ν2 = 52.6°

Co p by yri Lu ght leå © Un 201 ive 4, rsi ty o

r = a = h + R = 6971 km; θ = 12.68°; ν1 = 51.2°;

fT

Solution: With the orbits being circular, e = 0, a = r, and ω has no meaning since there is no perigee. By setting ω = 0 in Eq 5.29 and 5.30, ν becomes the angular distance from the ascending node to the intersection points. We have:

This makes sense if we examine the corresponding subsatellite traces: N

51.2°

52.6°

l

The arcsinus functions in Eq 5.29 and 5.30 also have additional solutions 180° apart, so we would expect another point of intersection at ν1 = 51.2° + 180° = 231.2°;

ν2 = 52.6° + 180° = 232.6°

This we know intuitively to be true. Since we are dealing with two circular orbits, the values of V1, V2, θ and ∆V remain the same for the second point, so both points are equally valid as concerns the propellant consumption.

5.4.3. Determination of Velocity Vectors

Having determined the magnitude of ∆V, we are now in a position to estimate the quantity of propellant needed to carry out the orbital manoeuvre, but we also need to find the

198

5 / Orbit Management direction of ∆V, i.e. the thrust vector ∆V. Knowing the direction is a prerequisite for orientating the rocket motor, and hence the spacecraft, in preparation for the manoeuvre.

olo gy .

As we have seen, ∆V = V2 – V1, where V1 and V2 are the velocity vectors for the old and the new orbits at the chosen point of intersection. The challenge is now to express the components of ∆V, V1 and V2 in a shared coordinate system. Different operators have different preferences, so on this occasion we will choose the same geocentric-equatorial inertial xyz-system that we used in Section 5.3.2 above.

hn

To facilitate the process, we will temporarily use the satellite and orbit-fixed coordinate systems lmn and uvw shown in Figure 5-20, and then rotate these systems to fit with xyz. l

v

u

Co p by yri Lu ght leå © Un 201 ive 4, rsi ty o

n and w point towards the reader.

ν

fT

r m

ec

η V

Figure 5-20 Temporary satellite and orbit-fixed coordinate systems.

We see immediately that

ruvw = r (cos ν, sin ν, 0)T

and

(5.36)

Vlmn = V (cos η, sin η, 0)T

where r and V are available from Eq 2.3 and 2.13 as: r=

a (1 − e 2 ) 1 + e cos ν

and

2 1 V = µ −  r a

and where ν and η have been calculated in Eq 5.29 and 5.34 above. The V-vector is brought from the lmn-system to the uvw-system through the following rotation:

Vuvw

 cos ν − sin ν 0   cos η   cos ν cos η − sin ν sin η       = V  sin ν cos ν 0   sin η  = V  sinν cos η + cos ν sin η   0   0 1   0  0   

(5.37)

199

5 / Orbit Management We have now defined both r and V in the uvw-system of the old orbit. Since the new orbit has a different uvw-system, it remains to convert each to the shared xyz-system. Recalling the matrix manipulation method in Chapter 2, we find that

olo gy .

Vxyz

(5.38)

0 0  cos ω − sin ω 0 cos Ω − sin Ω 0 1    =  sin Ω cos Ω 0  0 cos i − sin i    sin ω cos ω 0  Vuvw  0 0 1 0 sin i cos i   0 0 1 

(5.39)

hn

rxyz

0 0  cos ω − sin ω 0 cos Ω − sin Ω 0 1    =  sin Ω cos Ω 0  0 cos i − sin i    sin ω cos ω 0  ruvw  0 0 1  0 sin i cos i   0 0 1

ec

Vectors ruvw and Vuvw are available from Eq 5.36 and 5.37.

fT

Inserting the COE, along with η, for the old and the new orbits, we arrive at V1, V2 and ∆V.

Co p by yri Lu ght leå © Un 201 ive 4, rsi ty o

Numerical example: Let us continue with the earlier example, in which a polar circular orbit is manoeuvred to become sun-synchronous. Since both orbits are circular, it follows that e1 = e2 = 0, and η1 = η2 = 90°. Recall that ω1 = ω2 = 0°, that i1 = 90°, i2 = 97.8°, and that ν1 = 51.2°, ν2 = 52.6°. Let us assume that Ω1 = 0°, Ω2 = 10°, which gives us our ∆Ω = 10°. According to Eq 5.37:

(Vuvw)1 = V1 (-sin ν1, cos ν1, 0)

T

and, according to Eq 5.39:

 − sin ν1  1 0 0 1 0 0  1 0 0  − sin ν1            (Vxyz)1 = V1 0 1 0  0 0 − 1  0 1 0  cos ν1 = V1  0     0 0 1  0 1 0  0 0 1     cos ν         0   1 

Let us test the validity of this equation by setting ν1 to 0° and to 90°, as in the figures below.

z

z

V

V

r

r

x

T

x

The left figure shows ν1 = 0°, and therefore (Vxyx)1 = V1 (0, 0, 1) , as expected. In the rightT hand figure we have ν1 = 90°, which gives us (Vxyx)1 = V1 (-1, 0, 0) , again as expected. Inserting the true value ν1 = 51.2° yields:

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5 / Orbit Management

 − 0.78   − 5.89      Vxyz = V1  0  =  0  , with V1 = V2 = 7.56 km/s. 1  0.63   4.74 

)

olo gy .

(

We may now proceed with calculating (Vxyx)2.

(Vxyz )

2

0 0 cos 10° − sin 10° 0 1  1 0 0  − sin 52.6°       = V2 sin 10° cos 10° 0  0 cos 97.8° − sin 97.8°  0 1 0  cos 52.6°    0      0 0 1  0 sin 97.8° cos 97.8°  0 0 1     z

V1 V2 y

x

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 − 0.77   − 5.81    

ec

52.6°

fT

51.2°

(Vxyz )

2

hn

∆V

= V2 − 0.22 = − 1.66      0.60   4.55 

Lastly:

 0.08    ∆V = V2 − V1 = − 1.66    − 0.19 

Check:

∆V =

Check:

θ = cos (V1•V2) = 12.68° (same as in the earlier example).

∆V x

2

+ ∆V y

2

+ ∆V z

2

= 1.67 km/s (same as in the earlier example).

-1

5.4.4. Propellant Consumption

Newton’s Second Law states that the force acting on a moving body equals the time derivative of its linear momentum, i.e. F=

d (mV ) dV dm =m +V dt dt dt

(5.40)

If we apply this equation to a burning rocket and imagine ourselves sitting astride it, we observe two things: the expulsion of propellant mass through the nozzle at a constant velocity Ve (“action”), and an acceleration of the rocket a = m dV/dt (“reaction”). Seen from where we are sitting, the action and reaction forces are in balance. Returning to Eq 5.40:

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F =m

dV dm + Ve = 0; dt dt

m

dV dm = −Ve ; dt dt

dV = −Ve

dm m

m 0 − ∆m

∆V = −Ve

∫

m0

 m0  dm m 0 − ∆m  = −Ve [ln m]m 0 = Ve ln m  m0 − ∆m 

olo gy .

After integration: (5.41)

hn

This is known as the Tsiolkovsky rocket equation. Here m0 is the total rocket mass before the burn, and ∆m is the amount of propellant consumed during the burn to achieve the velocity increment ∆V. Note that we have assumed the velocity Ve of the expelled propellant mass to be constant, which is approximately true in real life.

∆V −    g Isp    = m0 1 − e      

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fT

∆V −  ∆m = m0 1 − e Ve  ∆V −   ∆m = m0 1 − e g Isp      ∆V

− m0 − ∆m = e Ve ; m0

ec

Moving along, Eq 5.41 can be written as follows:

(5.42)

Here we have introduced Isp which is known as the specific impulse of the rocket motor. The specific impulse is defined as:

I sp =

F dt Ve F = = g dm / dt g dm g

(5.43)

where F is the thrust, dm/dt is the (constant) propellant consumption, and g is the mean acceleration of gravity (= 9.81 m/s2). Isp is measured in seconds and is an indication of the motor’s efficiency, i.e. how much thrust the motor delivers in relation to the propellant consumption. It is basically dependent on the choice of propellant and of the altitude (Chapter 6). Note that, in some literature, the “g” is omitted in the definition of Isp, with the result that the values of Isp are almost 10 times greater, and the unit becomes Ns/kg. Given the rocket motor’s Isp, Eq 5.42 yields the propellant mass ∆m in kg needed to achieve a certain ∆V. Note that ∆V must be expressed in metres per second (rather than km/s), so as to be compatible with the unit of g. Figure 5-21 shows the propellant consumption of a satellite with m0 = 1000 kg for various values of ∆V and Isp. Since ∆m is directly proportional to m0, it is easy to find the propellant consumption for other initial satellite mass figures.

202

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olo gy .

5 / Orbit Management

ec

Figure 5-21 Propellant consumption as a function of ∆V and Isp for a satellite mass m0 = 1000 kg.

Co p by yri Lu ght leå © Un 201 ive 4, rsi ty o

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For example, if we were to perform the manoeuvre in the above example using a conventional hydrazine thruster (Isp ~300 s), we would require ∆m = 433 kg to impart a ∆V of 1.67 km/s on a satellite weighing m0 = 1000 kg before the burn. This illustrates how propellant-guzzling orbital manoeuvres can be, even when their magnitudes are modest.

Q: Since the acceleration of gravity g decreases with the square of the distance from the earth, must I take that into account when using Eq 5.42?

A: No, because here g is purely a form factor to convert the thrust force F from Newton to kilogrammes. Note that Isp has the unit “seconds,” which may seem curious for a measure of efficiency. This is an artefact of the definition of Isp and bears no intuitive relationship to time.

5.4.5. Orbit Control Strategies

Because orbit control manoeuvres consume a great deal of onboard propellant, many amateur satellite missions (e.g. CubeSats) avoid them altogether. Carrying propellant not only increases the complexity and cost of the satellite itself, but the extra propellant mass also influences the launch cost. There are, however, many scenarios where active orbit control is unavoidable, e.g.:

•

when the launch vehicle is unable to deliver the satellite to its operational orbit;

•

to maintain a GEO satellite at its designated orbital location;

•

to move a GEO satellite to a different orbital location;

•

to maintain the relative phasing between LEO satellites in a constellation;

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5 / Orbit Management •

to maintain correct phasing of sun-synchronous orbits; and

•

to remove a satellite from orbit before the end-of-life.

5.4.5.1.

olo gy .

Each of these scenarios will be analysed in the following.

Transfer from GTO to GEO

GEO

ec

∆V

hn

Many of the current commercial launch vehicles leave their GEO payloads stranded in a highly elliptic geostationary transfer orbit (GTO). The apogee height of the GTO is usually at GEO altitude (35,786 km), while the perigee stays in the region of 250 – 500 km. It is therefore up to the satellite to carry its own apogee motor to circularize the GTO at apogee to form a GEO – see Figure 5-22.

Co p by yri Lu ght leå © Un 201 ive 4, rsi ty o

fT

GTO

Figure 5-22 Simple transfer from GTO to GEO.

If the GTO and the GEO are in the same plane, the magnitude of ∆V is simply the difference between the velocity at GEO less the velocity at GTO apogee, i.e.

∆V = VGEO – VGTO = 3.07 – 1.61 = 1.46 km/s

assuming a GTO perigee height of 300 km. Note, however, that the GTO inclination is at least as high as the latitude of the launch site (Section 11.4), whereas the required GEO inclination is near zero. It is therefore necessary to have the apogee motor perform a combined circularizing and inclination-reducing manoeuvre. The latter component is known as a dog-leg manoeuvre because of the way the orbit is kinked like the hind leg of a dog.

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5 / Orbit Management

Figure 5-23 Dog-leg manoeuvre.

hn

olo gy .

Dog-leg manoeuvre

fT

ec

The manoeuvre shown in Figure 5-22 is typical for a solid-propellant apogee kick motor (AKM) with its short, powerful thrust. The difficulty with this type of motor is that the slightest error in attitude, thrust level or thrust duration translates into GEO imperfections which will subsequently have to be corrected using the satellite’s micropropulsion system (i.e. small thrusters for attitude and orbit adjustments) – a process which is both inefficient and time-consuming.

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A better solution is to use a restartable liquid-propellant motor, also known as a liquid apogee engine (LAE). Such an engine allows the GTO to be raised to GEO in stages, so that operators have time to make careful interim assessments before commencing the next stage – Figure 5-24. GEO

∆V1,2,3

GTO3

GTO2

GTO1

Figure 5-24 Staged transfer from GTO to GEO.

5.4.5.2.

GEO Station-Keeping

Section 2.5.1.2 described the perturbations that strive to dislodge a GEO satellite from its designated orbital position. The main perturbations are of two kinds: the ones that pull the satellite in an east-west direction along the equator (E/W drift), and the ones that make

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5 / Orbit Management the inclination grow (N/S drift). Both kinds have to be controlled using the onboard propulsion system.

olo gy .

In the case of E/W drift, the radio frequency licensing rules require each GEO satellite operator to maintain the spacecraft within a longitude window of ±0.1° (or in some cases ±0.05°), so as to avoid radio interference and possibly collision with adjacent satellites.

Any N/S drift complicates the ground segment, in that TT&C stations and user terminals require steerable antennas to follow the satellite’s diurnal figure-8 movement. Most communications satellites in GEO are therefore maintained within ±0.1° on either side of the equator, which is the same as saying that the inclination i is < 0.1°.

hn

East-West Station-Keeping

ec

The rate of natural E/W drift depends on where the satellite is located, i.e. on its longitudinal distance ∆Λ from the stable and unstable points along the equator. Figure 2-78 is reproduced here for convenience.

fT

20

10

Co p by yri Lu ght leå © Un 201 ive 4, rsi ty o

E/W drift acceleration x 10 -4 (deg/day 2 )

15

5

Stable

Unstable

Stable

Unstable

0

-5

0

60

120

180

240

300

360

-10 -15 -20 -25

East longitude Λ (deg)

Figure 5-25 GEO satellite east/west acceleration.

If ∆Λ = 0°, the satellite will experience no E/W drift, and no action is required. However, in the worst case the satellite is positioned at ∆Λ ≈ 45°, with angular accelerations in the region of 15 – 20 · 10-4 degrees per day2. The most efficient remedy is to inject the satellite at, say, the -0.1° end of the E/W window and give it a nudge towards the +0.1° end, but only enough to make the satellite turn around and head back to the -0.1° end, where the process is repeated (Figure 5-26).

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5 / Orbit Management Longitude +0.1° ∆φ time

-0.1° Nudge

Nudge

Nudge

olo gy .

Nominal ∆φ

Nudge

Figure 5-26 E/W station-keeping strategy.

∆φ

Λ

(days)

(5.44)

fT

∆t = 4

ec

hn



. It can be shown Let us denote the window half-size ∆φ and the angular acceleration Λ that the time ∆t needed for the satellite to complete a round-trip within the window (width = 2 ∆φ) is:

and the corresponding velocity increment ∆V for the manoeuvre is given by:

Co p by yri Lu ght leå © Un 201 ive 4, rsi ty o



= 45.44 ∆φ (m/s) ∆V = 11.36 ∆φ ⋅ Λ ∆t

(5.45)



= 20 · 10-4 deg/day2 (satellite at 120°E), For example, with a worst case acceleration of Λ and with ∆φ = 0.1°, it will take 28 days before this particular satellite needs another nudge. The manoeuvre is performed by temporarily increasing or decreasing the semimajor axis a in order to generate a westbound or eastbound nudge, respectively. The corresponding ∆V = 0.16 m/s, and the propellant mass ∆m = 0.05 kg for a 1-ton satellite with Isp = 300 s.

Figure 5-27 shows the required ∆V per nudge as a function of the satellite’s longitudinal position. The frequency, with which the manoeuvres need to be performed to stay within the window, is shown in Figure 5-28, and the propellant consumption ∆m per year for a 1000-kg satellite is given in Figure 5-29. The latter quantity is calculated by multiplying the ∆m per manoeuvre by 365.25 days and dividing by ∆t.

207

180 150

olo gy .

120 90 60 30 0 0

60

120

180

240

hn

Duration ∆ t between manoeuvres (days)

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300

ec

East longitude Λ (deg)

360

fT

0.18 0.16 0.14

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Velocity increment ∆ V (m/s)

Figure 5-27 Time ∆t between “nudges” to correct for E/W drift.

0.12 0.10 0.08 0.06 0.04 0.02 0.00

0

60

120

180

240

300

360

East longitude Λ (deg)

Propellant mass ∆ m per year

Figure 5-28 Delta-V required for E/W correction.

0.8 0.7 0.6 0.5

0.4

0.3 0.2 0.1 0.0

0

60

120

180

240

300

360

East longitude Λ (deg)

Figure 5-29 Propellant consumption ∆m per year for E/W corrections pertaining to a 1000-kg satellite with Isp = 300 s.

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5 / Orbit Management

North-South Station-Keeping

olo gy .

Section 2.5.1.2.2 described the drift of GEO inclination, which translates to a diurnal north-south oscillation of the satellite around the equator.

N/S station-keeping aims to maintain the GEO satellite within an inclination window. The polar diagram in Figure 5-31 shows a window of 3 degrees. As in the case of E/W stationkeeping, we can buy time by injecting the satellite at i = -3° and letting it drift to +3°.

hn

Purists frown at the mention of negative inclination, since a negative value turns the defining ascending node into a descending node. By examining Figure 5-30, we realize that a negative inclination at one node is the same as a positive inclination at the opposite node. Ascending node

fT

ec

Ascending node

+i

-i

Co p by yri Lu ght leå © Un 201 ive 4, rsi ty o

+i

Descending node

Figure 5-30 The inclination is defined as the positive angle between the orbit and the equator to the right of the ascending node.

The i-Ω relationship is illustrated in Figure 5-31 (reproduced from Figure 2-79 for convenience). Note that the (positive) inclination begins to decrease if the ascending node is located in the range 180° < Ω < 360°, and increases for all other initial values of Ω. When i is at its closest to zero, Ω abruptly swings around almost 180°. This tallies with Figure 5-30 above.

In the 3° inclination window shown in Figure 5-31, the longest drift trajectory is the one that begins at Ω = 280°, passes through i = 0°, and crosses the opposite window boundary at Ω = 80°. In other words, the frequency of N/S station-keeping manoeuvres is minimized if we launch the satellite into a GEO with Ω = 280°. This is precisely the strategy adopted by some GEO satellite operators. With an inclination window of 3° and a drift rate di/dt ≈ 0.85°/year, the satellite will remain within this particular window during approximately 7 years.

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5 / Orbit Management Ω =180° 210°

150°

120° i = 1°

2°

olo gy .

240°

3°

270°

90°

60°

330°

hn

300°

30°

ec

Ω = 0°

fT

Figure 5-31 N/S station-keeping strategy.

At the end of the 7-year period, it is time for the manoeuvre. We have three choices:

Co p by yri Lu ght leå © Un 201 ive 4, rsi ty o

a) Make a pure ∆i = 2·3° = 6° manoeuvre, in which case the initial Ωnew = Ωold + 180° = 260°, which is different from the ideal initial value of Ω = 280°. The subsequent drift trajectory will therefore not go through i = 0°, and the need for N/S manoeuvres will become more frequent.

b) Combine the ∆i = 6° with a ∆Ω = 280° - 80° = 200° such that Ωnew = 280°. c) Start out with Ω = 270°, such that Ω = 90° seven years later, thereby allowing a return to Ωnew = 270° by means of a pure ∆i manoeuvre. However, with this strategy the inclination will never reach 0°.

The three strategies are illustrated in Figure 5-32. Whichever strategy is adopted depends on factors such as propellant consumption, size of the inclination window, and expected satellite lifetime.

(a)

(b)

Figure 5-32 Three methods of performing N/S station-keeping.

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(c)

5 / Orbit Management

To compute the propellant mass consumption, we return to Eq 5.28, 2.14, 5.27 and 5.42: θ = cos-1[cos i1 cos i2 + sin i1 sin i2 cos ∆Ω]

(5.46)

olo gy .

µ

V1 = V2 =

(5.47)

a 2

2

∆V = V1 + V2 − 2V1V2 cosθ

(5.48)

   ∆m = m 0  1 − e (5.49)     In Figure 5-33 we have chosen an inclination manoeuvre with i1 = i2 ≤ 3°, ∆Ω = 180°, m0 = 1000 kg and Isp = 300 s, i.e. typical values for a small GEO mobile communications satellite using bipropellant hydrazine. Notice how costly N/S station-keeping is compared to E/W position maintenance. This is generally true for out-of-plane manoeuvres, in comparison to in-plane manoeuvres.

ec

fT

120 100 80

Co p by yri Lu ght leå © Un 201 ive 4, rsi ty o

Propellant consumption ∆ m (kg)

hn

∆V − g Isp

60 40 20

0

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Inclination half-window ∆ i (deg)

Figure 5-33 Propellant consumption ∆m per manoeuvre for N/S station-keeping pertaining to a 1000-kg GEO satellite with Isp = 300 s.

The propellant consumption also depends on the size of the inclination half-window ∆i. Taking the drift rate to be di/dt = 0.85°/year, manoeuvres have to be performed at the following time intervals: Interval =

2∆i 2 = ∆i = 2.35 ∆i (years) di 0.85 dt

(5.50)

Over a 7-year satellite lifetime it is therefore necessary to perform N manoeuvres, where: N = 7/Interval

Using the above mass prediction equations, along with the physical data for our sample GEO satellite, we find that the amount of propellant consumed for the total number of N manoeuvres is as shown in Figure 5-34.

211

109 108 107 106 105 104 103 102 0.5

1.0

1.5

2.0

Inclination half-window ∆ i (deg)

2.5

3.0

hn

0.0

olo gy .

Propellant mass consumption ∆ m (kg)

5 / Orbit Management

ec

Figure 5-34 Propellant consumption ∆m during 7 years for N/S station-keeping pertaining to a 1000-kg GEO satellite with Isp = 300 s.

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We observe that there is a modest price to be paid in terms of propellant mass for maintaining a tight inclination window – and this in addition to the operational inconvenience and reliability hazard. Even so, most GEO satellite operators prefer a tight window to avoid having steerable ground antennas, or to ensure consistent quality of earth images. 5.4.5.3.

GEO Satellite Relocation

Some years after launch, GEO satellite operators often move their spacecraft to different orbital locations. The reason may be a desire to change the service coverage area, or else to replace a defunct satellite with a new one. The manoeuvres need to be carried out quickly to minimize service interruptions. If the relocation distance is significant – say, 90° or more – an angular velocity of up to 3° per day is common. The faster the relocation speed, the higher the price in terms of propellant consumption. Let us prove this mathematically.

To achieve an eastbound drift, the angular velocity of the range vector has to be greater than that of GEO, i.e. the orbital period must be shorter. The opposite is of course true if we want to accomplish a westbound drift. Changing the period τ means altering the semimajor axis a, since τ = 2πa(a/µ)½. The strategy outlined in the Section 5.4.1.7 is preferred, as illustrated in Figure 5-35 below.

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5 / Orbit Management V2

Orbit 2

}

Orbit 1

∆ V 21

olo gy .

V1

V2

Orbit 2 Orbit 1

hn

V1

ec

}

∆ V 12

fT

Figure 5-35 Satellite westbound relocation strategy (not to scale).

Co p by yri Lu ght leå © Un 201 ive 4, rsi ty o

The relocation takes place in two stages. In stage one, a suitable set of thrusters is fired so as to achieve a slightly elliptic orbit with a different a. In stage two the process is reversed to re-establish the original GEO, but with a new true anomaly ν. The longitudinal drift rate dΛ/dt is proportional to the difference in orbital period and semimajor axis as follows:

  a 3 / 2  dΛ τ1 − τ 2 = ⋅ 360° = 1 −  2   ⋅ 360° dt τ1   a1  

deg/day

(5.51)

Recalling that an eastbound drift rate counts as positive, and choosing a westbound drift rate of -3°/day for our example, Eq 5.51 gives us a2 as:

− 3°   a 2 = a1 1 −   360° 

2/3

= 42,398 km

since a1 = 42,164 km for GEO. The corresponding velocities V1 and V2 at the thruster firing point are: 2 1  2 1 V1 = µ −  = µ −  =  r a1   a1 a1 

µ = 3.0747 km / s a1

2 1  2 1 V2 = µ −  = µ −  = 3.0831 km / s  r a2   a1 a 2 

(5.52)

and therefore: ∆V21 = V2 – V1 = 8.5 m/s. The same ∆V is required to stop the drift, albeit in the opposite direction (∆V12), so the total relocation impulse is ∆Vreloc = ∆V21 + ∆V12 = 213

5 / Orbit Management 17 m/s. The corresponding propellant mass expenditure for our 1000-kg sample satellite is ∆m = 5.4 kg.

∆Vreloc = 5.66 dΛ/dt

olo gy .

By combining Eq 5.51 and 5.52 and doing some elaborate algebra, we arrive at the following simple and useful equation for GEO satellite relocations: m/s

(5.53)

with dΛ/dt in degrees per day. As a test, set dΛ/dt = 3°/day, which yields ∆Vreloc = 17 m/s, i.e. the same value as above.

∆Λ dΛ dt

days

(5.54)

ec

Treloc =

hn

If the longitudinal distance to be covered is ∆Λ degrees, the time Treloc required to perform the relocation is:

Satellite Re-Phasing

Co p by yri Lu ght leå © Un 201 ive 4, rsi ty o

5.4.5.4.

fT

For example, a relocation of a GEO satellite from longitude 0° to 90°E at 3°/day would take 30 days.

Adjusting the relative phase between satellites is particularly important in constellations. By “constellation” we mean a large number of identical satellites in parallel orbits designed to offer global coverage. Examples are the mobile communication satellite systems Iridium and Globalstar.

Figure 5-36 Satellite constellation with global coverage.

For the earth coverage to be truly global, it is essential that the relative phasing between the satellites be maintained. The phasing might be upset by small launch injection errors or by asymmetric luni-solar perturbations. To re-align the satellites, the same technique is used as in Section 5.4.1.7 (“Changing ν”). This means that the semimajor axis a of the target satellite’s orbit is altered temporarily so as to modify the orbital period τ. When the satellite has caught up with its intended phase, the original orbit is restored.

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5 / Orbit Management 5.4.5.5.

Maintaining a Sun-Synchronous Orbit

5.4.5.6.

olo gy .

Recall from Sections 2.3.5.1 and 2.5.2 that sun-synchronism depends on three orbital elements being properly tuned, namely a, e and i. If e changes over time due to solar pressure on the satellite, the orbital nodes will no longer remain perfectly synchronised with the apparent movement of the sun vector. In order to rectify the situation, it is necessary to adjust the nodes periodically, as discussed in Section 5.4.1.5. Transfer to Graveyard Orbits

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When a satellite is approaching the end of its operational life, it is considered good practice to remove it. In the case of the highly congested GEO, the removal is important to avoid future collisions with adjacent satellites. Collisions are possible as defunct satellites lose their E/W station-keeping ability; expelling them to a higher or lower orbit reduces that risk. LEO satellites can be removed by changing the orbit such that they reenter the earth’s atmosphere and burn up.

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GEO Satellites

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The graveyard orbit (more properly known as the disposal orbit) of choice is a circular orbit some 300 km higher than GEO (a higher orbit is preferred over a lower orbit to avoid interference with satellites in GTO and obstructing the RF signals of other GEO satellites). The corresponding manoeuvre is carried out by firing the onboard thrusters in several graveyard burns. Since the new orbit should not intersect GEO at any point, it is necessary to perform two Hohmann transfers, as illustrated in Section 5.4.1.1 and Figure 5-37. V2

}

V1

∆ V12

V3

}

∆ V34

V4

Figure 5-37 Hohmann transfer to graveyard orbit.

Let V1 = 3.075 km/s denote the velocity of GEO and V2 = 3.080 km/s the velocity at the perigee of the transfer orbit (given that a1 = 42,164 km and a2 = 42,164 + ½ · 300 = 42,314 km). Since the transfer is co-planar, we have ∆V21 = V2 – V1 = 5.45 m/s. Similarly, let V3 = 3.058 km/s denote the velocity at the apogee and V4 = 3.064 km/s the velocity of the circular graveyard orbit (given that a2 = 42,314 km and a3 = 42,464 km). Then ∆V43 = V4 – V3 = 5.44 m/s.

215

5 / Orbit Management The total manoeuvre to the +300 km graveyard orbit requires ∆Vgrave = ∆V21 + ∆V43 = 10.88 m/s, and the corresponding propellant consumption for our 1000-kg satellite is ∆m = 3.69 kg.

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An international agreement exists as to how high the graveyard orbit should be for a given satellite, depending on the propensity of its orbit to become deformed (i.e. eccentric) over time under the influence of solar pressure (see Section 12.3.9), such that its perigee touches on GEO. Intuitively, we can accept that this propensity is proportional to the illuminated surface area of the satellite, and inversely proportional to its mass. Thus the minimum height ∆H (in km) is obtained from the following equation: A ∆H ≥ 235 + 1000 Cr (5.55) m with A = projection of surface area exposed to the sun (m2) m = satellite mass (kg) Cr = reflectivity coefficient.

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LEO Satellites

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Cr is a function of the satellite’s specular and diffuse reflexion coefficients (β and γ) and has a value between 1 and 2. For example, a large communications satellite at the end of its propellant life may have the following characteristics: m = 2000 kg, A = 50 m2 and Cr = 1.5. Eq 5.53 then yields ∆H = 272 km.

Satellites with a semimajor axis a < 6700 km (or h ≈ 300 km circular) re-enter the earth’s atmosphere within days or weeks. Higher orbits take many months or years to decay – see Section 12.3.3. As in the case of GEO, there exists an international recommendation that non-GEO satellites be made to re-enter the earth's atmosphere within 25 years from the date they cease to function. A satellite in high LEO can be de-orbited forcibly by firing the onboard thrusters to create a low enough perigee for re-entry. In contrast to GEO, the nominal LEO and the elliptic graveyard orbits intersect, so we only need to perform a single-step Hohmann transfer (Figure 5-38).

Figure 5-38 De-orbiting of a LEO satellite through re-entry. The original orbit may be circular or elliptic.

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5 / Orbit Management

Approximate lifetime (days)

10

20

40

80

140

160

180

200

150

300

500

900

78 76 74 72 70 68 66 64 62 60 220

240

280

300

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Perigee height Pe (km)

260

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120

1400

hn

Propellant mass ∆ m (kg)

80

1

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Figure 5-39 shows the propellant mass needed to achieve various perigee heights and attendant residual lifetimes. In this example we have again used our sample 1000-kg satellite with its Isp = 300 s, starting out from a 1000 km circular orbit.

Figure 5-39 Propellant mass and lifetime as a function of de-orbit perigee height for a sample spacecraft weighing 1000 kg in a 1000 km operational orbit.

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To minimize the propellant consumption, there is an incentive to leave the perigee as high as possible. On the other hand, for a graveyard manoeuvre to make sense, the satellite should re-enter the atmosphere within a reasonable time. Taking 1 year as an example, the perigee must be lowered to 240 km in our example (with the apogee at 1000 km – see Figure 12-7). It is then up to the operator of our sample satellite to ensure that at least 66 kg of propellant mass remains when the manoeuvre is initiated.

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5 / Orbit Management

olo gy .

Historical notes: There has been speculation that if two GEO satellites were to collide and break up, the hundreds of resulting fragments would have enough kinetic energy to blow apart satellites further along – and so forth, until the chain reaction has converted all the satellites around the globe to a veritable Saturn ring of man-made debris. This prospect is known as the Kessler Syndrome and was the inspiration for the 2013 movie Gravity starring Tom Hanks and Sandra Bullock.

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5.4.6. Sample Propellant Budget

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In the period 1997 – 1999, nearly 100 Iridium satellites for mobile communications were launched into 780-km circular orbits. After the company went bankrupt, preparations were made to de-orbit all the satellites using the technique described above. At the eleventh hour the company was rescued, and the satellites were allowed to continue operating in their orbits.

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The aim of the propellant budget is to establish whether the satellite’s lift-off mass (i.e. the dry mass plus the propellant mass) is within the lift capability of the favoured launch vehicle(s), and whether the intended operational lifetime is in fact achievable.

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Let us assume that our sample GEO satellite (m = 1000 kg, Isp = 300 s) is to be launched into GTO (Pe = 300 km, Ap = 35,786 km), from where it will use its own apogee motor to reach GEO. The satellite will be positioned initially at longitude 78°E. Five years later it is to be relocated to longitude 75°E in a manoeuvre that must take no longer than 3 days. After another 5 years the satellite is expected to be running low on propellant and should therefore be expelled into a circular graveyard orbit with a radius 300 km higher than GEO. The propellant consumption for attitude control is expected to be 4 kg over the 10-year lifetime. The E/W station-keeping half-window is 0.1°, and the N/S half-window is 3°. Our sample satellite is initially placed at the start of both windows, as discussed earlier. The satellite is lucky to be located near one of the stable points along the equator (longitude 75°E). Consequently, at the initial location of 78°E an E/W correction is required only every 90 days, and no correction at all is required at the final location. The inclination is allowed to drift freely for 7 years within the 3 deg limits. The following propellant budget gives a breakdown of the corresponding ∆V and ∆m for the sample satellite. Note that the satellite mass at the start of each manoeuvre is reduced as a result of the propellant consumption during the previous manoeuvre. Nearly half the lift-off mass (393 kg) is made up of the propellant needed to perform the transfer from GTO to GEO. This is typical for GEO satellites launched into GTO.

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5 / Orbit Management

m0 (kg) = Isp (s) = µ (km3/s2) =

GTO

GEO

1000

Ap (km) =

35786

a (km) =

42164

300 398601

Pe (km) =

300

V (m/s) =

3074.7

ra (km) =

42164

Λ1 (deg) =

78.0

olo gy .

Constants

R (km) =

6378

rp (km) =

6678

Λ2 (deg) =

g (m/s2) =

9.81

a (km) =

24421

τ (s) =

Vap (m/s) =

1607.8

τ (h) =

Vpe (m/s) =

Graveyard

75.0

86163.5 23.93

2

2

2

2

EW d Λ1/dt =

10151.6

-0.0002

ra (km) =

42464

τ (s) =

37980.1

EW d Λ2/dt =

0

rp (km) =

42164

τ (h) =

10.55

EW ∆Λ (deg) =

0.1

a (km) =

42314

∆V (m/s) =

1466.8

EW ∆t1 (days) =

90.05

Vap (m/s) =

3058.3

Vpe (m/s) =

3080.1

Relocation

∆V (m/s) =

10.89

dΛ/dt (deg/day)

1.0

∆V (m/s) =

5.66

m0 (kg)

∆m (kg)

0.003

hn EW ∆V1 (m/s) =

EW ∆V2 (m/s) = NS ∆i (deg) =

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∆V (m/s)

∞

0.05

3

NS ∆t (years) =

7.06

NS ∆V (m/s) =

321.8

Comment

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Quarter

EW ∆t2 (days) =

0

GTO - GEO

1466.84

1000

392.51

1

E/W st.-keeping

0.05

607.49

0.01

2

E/W st.-keeping

0.05

607.48

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0.01

3

E/W st.-keeping

0.05

607.47

0.01

4

E/W st.-keeping

0.05

607.46

0.01

5

E/W st.-keeping

0.05

607.45

0.01

6

E/W st.-keeping

0.05

607.44

0.01

7

E/W st.-keeping

0.05

607.43

0.01

8

E/W st.-keeping

0.05

607.42

0.01

9

E/W st.-keeping

0.05

607.41

0.01

10

E/W st.-keeping

0.05

607.40

0.01

11

E/W st.-keeping

0.05

607.39

0.01

12

E/W st.-keeping

0.05

607.38

0.01

13

E/W st.-keeping

0.05

607.37

0.01

14

E/W st.-keeping

0.05

607.36

0.01

15

E/W st.-keeping

0.05

607.35

0.01

16

E/W st.-keeping

0.05

607.34

0.01

17

E/W st.-keeping

0.05

607.33

0.01

18

E/W st.-keeping

0.05

607.31

0.01

19

E/W st.-keeping

0.05

607.30

0.01

20

E/W st.-keeping

0.05

607.29

0.01

No E/W station-keeping

21

Relocation

5.66

607.28

1.17

at long. 75°E.

28

N/S st.-keeping

321.83

606.12

62.79

N/S station-keeping

40

Graveyard burn

10.89

543.33

2.01

after 7 years.

Attitude control

22

541.32

4.03

Spread over 10 years.

Total:

1828.23

Check total:

462.71

462.71

Using total ∆V

Table 5-2 Sample propellant budget.

219

5 / Orbit Management The “check total” entry at the bottom of the table is interesting, in that the sum total of ∆V (= 1822.79 m/s) has been inserted directly in Eq 5.42 to yield ∆m = 461.71 kg. This suggests that, when calculating the total ∆m over the satellite’s lifetime:

olo gy .

a) there is no need to calculate ∆m at each step along the way; and

b) it makes no difference for the total ∆m in which order the manoeuvres are performed.

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Note, however, that these statements are only true if the Isp is identical during all the manoeuvres. This is not strictly the case in most satellite missions, partly because some manoeuvres are performed in pulsed rather than continuous thrust mode, and partly because the pressure in the propellant tanks decreases as propellant is expelled. But if the Isp does not vary too much, the two statements are good enough for making initial propellant consumption estimates.

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Q: Statement (b) above seems to go against intuition, since one would assume that performing a large manoeuvre towards the end, when the satellite is lighter, is more economical overall.

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A: Here is the mathematical proof that statement (b) is correct. Assume that we wish to undertake two manoeuvres in sequence, namely -∆V1/(g Isp)

and

a1

∆m1 = m1 [1 – e ] = m1 [1 – e ] -∆V2/(g Isp) a2 ∆m2 = m2 [1 – e ] = m2 [1 – e ]

a1+a2

with m2 = m1 – ∆m1. If we can prove that ∆mtot = ∆m1 + ∆m2 = m1 [1 – e -(∆V1+ ∆V2)/(g Isp) = m1 [1 – e ], then clearly the order is unimportant. For the 1st manoeuvre: For the 2nd manoeuvre:

]

a1

∆m1 = m1 [1 – e ] a2 ∆m2 = (m1 - ∆m1) [1 – e ] a1

a2

a1

a2

Therefore: ∆mtot = ∆m1 + ∆m2 = m1 [1 - e ] + m1 [1 - e ] - m1 [1 - e ] [1 - e ] a1 a2 a1 a2 a1 a2 a1 a2 = m1 [1 - e + 1 - e - 1 + e + e - e e ] = m1 [1 - e e ] a1+a2 = m1 [1 - e ] which is the proof we are after. By extension, for n manoeuvres we have: -(∆V1 + ∆V2 + ∆V3 + .... + ∆Vn)/(g Isp)

∆mtot = m1 [1 – e

]

Here again, the order of the various ∆V is unimportant for ∆mtot .

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5 / Orbit Management

5.5.

Solved Problems

See Appendix C for solutions.

r = (4000, 0, 6000)T km;

V = (-6, 0, 4)T km/s.

Calculate the apogee and perigee heights.

olo gy .

5.5.1 Through ranging and range rate measurements, a ground station has computed the following values for the radius vector and the velocity vector of a satellite at a certain moment in time:

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5.5.2 A satellite with a mass m = 500 kg is travelling in a circular polar orbit (i = 90°) at an altitude h = 600 km above the earth’s surface. Without changing the altitude or the inclination, the orbit is to be modified by initially rotating the nodes 2 degrees eastward (i.e. ∆Ω = 2°). Subsequently, the inclination i is to be altered to make the orbit sunsynchronous, without any further rotation of the nodes. What is the total propellant consumption ∆mtot for these two manoeuvres? The earth’s radius R = 6371 km, the gravitational parameter µ = 398601 km3/s2, and the thruster onboard the satellite has a specific impulse Isp = 300 s.

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5.5.3 Returning to Problem 5.5.2, how much propellant would be saved if both manoeuvres were performed simultaneously? 5.5.4 The thruster in Problem 5.5.3 develops 10 N of thrust. What is its propellant consumption rate dm/dt, and how long will the manoeuvre take, expressed in cumulative burn time? 5.5.5 A geostationary satellite with a mass m = 4000 kg is located at longitude 120°W and is to be maintained at that position within a longitude window of ± 0.1°. How frequently must an east-west correction be performed, and how much propellant does the first such correction consume? The specific impulse Isp = 300 s.

5.5.6 What is the worst-case ∆V per year required for east-west station-keeping of a GEO satellite within the statutory ± 0.1° longitude window? Same question for northsouth station-keeping?

221

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6 / Propulsion

6. Propulsion Introduction

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6.1.

The propulsion subsystem allows the satellite to change its attitude and orbit. It is made up of thrusters (large and small rocket motors) and associated propellant storage, feed, filters and valves.

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The subsystem is responsible for a substantial portion of the satellite’s lift-off mass, and hence its launch cost. As demonstrated in Section 5.4.6, nearly half of the mass of a GEO satellite consists of propellant needed to take it from the geostationary transfer orbit to the geostationary orbit. The remaining propellant is spent over a number of years on out-ofplane and in-plane orbital manoeuvres, and to a lesser degree on attitude control.

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More than any other spacecraft constituent, the propulsion subsystem defines the satellite’s operational lifetime. When the onboard propellant is nearing depletion, steps are taken to terminate the mission in an orderly fashion.

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Satellite engineering employs two different propulsion technologies, namely thermodynamic and electrodynamic propulsion. Each technology has several subcategories (Figure 6-1). Sometimes more than one category exists onboard a single spacecraft.

Propulsion

Thermo-dynamic

Cold gas

Compressed Vaporizing liquid

Electro-dynamic

Chemical

Solid propellant Liquid monopropellant Liquid bipropellant

Electrothermal

Resistojet Arcjet

Electromagnetic MPT PIT PPT

Electrostatic Ion HET FET

Figure 6-1 Propulsion technologies for satellite applications.

In the following sections we will examine design and operational aspects of the most common propulsion variants. We will conclude with suggestions for thruster layout possibilities to gain optimum cost-effectiveness. 223

6 / Propulsion

The dynamics of attitude and orbit control are dealt with in Chapters 4 and 5.

Thermodynamic Propulsion

olo gy .

6.2.

Most thermodynamic propulsion relies on the compression of cold gas or the expansion of heated gas to develop thrust. By contrast, electrodynamic thrust is generated by accelerating charged atoms or molecules, with the exception of electrothermal thrusters which combine thermodynamic and electrodynamic technologies (see Section 6.3).

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6.2.1. Propellant Characteristics

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Propellant

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A chemical bi-propellant has two constituent parts: fuel and oxidizer. The fuel is the component that provides most of the propulsive energy, while the oxidizer enables the fuel to combust. Frequently the word “fuel” is used synonymously with “propellant,” probably because what we put in our automobiles is indeed fuel (i.e. petrol), while the engine draws the oxidizer (oxygen) from the ambient atmosphere. A rocket which operates in vacuum has to carry its own oxidizer along with the fuel, so it is helpful to make the distinction between propellant, fuel and oxidizer:

Fuel

Oxidizer

Figure 6-2 Propellant and its parts.

Chemical monopropellants, cold gas and electrodynamic propellants only contain a single propulsive component, i.e. they do not require an oxidizer to combust or accelerate.

The most important figure of merit of a particular propellant is its specific impulse Isp, defined as the impulse per kg of expended propellant (Eq 5.43):

I sp =

Ve F ∆t F = = g g m
g ∆m

(seconds)

(6.1)

where Ve is the exit velocity of the combusted propellant (in m/s), F is the thrust (Newton), m
= ∆m/∆t is the propellant mass flow (kg/s) and g is the acceleration of the earth’s gravitation (= 9.81 m/s2). The fact that Isp is measured in seconds is an artefact of its definition (ratio of kg and kg/s), and it offers no intuitive relationship to time. Assuming that the thrust F is reasonably constant during a burn (which is usually the case), the total impulse It imparted on the satellite is:

224

6 / Propulsion

I t = F ∆t = gI sp ∆m

(6.2)

olo gy .

with ∆t being the burn time and ∆m the propellant mass consumed. Isp can be thought of as a measure of the rocket motor's “fuel consumption,” with the difference that it measures the thrust per spent kilogramme rather than miles per gallon. The higher the Isp, the more thrust we get out of a given quantity of propellant (Eq 6.1), or the less propellant we have to carry onboard for a given impulse requirement (Eq 6.2). The velocity increment resulting from a given impulse may be derived from Eq 5.49:

     

hn

(6.3)

ec

   m0  m0  = Ve ln ∆V = Ve ln  F∆t  m0 − ∆m   m0 − Ve 

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Solid propellant rockets are often referred to as “motors,” while liquid propellant and electric rockets are called “engines.”

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Table 6-1 gives some typical Isp values for different types of propellant, along with some other characteristics that influence the designer’s choice.

Cold gas Monopropellants Solid propellants

Isp in vacuum 30

Thrust (N) 0.1 - 250

Handling hazard Low

Ease of storability Very easy

250 – 290

100 - 106

High

Easy

Hydrazine

200 - 250

0.5), the diameter must be greater than the wavelength. It therefore makes sense to look for other antenna types which are inherently less directional than the parabolic reflector. Table 7-4 provides an inventory of the most common antenna types for TT&C.

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7 / Tracking, Telemetry and Command (TT&C)

θ3dB (deg)

Max gain

D = diameter

Helix (endfire)

2

15.3η

Nd  πD   ÷ λ  λ 

16.5°

λ λ  ÷ Nd  D 

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hn

N = number of turns d = distance between turns D = diameter

olo gy .

λ  70° ÷ D

2

 πD  η ÷  λ 

Parabolic reflector

λ  70° ÷ D

2

 πD  η ÷  λ 

Circular horn D = aperture diameter a = horn length

D2 3λ

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fT

a=

Quarter-wavelength dipole

2 dB

~ 60° x 360°

h = height = λ/4

Table 7-4 Antenna types for TT&C.

The first three antenna types feature a pear-shaped radiation pattern, as in Figure 7-35. The dipole produces a kidney-shaped pattern. There are many other antenna types whose patterns suit particular applications, such as the toroidal pattern antenna (Figure 7-37). The shape represents the wavefront contour at a particular moment in time.

Figure 7-37 Toroidal antenna radiation pattern.

At the risk of over-simplifying, one can say that the parabolic reflector (whose feed is typically a small horn) is best suited for narrow beams (i.e. high gain) over a wide range of frequencies. The helix offers a fairly wide beam and is more compact than the parabola 282

7 / Tracking, Telemetry and Command (TT&C)

7.3.4.4.

olo gy .

and the horn below 4 GHz. The horn complements the helix by being the most compact at frequencies above 4 GHz. The dipole antenna pattern is almost hemispherical, so with two dipoles mounted diagonally opposite each other on the satellite's body, it is possible to create something resembling an omnidirectional antenna pattern. Antenna Polarization

hn

A radio signal carrier travels through the atmosphere and free space in the form of two plane, sinusoidal waves, one representing the electric field (the E-field) and the other the magnetic field (the H-field); hence the term electromagnetic radiation. The electric and magnetic field planes are perpendicular to each other and to the direction of travel. The two are inseparable, so in the following discussion regarding polarization we will focus on the E-field and ignore the H-field.

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We all know about polarized sunglasses and how they reduce the glare from sunlight reflexions against water surfaces. Before striking the water, the light is unpolarized or, rather, it may be represented by sinusoidal light waves embedded in an infinite number of different planes. When the light is reflected against the water (assuming that the surface is smooth), all the planes except the horizontal one are dispersed, so that the reflected light becomes horizontally polarized. Since the lenses of the sunglasses are polarized vertically, they filter out the horizontally polarized light, thereby eliminating most of the annoying glare. (But check what happens if you tilt your head 90 degrees!) A steadily increasing number of people want to transmit and receive radio signals, to the point where the radio spectrum is becoming very crowded. If it were possible for two users to share the same frequency without interfering with each other, the usable size of the radio spectrum would double. There are in fact several frequency sharing techniques. One of them is called polarization discrimination, i.e. sharing is possible by having two users employ opposite polarization, whereby the “glare” of one transmission is ignored by the other. It is the task of the two transmitting antennas to generate the opposing polarizations, and of the two receive antennas to discriminate between the two.

A typical satellite antenna feed is capable of producing two perpendicular E-fields, and to vary the relative phase and amplitude between the two sine waves. Their projections are shown in Figure 7-38, borrowed from www.wikipedia.com. If their phase and amplitude remain the same, the resultant of the two tracing vectors grows and contracts along a straight line, whereas if the phase of one component is shifted by 90 degrees relative to the other, the resultant vector describes a circle (Figure 7-39). In the first example we have linear polarization, while in the second case we talk about circular polarization. (In the latter mode, we can also achieve elliptic polarization by giving the two components different amplitudes, or by choosing a phase angle other than 90 degrees.)

283

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7 / Tracking, Telemetry and Command (TT&C)

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Polarizers

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Figure 7-38 Linear, circular and elliptic polarization.

Orthogonal waveforms in phase

Orthogonal waveforms out of phase

Figure 7-39 Creating linear and circular polarization through waveform phase shift. The “fat” arrows are the resultant vectors of the “thin” phase vector pairs in the two perpendicular E-planes.

Linear polarization can be either horizontal (HP) or vertical (VP), while circular polarization is either left-handed (LHCP) or right-handed (RHCP). Figure 7-40 illustrates horizontal and vertical linear polarization in separate radio signals radiating towards a satellite on the same frequency from two transmitters on the earth. Because the polarization is perpendicular, the two signals do not interfere with each other at the satellite receiver, provided the receiving antenna is able to discriminate between the two polarizations. By convention, the horizontal polarization plane is parallel with the equatorial plane, while the vertical polarization plane contains the earth’s spin axis.

284

Figure 7-40 Linear polarization.

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7 / Tracking, Telemetry and Command (TT&C)

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Alternatively, if one satellite is only interested in the horizontally polarized signal, it will make sure that its receive antenna is also horizontally polarized. Another satellite may then be tuned into receiving the signal with vertical polarization. Here again, there will be no interference even if both signals use the same frequency.

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The only problem with linear polarization is that, by geometrical necessity, the relative tilt of a given E-plane changes as the satellite passes overhead. Unless the transmit and receive antennas track each other’s polarization accurately (e.g. by slowly rotating the receiving antenna’s feed), the latter antenna will also pick up part of the other signal, i.e. there will be radio interference. The advantage of circular polarization is that polarization tracking is not necessary, as long as the antennas at both ends are tuned in to the same orientation (RHCP or LHCP). For this reason, circular polarization is more common than linear in radio links for TT&C. 7.3.5. Noise and Attenuation 7.3.5.1.

Signal Noise

In the above preliminary link budget for telemetry, the noise temperature T was set at 80 K without further justification, and was converted to noise power via Eq 7.19: N = kTB. Since N is a determining factor for the quality of the link, and hence for the design of the TT&C subsystem, it deserves more careful attention. An in-depth analysis is offered by Gordon & Morgan [7], from where Figure 7-41 has been adapted. The figure illustrates how the various sources of radio noise are viewed by the ground station antenna’s main lobe and side-lobes. The main sources are the sun, the moon, precipitation, surrounding ground objects and the ground itself. The noise is additive and has the effect of corrupting the telemetry signal from the satellite, as represented by the ratio C/N in Eq 7.15.

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7 / Tracking, Telemetry and Command (TT&C)

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Figure 7-41 Sources of noise. (Adapted from Gordon & Morgan [7] by permission of John Wiley & Sons, Inc.)

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Let us now examine the case of the telecommand uplink, as perceived by the satellite receiver. A high-gain satellite antenna looking down at the earth “sees” the earth’s disc and perhaps some surrounding space. The earth radiates at a temperature T ≈ 293 K, while the cosmic radiation from the surrounding space is negligible. The corresponding noise power N is distributed fairly evenly across the radio spectrum. Let us call this temperature Tant, since the antenna is the collector of radiation from the earth. Add to this a resistive noise contribution from the signal line (coax or waveguide) between the antenna and the receiver, as well as from the receiver itself, and we obtain the system noise temperature at the output from the telecommand receiver as:

Tsys =

Tant L −1 + 290 line + Trx Lline Lline

(7.26)

Lline ≥ 1 and is close to 1 if the line is short, making the middle term negligible, so there is an incentive to keep the signal line as short as possible. Trx depends on the chosen frequency and on the receiver technology used (Figure 7-43). In the 2.0 – 2.3 GHz TT&C band, Tsys lies typically in the range 500 – 1000 K in the telecommand uplink, and around 100 – 200 K in the telemetry downlink. The received noise analysis of the telemetry downlink is more complicated, as evidenced by Figure 7-41. While antenna designers work hard to suppress side lobes, some inevitably remain, and these pick up noise from the sun, the moon, outer space, the ionosphere, the troposphere, precipitation, the surrounding topography (buildings, forests, etc.), and from the earth itself. The detailed breakdown and weighting of these components is beyond the scope of this book, but Figure 7-42 offers an indication of the galactic and tropospheric noise

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7 / Tracking, Telemetry and Command (TT&C)

olo gy .

temperatures at various frequencies and elevations above the local horizon. The curves ignore the influence of the sun in Figure 7-41, since the receive antenna on the ground is likely to be large with a high gain and narrow beamwidth, and the operator will therefore avoid using the antenna when the sun is just behind the satellite. Ttrop (K)

10

10° 30° 60° 90°

ec

δ= 0°

fT

100

hn

1,000

Co p by yri Lu ght leå © Un 201 ive 4, rsi ty o

Noise temperature T sys (K)

10,000

1

0.1

1

10

100

Frequency f (GHz)

Figure 7-42 Galactic and tropospheric noise temperatures at various ground antenna elevations δ.

From Figure 7-42 we may conclude that T = 80 K at δ = 10° worst case was too conservative a selection in the above preliminary link budget. A more accurate value would have been 25 K at the 2 GHz frequency. Noise contributions from the sun and the moon must be considered if the antenna beamwidth is large, since one can no longer afford to be as selective with its usage as with a narrow-beam antenna. The sun’s surface temperature is T0 = 5,805 K, while the moon’s temperature is T0 ≈ 200 K. The noise components are proportional to the square of the fraction α that the heat source occupies within the antenna’s beamwidth θ3dB, i.e. Thotbody

 α = T0   θ 3dB

  

2

(7.27)

where T0 is the temperature of the source (αsun = αmoon = 0.53°). For example, an antenna 2

 0.53  with θ3dB = 10° will attract Tsun = 5805  = 16.3 K. The equivalent value from the  10  moon is Tmoon = 0.5 K, i.e. negligible in this example.

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7 / Tracking, Telemetry and Command (TT&C) By comparison, a low-gain antenna with θ3dB = 30° would only pick up 2 K from the sun, which is to say that low-gain antennas are more or less immune from solar and lunar noise influences.

olo gy .

Ground station receivers contribute noise depending on the technology used and whether or not the receiver is cooled. Figure 7-43 illustrates the magnitudes involved.

Tunnel diode FET

hn

Bipolar Uncooled parametric

fT

ec

Cooled parametric

Co p by yri Lu ght leå © Un 201 ive 4, rsi ty o

Figure 7-43 Ground station receiver noise.

7.3.5.2.

Noise and Attenuation due to Rain

In Eq 7.25 we included a loss factor L to Eq 7.15 to account for the weakening of the received carrier power C due to rain, mismatched polarization, inaccurate antenna pointing, etc. Figure 7-36 showed the atmospheric attenuation in clear sky conditions. Rain is in a class by itself, in that it both attenuates the carrier (reflected in C) and contributes to the noise temperature (included in N), according to the equation: Lrain =

270 270 − Train

(7.28)

with Train measured in Kelvin. Figure 7-44 shows how the attenuation loss Lrain and the corresponding noise temperature Train depend on the rain rate measured in mm/h, on the assumption that the cloud temperature is 290 K and that the antenna boresight elevation is δ = 30° above the local horizon.

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100

f GHz

olo gy .

10

250 200

10

100

8 1

6

50

hn

Attenuation Lrain (dB)

Train (K)

14 12

0.1 20

30

40

50

60

Rain Rate (mm/h)

70

80

90

10

100

fT

10

ec

4

Co p by yri Lu ght leå © Un 201 ive 4, rsi ty o

Figure 7-44 Attenuation Lrain and noise temperature due to rainfall at 30° elevation above the local horizon.

The thickness at zenith hrain(90°) of the “rainy” part of the atmosphere is approximately 3 km (i.e. δ = 90°). The elevation-dependent thickness is obtained from Eq 7.21:

  R + 3  2  2 hrain (δ ) = R    − cos δ − sin δ    R  

(7.29)

with hrain(δ) in km. For example, hrain(30°) = 6 km. Therefore, to obtain the attenuation at other elevation angles, it is sufficient to divide the attenuation values in Figure 7-44 by 6 (which gives the attenuation in dB/km), and then multiply the result by hrain(δ) at the desired elevation δ. Thus:

[Lrain (δ )]dB

7.3.5.3.

=

[Lrain (30°)]dB 6

hrain (δ )

(7.30)

Line Noise and Attenuation

At frequencies below 4 GHz, coaxial cables are used to link the transmitter and the transmit antenna, as well as to link the receive antenna and the receiver. Waveguides (rectangular tubes without central conductor) are preferred at higher frequencies, given that their cross-sectional dimensions are proportional to the wavelength. A typical coaxial cable causes a signal loss of 0.5 dB per metre of cable. A 100 cm cable therefore represents a loss [Lline] = 0.5 dB, corresponding to Lline = 1.1. This loss also gives rise to an equivalent noise temperature, whereby

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7 / Tracking, Telemetry and Command (TT&C)

 1   Tline ≈ 2901 − L line  

olo gy .

7.3.5.4.

(7.31)

Polarization Loss

Lpol = cos2Φ, or

[Lpol]dB = 20 log (cosΦ)

hn

If the transmitting and the receiving antennas have their polarization perfectly aligned, there will be no polarization loss, i.e. Lpol = 1, or [Lpol] = 0 dB. If, on the other hand, the polarization of the two antennas is orthogonal, there will be no communication at all, with Lpol = ∞. Mathematically, the loss may be expressed as a function of the misalignment angle Φ: (7.32)

Φ = 17°/f 2

fT

ec

Even if the transmitting and receiving antennas are geometrically aligned, a certain misaligment may occur due to so-called Farady rotation in the ionosphere. The phenomenon is most severe during the local daytime. The average daytime Faraday rotation Φ may be estimated from the following equation: (7.33)

Co p by yri Lu ght leå © Un 201 ive 4, rsi ty o

with the frequency f expressed in GHz. For example, a 2 GHz telecommand signal with vertical polarization at the transmitter may suffer a rotation of up to 4° by the time it has travelled through the ionosphere and reached the receiver, resulting in a loss of usable power flux Lpol according to Eq 7.32. Occasionally, rotations as high as 30° are measured. Circular polarization eliminates this problem and is therefore used at the lower GHz frequencies. At higher frequencies, Faraday rotation is no longer an issue, and linear polarization is therefore preferred, because linearly polarized antenna feeds are easier to build. As discussed in Section 7.3.4.4, circular polarization is achieved by combining orthogonal E-field waveforms with a relative phase shift of +90° or -90°. One phase shift yields RHCP, while the other shift produces LHCP. Here again, a transmitting and a receiving antenna will communicate perfectly if both adhere to the same sense of circular polarization, and not at all if their polarization is in the opposite sense. In practice, the polarizations at both ends of the radio link are seldom perfectly aligned. One reason is that the relative orientation of the satellite and the ground station changes as the satellite passes overhead, causing a time-dependent misalignment between linear polarizations, and elliptic deformations of circular polarizations. Another reason is the Faraday rotation of linear polarization planes mentioned earlier. As a result, an attenuation Lpol due to polarization mismatch may occur and must be estimated. For example, a worst-case 30° Faraday rotation of a signal in linear polarization would cause a loss of 1.25 dB. The polarization isolation between an antenna with linear polarization and one with circular polarization is 3 dB. In plain language, this is to say that if an antenna transmits in linear polarization and another antenna receives in circular polarization, only half the signal power is received as compared to a perfect polarization match.

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Returning to Eq 7.25, we are now in a better position to estimate three important loss factors:

or, in dB:

olo gy .

L = Latm·Lrain·Lpol [L] =[Latm] + [Lrain] + [Lpol]

as it relates to our particular TT&C frequencies and circumstances. Recall that these losses are in addition to the free space loss {λ/(4πd)}2 which is already accounted for in Eq 7.25. Antenna Pointing Loss

hn

7.3.5.5.

θ 3dB ≈ 70°

λ

[L ]

  

2

(7.34)

Co p by yri Lu ght leå © Un 201 ive 4, rsi ty o

po int dB

 ε = 12  θ 3dB

fT

(in degrees) D An antenna pointing error ε will lead to a signal loss:

ec

Equation 7.22 gave us the antenna beamwidth as

For example, we know from Eq 7.22 that a parabolic dish receive antenna with a diameter Dr = 3 m, operating at 2 GHz, has a half-power beamwidth θ3dB = 3.5°. If such an antenna were to be mispointed by the same amount (ε = 3.5°), it would experience a signal loss of 12 dB. The signal does not disappear completely, because there is still some antenna coverage left beyond the half-power beamwidth. 7.3.5.6.

Implementation Losses

Losses in the range 0.5 – 3 dB are encountered in various hardware onboard the satellite as well as on the ground. Examples are switches and filters, which should be avoided whenever practicable. These losses are frequency-dependent. Figure 7-45 shows losses in a typical satellite S-band circuit where telemetry and telecommand signals are routed to and from redundant transmitters and receivers.

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7 / Tracking, Telemetry and Command (TT&C)

Telemetry transmitter 2

0.3 dB

0.8 dB

Diplexer (filter) Telecommand receiver 1

olo gy .

Telemetry transmitter 1

1.2 dB

0.3 dB

hn

Telecommand receiver 2

Summary of Sources of Noise and Losses

fT

7.3.5.7.

ec

Figure 7-45 Losses in switches, diplexers and antenna signal lines.

Co p by yri Lu ght leå © Un 201 ive 4, rsi ty o

We are now in a positon to restate Eq 7.25 as follows, taking into account the various sources of noise and attenuation discussed earlier: Pr C Pt Gt G r 1  λ  = =   N N kTs B Latm Lrain L pol Lline1 Lline 2 L point1 L point 2 ⋅ ⋅ ⋅ ⋅ ⋅  4πd 

2

(7.35)

Here, Ts is the sum of all noise temperature contributions. Lline1 refers to losses in the coax or waveguide that links the transmitter with the transmit antenna, while Lline2 is the equivalent loss on the receiver side. Similarly, Lpoint has to be accounted for on both the transmit and receive side relative to the range vector.

Figure 7-46 gives an overview of all the noise and loss sources that should be accounted for in a comprehensive link budget. This figure also suggests the existence of other noise and loss factors that apply in special cases. The preliminary link budget in Table 7-2 is updated in the following, using Eq 7.35 with the various noise and loss elements filled in from Table 7-5 below. The transmit power Pt is boosted from 0.5 W to 5 W to avoid arriving at a negative link margin. The loss and noise temperature values in Table 7-5 are representative of a geostationary S-band telemetry system using a 0.2 m satellite telemetry antenna and a 2 m ground station receive antenna at 10° elevation.

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7 / Tracking, Telemetry and Command (TT&C)

TLO Lmod Lline1 Tline1 Lpoint1 Lpol Latm

Atomspheric loss

Galactic & tropospheric noise temperature

Ttrop

Eq 7.31 Eq 7.34 Eq 7.32 Figure 7-36 Figure 7-44

3.0

0.3 0.0

Eq 7.27

Lpoint2 Lline2 Tline2

Eq 7.34

0.2

Eq 7.31

0.5

22

32 30

Receiver phase jitter noise Subcarrier & demodulator losses Carrier circuit loss Waveform distorsion loss

Tphase Lsub Lcirc Lwave

Implementation loss (switches, etc.)

Limpl

Ranging interference1)

Lrang

4.0

Ls, Ts

14.5

1)

0 3

100

Figure 7-43

Figure 7-45

160

0.0

Thotbody

Trec

Receiver noise

3.5

Figure 7-42

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Hot body noise (sun, moon and surroundings) Receive antenna pointing loss (0.5 deg) Receive antenna circuit (feed or line) loss/temp

Noise temp (K) 0

0.0

fT

Cosmic noise temperature

Lrain Train Tcosm

Rain loss/temp

Loss (dB)

olo gy .

Local oscillator phase noise Modulation loss Transmit antenna circuit (feed or line) loss/temp Transmit antenna pointing loss (0.5 deg) Polarization loss (alignment and Faraday combined)

Fig or Eq

hn

Param

ec

Source

0 1.0 0.0 0.0 2.0

Loss due to power-robbing when telemetry transmission and ranging occur simultaneously.

Total:

347

Table 7-5 Overview of sources of S-band TT&C losses and noise temperatures.

293

olo gy . hn Polarization loss

Legend:

ec

Transmit antenna Antenna circuit loss

Modulation loss

Power amplifier

Carrier

Square/sine wave subcarrier

Atmospheric loss; Scintillation fades; Rain loss

Loss

Noise

fT

Pointing loss

Local oscillator phase noise

Phase modulator

Subcarrier modulator

Free space loss

Both

Receive antenna

Feeder loss

Signal interference

Pointing loss

Receiver

Waveform distortion loss

Carrier circuit loss

Receiver noise

Phase jitter loss

Demodulator

Ranging inteference

Subcarrier & demodulator losses

Bit synchronizer and detector

INFORMATION SINK

Cosmic noise; Hot body noise; Weather noise & losses

294

Local oscillator phase noise

INFORMATION SOURCE

Figure 7-46 Typical causes of RF loss and noise. (Source: CCSDS 401.0-B)

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7 / Tracking, Telemetry and Command (TT&C)

7 / Tracking, Telemetry and Command (TT&C)

olo gy .

34.8

10.3 2.4 7.9 dB

7.0 -3.5

160.0 0.2

Link margin

How? Input Input Figure 7-31 Input Input Eq 7.13 Input Figure 7-33 How? Input Eq 7.17 Input Eq 7.21 Input Input Eq 7.31 Input Eq 7.16 Eq 7.34 Input Figure 7-36 Figure 7-44 Figure 7-44 [λ/(4πd)]2 Input Figure 7-42 Eq 7.27 Input Eq 7.18 Eq 7.34 Input Eq 7.31 Figure 7-43 Figure 7-45 Input dB: Σ(13..34)-(17)-(28) Linear: Σ(15..32)-(16)-(27) Input (36)+(37) (35)-(38) (39)-(1) (39)-(3) (40)-(7)

hn

Linear 2200.0 0.136 10.0 40586.0 5.0

dB 37.0

ec

Linear 5000 QPSK 3000 1.00E-05 Concatenated

11.4 0.0 -3.0 -0.3 0.0

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(9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21) (22) (23) (24) (25) (26) (27) (28) (29) (30) (31) (32) (33) (34) (35) (36) (37) (38) (39) (40) (41)

Required signal quality ρ (bps) Modulation B (Hz) BER Coding Req Eb/N0 PSK Req Eb/N0 Concatenated Coding gain Achieved signal quality f (kHz) λ (m) δ (deg) d (km) Pt (W) Lline1 Tline1 (K) Dt (m) Gt Lpoint1 Lpol Latm Lrain Train (K) Lfreespace Tcosm (K) Ttrop (K) Thotbody (K) Dr (m) Gr Lpoint2 Lline2 Tline2 (K) TRx (K) Limpl Lrang C = Pr (W) Total noise temp Ts (K) Boltzmann constant N0 (W) Ach C/N0 Ach Eb/N0 Ach C/N

fT

No. (1) (2) (3) (4) (5) (6) (7) (8)

0.0

-191.5

3.0 22.0 100.0 2.0

31.4 -0.2 -0.5

32.0 30.0

3.04E-16 347.0 1.38E-23

-2.0 -4.0 -155.2 25.4 -228.6 -203.2 48.0 11.0 13.3 8.6

Table 7-6 Complete link budget.

The link margin of 8.6 dB well exceeds the minimum recommended margin of 3 dB.

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7 / Tracking, Telemetry and Command (TT&C)

7.4.

Solved Problems

See Appendix C for solutions.

olo gy .

7.4.1 With what accuracy (expressed in percent of full scale) is it possible to interpret an analogue telemetry read-out after A/D conversion using 8 bits? 7.4.2 Modern satellites contain an onboard clock consisting of a 48-bit digital counter driven by a 65,536 Hz oscillator. If the clock is set to zero on 1 January 2000, what date and year will it restart from zero? A year has 365.25 days.

ec

hn

7.4.3 A satellite in a 600 km circular orbit is specified to transmit telemetry to the ground in BPSK at a bit rate ρ = 10,000 bps with a bit error rate BER = 10-6. The carrier frequency f = 2 GHz. Determine the EIRP of the satellite’s transmitter, assuming a ground station G/T of -10 dB. Allow a 6 dB link margin to account for atmospheric and circuit losses.

fT

The earth’s radius R = 6371 km. The Boltzmann constant k = 1.38 · 10-23 J/K.

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7.4.4 What should the gain Gt of a telemetry transmit antenna be on a GEO satellite if it is to cover the entire earth’s disc? The frequency is 2 GHz, and the antenna efficiency η = 0.65. 7.4.5 What is the coding gain for a QPSK signal that undergoes concatenated forward error correction? The chosen Reed-Solomon code rate R = 223/255, and the Viterbi coding has a code rate R = 1/2 with a constraint length K = 7.

7.4.6 Estimate the noise temperature and the signal losses experienced by a ground station that receives telemetry at 12 GHz from a geostationary satellite appearing 30 deg above the local horizon. The beamwidth of the ground station antenna is 2 deg. The maximum rain rate in the region is 70 mm/h. Assume that the satellite’s antenna is accurately pointed towards the ground station, that the ground station’s antenna is mispointed by 1 deg, and that both the satellite and the ground station use horizontal linear polarization. Disregard all losses behind the respective antennas.

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8 / Onboard Data Handling (OBDH)

8.1.

olo gy .

8. Onboard Data Handling (OBDH) Introduction

hn

Up until the late 1970s, satellites possessed only a rudimentary data handling capability, while most complex operations were planned and executed from the ground. A degree of attitude control autonomy was then introduced, telemetry formatting became more adaptive to operational needs, and high-capacity electronic mass memories became available.

ec

As the satellites grew larger and more complex, so did the wire harnesses that relayed electric power and electronic signals between all the corners of the spacecraft. As a rule of thumb, the wire harness represented up to 10% of the satellite’s entire dry mass. The sheer bulk of the harness posed additional problems, as did its serviceability.

Co p by yri Lu ght leå © Un 201 ive 4, rsi ty o

fT

In this environment it became increasingly urgent to revisit the division of executive authority between the ground and the satellite, as well as the methods of onboard data handling. The result is the onboard data handling (OBDH) subsystem, made possible by the development of ever increasing computing power within a given weight and volume constraint.

The OBDH subsystem is the signal mastermind. At the heart of the OBDH is a central processing unit (CPU) and an associated mass memory; together they constitute the brain of the satellite. In the distributed bus configuration (Section 8.4), the CPU is served by a number of remote terminal units (RTU) – a set of mini-processors out in the field that perform local housekeeping tasks to make the dialogue run smoothly between the functional units and the CPU. By functional units we mean instruments, sensors, relays, actuators, storage devices, etc. An intensive debate took place all through the 1970s and 80s whether it was wiser to keep the primary operational authority on the ground or to transfer it to the satellite. The hesitation was natural, given the relatively high mass, limited intelligence, low capacity and poor reliability of computers at the time. Meanwhile, these problems have been largely resolved, and the spacecraft designers’ appetite to make satellites almost fully autonomous is growing fast. There are several good reasons for shifting operating responsibility from the ground to the satellite. As highlighted in Chapter 7, the OBDH makes optimum use of onboard resources during routine collection, forwarding and interchange of data between the satellite and the ground, as well as within the satellite itself. It provides a quicker response time in critical situations, e.g. switching over to backup equipment when a primary unit fails, or placing the satellite in a safe mode pending detailed anomaly investigations. It also translates into cost savings by reducing the number of spacecraft controllers needed on the ground to monitor and control the satellite around the clock.

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8 / Onboard Data Handling (OBDH)

8.2.

Onboard Computer

olo gy .

The onboard computer has a similar architecture to that of a PC. It typically consists of a CPU, a memory bank and an input/output (I/O) inteface. The CPU is in charge of • boot-up and initialization of application software • execution of application software supported by the operating system • interrupt management • error detection and alarm signal generation.

CPU

fT

ec

hn

The CPU is made up of an arithmetic & logic unit (ALU), a control unit and a set of data registers. In a nutshell, the ALU performs both arithmetic (addition, subtraction, multiplication, division, trigonometry, square roots, etc.) and logic operations (type AND, OR, XOR and NOT). The Control Unit interprets software code into basic instructions that the ALU can process. The data registers serve as temporary "scratch pads" for the ALU; they contain limited amounts of data needed by the ALU for frequent and rapid retrieval.

Arithmetic & Logic Unit (ALU)

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Control Unit

Registers

Telemetry Telecommand Payload data

Input / Output Controller

Mass Memory

RAM

EEPROM

Memory Bank

Figure 8-1 Onboard coputer architecture.

The main customers of the CPU are the payload, the TT&C, the attitude and orbit management, the power management and the thermal management subsystems. Some common applications: • • • • • • •

298

Steering phased array antennas; Re-allocating payload transponder channels and power levels to different spotbeam antennas; Initiating the deployment of antennas and solar panels; Keeping the solar panels pointed at the sun; Regulating power by switching solar cell strings; Managing battery charge and discharge rates; Activating local heaters as needed in the payload, battery and propulsion subsystems;

8 / Onboard Data Handling (OBDH)

• • •

• •

Executing attitude and orbit manoeuvres according to instructions preloaded from the ground; Performing autonomous attitude and orbit determination based on received GPS signals; Switching between redundant units in case one of them fails; Interrupting the mission in case of major onboard anomalies and placing the satellite in a safe attitude and configuration, pending diagnostic analysis and remedial action from the ground; Receiving, unpackaging and executing telecommands; Collecting, packaging and relaying telemetry data to the RF portion of the TT&C subsystem.

olo gy .

•

fT

ec

hn

In addition, the CPU provides an onboard clock and synchronization reference. The clock is typically a 48-bit counter driven by a crystal oscillator at 65,536 Hz (= 216 Hz). The clock thus completes a complete cycle every 248/216 = 232 seconds ≡ 136 years, which is well beyond the expected lifetime of any satellite. The correlation between the onboard clock count and UTC is known on the ground from telemetry, thereby allowing spacecraft controllers to instruct the processor to execute certain functions at a given time (i.e. a given counter reading).

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Synchronization pulses are issued at frequencies of the order 1 Hz, 8 Hz, 256 Hz, etc., and are used to trigger recurring events onboard the satellite. Examples include turning the solar panel stepper motor and taking scientific samples. Although computers act quickly and accurately, they only do what they are told, and that remains a lingering weakness. The modus operandi of the CPU is pre-defined in the operating system and application software. This pre-definition is only as good as the ability of its designers to predict every conceivable operational scenario. Inevitably situations arise which the designers did not foresee, and it is at such times that human judgment becomes indispensable. Human judgment may be introduced by making the onboard computers reprogrammable by telecommand. This approach has great potential, though it is too time-consuming to be used in emergency situations. There are ways to improve the tolerance of computers against failure. One approach is to install two or three identical processors. With two processors, it is possible for one of them to take over if the other one fails. For this to work, the redundant processor must intervene in the proper operational context, i.e. it has to mimic the performance of the primary processor at all times – a programming challenge with many pitfalls. The redundant processor can work either in hot redundancy, or else in cold redundancy under the authority of a small watchdog computer that acts as a memory of the main context.

With three redundant computers it is possible to introduce a majority vote scheme, whereby two processors in agreement will prevail over a disagreement with the third processor.

299

Historical note: The Space Shuttle orbiter employed four identical, redundant main processors with a sophisticated majority vote scheme. Three processors can out-vote the fourth processor, and two can even override the third if the fourth processor is manifestly malfunctioning and the third is in disagreement. At the limit, the crew could mobilize a fifth processor equipped with different software for independent analysis of the situation.

olo gy .

8 / Onboard Data Handling (OBDH)

Data Storage

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8.3.

fT

ec

hn

Nowadays many computer operations are performed by operational control procedures (OCP). These procedures are small pieces of code expressed in a high-level language, often called spacecraft control language (SCL), that are interpreted by the onboard software, allowing the OCPs to access a limited amount of the computer functions and interfaces. The OCPs typically execute routine tasks like battery monitoring or temperature control. They can also be used to handle spacecraft emergency cases and anomalies. If we compare with a typical PC, an OCP may be compared to a Java applet. The applet can execute rather advanced tasks, but the Java language prevents the applet from accessing vital computer resources.

As in the case of personal computers, a satellite CPU relies on a variety of memory chips to perform its calculations. These come in volatile and non-volatile versions. A volatile memory is erased - i.e. it loses all its data content - if the electric power supply is switched off for whatever reason, so it is only used for temporary data storage. Essential software, such as the operation system and application programs, must not be erased if power is lost and are therefore stored in non-volatile memory chips. A random access memory (RAM) is volatile and is used for rapid storage and retrieval of large amounts of data. A read-only memory (ROM) is non-volatile and is preprogrammed during manufacturing. A programmable ROM is called a PROM and may be programmed after manufacturing, but only once. Hence the operating system of a satellite computer is typically stored in a PROM which has been programmed before launch. The contents of an erasable PROM (EPROM) may be removed and then reprogrammed using an external programming device. This may seem lika a contradiction in terms, since a PROM is supposed to be read-only, but it is still non-volatile. Lastly there is the electrically erasable PROM, or EEPROM, which combines the advantages of nonvolatility and ability to be re-programmed in situ, e.g. by telecommand. EEPROMs therefore allow satellite operators to uplink new application programs to a satellite in order to rectify programming errors or change the spacecraft's modus operandi, but the data remains permanent in case of power loss. To make a (somewhat flawed) analogy with consumer electronics, the RAM is what determines how fast a home computer executes a program; a CD-R is a PROM while a CD-RW is an EPROM; and a USB memory stick is an EEPROM.

300

8 / Onboard Data Handling (OBDH)

olo gy .

So-called mass memories typically use RAM technology for storing large quantities of ephemeral data before transmission to the ground. In the early days of spacecraft engineering, satellites that collected large amounts of data used onboard tape recorders for storage. Whenever the satellites passed above designated ground stations, the tape recorders were played back in high-speed mode, and the data was archived on the ground. There were several difficulties with this technology. The tape recorders were heavy, power-hungry and unreliable (e.g. due to bearing seizure in vacuum), their storage capacity was limited, and they offered no random access to stored data.

ec

hn

The modern RAM eliminates the drawbacks of tape recorders. The earliest electronic memories found uses not only in scientific missions with high data volumes, but also in commercial store-and-forward applications. In store-and-forward, short messages in the field are uplinked to LEO satellites and subsequently downloaded over suitable ground stations for onward land transmission to the intended recipients – a kind of space-based email.

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Nowadays, satellite mass memories come with storage capacities of 100 Gbits or more, and data may be dumped at rate of several hundred Mbps. They are used for holding high-volume scientific and remote sensing data onboard satellites in LEO until the next available ground station pass.

Historical note: The spin-stabilized Meteosat weather satellites of the 1970s and 80s operated in GEO with a spin rate of 1 rps. Each satellite used an electronic mass memory to reduce the transmission bit rate, and hence the bandwidth, of images to the ground. The onboard radiometer swept across the earth’s disc once per spin revolution while recording 10,000 picture elements (pixels). Each pixel consisted of 8 bits, so a total of 80,000 bits was stored in a mass memory. Since the earth’s disc subtends 17.4° from GEO, the sweep took place during 17.4°/360° = 1/20 of a spin period. During the remaining 19/20 of the spin period, the stored pixels were transmitted to the ground. This “stretched” transmission bit rate therefore amounted to only 1/20 of what it would have been in a real-time mode, resulting in a massive saving of RF bandwidth.

8.4.

Data Distribution

As for data interfaces, there are two basic signal distribution architectures in common use, namely centralized and distributed. In the centralized layout, each functional unit is linked directly to the CPU, as illustrated in Figure 8-2. On a small scale, the centralized architecture is simple, reliable and easy to test, but it is also inflexible. The different functional units may require different interfaces, and adding another functional unit necessitates some reprogramming of the CPU. To communicate with each other, the units must take the detour via the central processor. This layout is mainly found on small satellites.

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Payload

Transmitters

CPU

Mass memory

Receivers

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Actuators

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Sensors

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Figure 8-2 Centralized architecture. (Source: T. Hult, Ruag Sweden.)

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In larger and more complex satellites, the amount of wire harness between the various functional units becomes an issue, as mentioned in the Introduction. They therefore favour a distributed layout, where units share a common data bus to communicate with the CPU and with each other (Figure 8-3). RTUs compile data to and from the various subsystems and interact with the CPU to ensure optimum prioritizing and packaging of two-way, high-speed data on the bus (typically 500 – 1000 kbps) along the lines described in Sections 7.2.1.2 and 7.2.2.2. The distributed layout is highly modular. With the functional interfaces being standardized, it allows units to be added or removed with relative ease. Most of the wire harness in the centralized layout is eliminated, allowing mass to be saved and reliability to be increased. Integration and test are also facilitated, though it is sometimes difficult to isolate a unit that is intent on disturbing the data flow. Mass Memory

Telemetry

CPU

Payload 1

Payload 2

Payload 3

RTU

RTU

RTU

RTU

RTU

RTU

Thermal Mgt

Power Supply

Central bus

Telecommand

Standard I/O Interfaces

Actuators

Figure 8-3 Distributed architecture. (Source: T. Hult, Ruag Sweden.)

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Attitude Processor

Sensors

8 / Onboard Data Handling (OBDH)

olo gy .

Note that a functional unit may be yet another processor, as in the case of the dedicated attitude processor in Figure 8-3. Attitude management is often the subsystem that places the greatest demand on computer capacity, so a separate processor may be justified on the grounds that the data processing is highly specialized and should enjoy a degree of autonomy.

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To take full advantage of the distributed architecture, it is essential that the participating units speak the same language, i.e. that they transmit and receive data according to an agreed protocol. Several different protocols have been invented, and the unit manufacturers have had to make a guess which one might be the most profitable to be compatible with. In the 1970s, the US Department of Defence established the so-called MIL-STD-1553B protocol which remains the most frequently used even today. A decade later, the German automotive industry developed the Controller Area Network (CAN) protocol, which is slowly finding favour among some European satellite builders. The two standards are basically incompatible. To make matters worse, there are yet more protocols being promoted on the market, each with its own advantages. There are two solutions to this dilemma: either build protocol translation circuits, or exclude incompatible units from participating in the data bus traffic.

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The telemetry and telecommand interface to the left of the CPU in Figure 8-3 deserves elaboration (Figure 8-4; see also Figure 7-1).

Transmitter

Telemetry Formatter

Diplexer

Mass Memory

CPU

Receiver

Telecommand Decoder

Standard I/O Interfaces

Command Pulse Generator

}

Payload data

Central bus Telemetry inputs

}

Telecommands

Figure 8-4 TT&C Data Handling. (Source: T. Hult, Ruag Sweden.)

In the above example we observe how data from the payload bypasses the central bus, because the data volume as well as the bit rate exceed its capacity. This is typical for some scientific and remote-sensing satellites that collect large quantities of data. The presence of a mass memory for storing a part of payload data suggests that the satellite travels in LEO rather than in GEO, so as to preserve collected data between passes over designated ground stations. The memory bypass link to the telemetry formatter indicates that some of the payload data is needed in real time. The bulk of platform subsystem data is routed to and from the CPU on the distributed data bus (the thick black line). Platform telemetry, as well as low-level on/off and 303

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proportional telecommands, are orchestrated by the CPU assisted by the standard I/O interface, while high-level on/off commands are energized by the command pulse generator (see also Section 7.2.2.2).

•

hn

• •

ec

•

What volumes of data are to be transferred (average and peak values)? What types of interfaces are provided by the most common sensors and actuators? How many nodes (users) will require their own addresses? Will the nodes communicate via a central processor and/or with each other? Will the data be formatted as single words, as large data blocks, or as a mix of the two? What types of disturbance may be expected on the bus, and how are they to be diagnosed? How much power will the bus and the nodes draw? What requirements are there with regard to electromagnetic compatibility (EMC) between the bus and the surrounding satellite electronics (Chapter 14)? Are there any suitable communication circuits and drivers available on the market?

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• • • • •

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When deciding on the choice between a centralized and a distributed architecture, the OBDH designer should ask himself the following questions:

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Technology advancements in the field of electronics are making the distributed architecture more accessible than before, and future small satellites are therefore likely to move from the centralized to the distributed architecture.

8.5.

Remote Terminal Unit

One might say that the CPU is the conductor of a symphony orchestra, the RTUs are the musicians, and the functional units are their instruments. The conductor is guided by a musical score (the operating system) to prompt the musicians in matters of pitch, rhythm, dynamics, timbre and texture (the application). He listens carefully to the quality of the performance and makes split-second decisions on subsequent measures. The RTUs have been instituted as local signal hubs for functional units, as well as for switching relays, thermistors and similar components scattered throughout the satellite. The aim is to reduce the length of signal cables and improve access for integration and repair. For example, an RTU might be mounted on the inside of a side wall. When folding open the wall, the technician only has to deal with the data bus and some power supply wiring, rather than a mass of fragile signal wires. RTUs come in many different guises and carry different names, depending on the mission requirements and the traditions of particular manufacturers. Some RTUs are stand-alone units while others are integrated into subsystems. Some are more intelligent than others. In its simplest form, an RTU may be viewed as a local telemetry encoder and telecommand decoder. As such, it performs A/D conversions and data (de)multiplexing. More advanced designs participate in the (un)packaging and execution of data in the wider context of packaged telemetry and telecommand described in Chapter 7.

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8.6.

OBDH Design Aspects

•

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•

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• •

What tasks will the subsystem perform? What is the expected nature and rate of the data flow? What is the most suitable architecture? Which tasks are to be executed in real time and deferred time? Which tasks are to be programmed in hardware, software and firmware? What operating system and programming language is to be used? How much processing and memory capacity will be needed, and what margins are prudent? How fast must the processors work? How will the processors handle predicted radiation, vibration and temperature extremes? Ditto regarding random fault events due to defective parts, single event upsets, or EMC transients? What are the volume, mass, power consumption and reliability constraints for the various OBDH units? How will the OBDH subsystem be tested? Which units exist off-the-shelf, how much do they cost, and what are the delivery times?

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• • • • • • •

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When defining the OBDH, it is essential to get it right from the start, as last-moment modifications can be extremely costly and time-consuming. A short definition checklist might look like this:

• •

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9. Structures and Mechanisms Introduction

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9.1.

The structure is the satellite’s backbone, and the mechanisms are its moving parts. Together they define the overall geometry of the satellite.

hn

In contrast to aircraft, which are immediately recognizeable, satellites come in every shape and size. This is not to say that there are no geometric constraints; quite on the contrary, the laws of physics impose restrictions on the spacecraft designer that are at least as severe as those affecting the aircraft builder.

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For starters, the satellite has to fit inside the available heatshield volume of the chosen launch vehicle. Consequently solar panels, antennas and other appendages often have to be stowed, i.e. folded up against the main structure of the satellite for later deployment.

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The static and dynamic launch loads, though well documented, are quite severe and represent a challenge for the designers of both the structure and the mechanisms. The temptation might be to resort to structural reinforcements, but the pressure is on the designers to do the exact opposite, namely to make the structure and the mechanisms as light as possible to save on launch cost. Once in orbit, the solar panels are expected to track the sun while the antennas are aimed at the earth. Given the complicated relative movements of the sun and the earth as seen from earth orbit, the satellite may have to be something of a contortionist to keep its appendages pointed correctly 24 hours a day (Figure 9-1). Daily transitions through the earth’s shadow give rise to sharp temperature gradients which cause thermal expansion and contraction, yet the structure must not deform to the point where scientific instruments, attitude sensors and narrow-beam antennas become misaligned.

Figure 9-1 Example of “contortionist” design of a hypothetical satellite in polar orbit. In this design, the solar panel is mounted at the end of a boom equipped with a steerable universal joint.

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The purpose of the structure is therefore to provide a sturdy mounting platform for the satellite’s functional units. It also serves as a common grounding reference for electrical equipment.

olo gy .

The most common examples of mechanisms are separation systems, relays, valves, deployable solar panels and antennas, and electric motors for steering solar panels and antennas. Statistically, mechanisms are more prone to failures than most other satellite equipment, and special attention is therefore given to their reliability and testing.

ec

hn

As mentioned earlier, much effort is devoted to making structures and mechanisms compact and lightweight, so as to keep launch costs down. This effort is anathema to achieving high reliability, since compactness is partly achieved by adding more mechanisms, and light structures are of course less sturdy and durable than reinforced ones. The paradox is resolved by employing a combination of advanced materials, careful mathematical modelling, and extensive testing.

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The former Soviet Union excelled in rocketry, and their launch vehicles Historical note: were among the most powerful and reliable in the world. The cost of launching was a moot point due to the arcane accounting methods used in the centralized economy. The lift-off weight of Soviet spacecraft was therefore not a critical design issue, which is why their satellites often appeared heavy and clumsy in Western eyes. In fact, the absence of Western-style mass constraints allowed Soviet designers to build satellites speedily and cheaply, in part because they could forgo most of the costly structural prototyping and qualification testing common in the West. It is true that Soviet satellites were not particularly reliable, but building and launching replacements was quick and inexpensive. There have been proponents in the West for adopting the Soviet approach to build “big, dumb satellites,” but our predilection for gold-plated solutions continues to make Western spacecraft and rockets exorbitantly expensive.

9.2.

Structures

9.2.1. Mechanical Configuration

As discussed in Chapter 4, most earth-satellites are either spin-stabilized or bodystabilized. The stabilization method has a major influence on the structural layout. A fast-spinning satellite is likely to have a cylindrical shape, with the solar panels forming the cylindrical skin. To mount the solar panels on a despun platform would not be practicable due to their large size. Attaching them to the walls of a spinning polygon rather than a cylinder might produce unacceptable modulation of the electric power output. A body-stabilized satellite is usually rectangular in shape. The antennas are fixed, deployed and/or steerable; the same is true for the solar panels. Monocoque (box) construction is widely employed to achieve structural strength and stiffness. Strength and stiffness are two different things. Strength is a measure of the structure’s ability to withstand a load condition without suffering a failure. Stiffness is 308

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what makes a structure resist deflections when loaded. A design may be stiff but not strong (e.g. a window pane), or it may be strong but not stiff (e.g. a mechanical spring).

olo gy .

A distinction is made between the primary structure and the secondary structure. The primary structure absorbs the main external loads (acceleration, vibration, thermal, etc.), and is usually made up of a central cylinder (the “spine”) in combination with flat panels. The secondary structure consists mainly of thermal closure panels. Figure 9-2 illustrates the above concepts in the case of a body-stabilized satellite.

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Antenna tower

Shear panels

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Core Structure

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Payload module

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Equipment platform

Equipment platform

Platform module

Figure 9-2 Typical structure of a body-stabilized satellite.

The central cylinder is usually referred to as the central tube. In the example shown, the central tube constitutes the primary load-bearing structure, along with the shear panels. In a geostationary satellite, the tube encapsulates the apogee kick motor, whose nozzle protrudes through the hole in the platform module. Inside the tube there are helium tanks for pressurizing the propellant tanks if the latter are mounted outside of the central tube. Another variant is to have one fuel tank and one oxidizer tank stacked in tandem inside the tube, with the pressurant tanks mounted on the outside. The separation of the equipment platforms into a payload module and a platform module has the advantage that they may be manufactured and tested independently by two different contractors. When assembled, the modules form a box that contributes to the monocoque strength and stiffness. The payload’s functional units are mounted on the inside walls of the payload module which is sometimes crowned by an antenna support structure. Similarly, the platform units are attached to the inside of the walls and floors of the platform module. 309

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While the load-bearing properties of a structure are of primary concern, the problems of access and handling also affect the satellite’s geometry and must not be overlooked. Access is important both to facilitate integration and to enable functional units to be replaced if necessary. As for handling, it must be possible to lift and rotate the satellite safely. Lifting and rotating occur during transport and testing, and also when measuring the satellite’s physical properties, notably its mass, centre of gravity and moments of inertia.

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9.2.2. Shapes and Materials

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So far, we have discussed the overall structure of the satellite. Each functional unit also offers structural challenges. Propulsion tanks must withstand internal pressure. Pipes have to be sized to prevent hydrodynamic resonance and be strapped down to mitigate vibration during launch. Electronic circuits need to be housed in a way to shield against vibration as well as radiation. Sensors require an unobstructed field of view and a solid foothold to prevent misalignment due to external loads. And the total assembly must meet system requirements in regard to volume, mass, centre of mass, and moments of inertia.

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The simultaneous requirements for structural strength and light mass limit the choice of materials. Panels, cylinders and cones are usually made of aluminium honeycomb sandwiched between thin aluminium or composite face sheets.

Figure 9-3 Cross-section of a honeycomb panel.

Supporting struts are also made of composite materials. By composites we mean polymer fibres (usually carbon) embedded in an epoxy, resulting in a light, stiff, directionally strong medium with minimal thermal expansion. Other materials in common use include aluminium, titanium, beryllium, steel, magnesium, and certain ceramics. The choice depends on the particular material’s strength-to-density ratio, stiffness, hardness, brittleness, thermal expansion coefficient, thermal and electric conductivity, toxicity, ease of manufacturing, and cost.

The strength and stiffness aspects deserve elaboration, because they provide a foretaste of the complexities of mathematical modelling. A strut or a panel subjected to compression may buckle (i.e. collapse). Buckling is, in most cases, catastrophic, because the structure’s resistance to the offending load diminishes at the onset of buckling. The propensity to buckle depends on the chosen material and the geometry of the article, and of course on the amount of force applied. The relevant parameters have been tabulated in various handbooks. For example, a straight rod will buckle when subjected to a critical buckling load Fc, defined by: 310

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Fc =

π 2 EI Le

(9.1)

2

olo gy .

where E is the material-dependent modulus of elasticity, or Young’s modulus; I is the area moment of inertia; and Le is the effective length.

The modulus of elasticity E is a material constant that defines the elastic relationship between stress σ and strain ε according to Hooke’s Law: σ=Eε

hn

(9.2)

σ = F/A

(N/m2)

ε = ∆L/L

(dimensionless)

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Now we need to define stress and strain:

(9.3)

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(9.4)

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where F is the applied compressive force, A is the cross-sectional area of the article, L is the article’s initial length, and ∆L is the amount of deformation under the influence of σ (Figure 9-4).

L

L

∆L

∆L

F

F

Figure 9-4 Elastic deformation under the influence of a compressive force (left) and a tensile force (right).

Note the difference between area moment of inertia, defined as I = ∫ x 2 dA , and mass moment of inertia defined as I = ∫ x 2 dm . (It is unfortunate that in most texts, both types

are given the symbol I.) The effective length Le is related to the actual length L of the rod but is scaled depending on the rod’s end attachments (Figure 9-5):

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F

F

F

olo gy .

F

F Le = L

F Le = 0.7 L

F Le = 0.5 L

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F Le = 2L

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L

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Figure 9-5 Effective length Le depending on end attachments.

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The same article will behave quite differently when subjected to tensile forces. There are three stages of tensile deformation, namely the elastic phase, the yield phase and the moment of rupture. These phases are illustrated in Figure 9-6 for three different materials exposed to a given stress level σ. Stress σ (N/m2) Y= U

Y

U

P

P

U

Y

P

Strain ε

Figure 9-6 Stress–strain curves for three different materials.

Stage P is known as the proportional or elastic phase. As the names suggest, the article will return to its original length once the force F is removed. Stage Y is the yield phase, during which the article undergoes plastic deformation. If the article is stretched to this extent and the force is removed, a permanent deformation will remain. At point U the article breaks, and we have the ultimate failure point. The steep curve to the left typifies brittle materials with its minimal strain under the influence of stress, and with its almost non-existent yield phase. For example, porcelain is 312

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olo gy .

brittle and will break with no noticeable yield. The middle curve is typical of soft metals like aluminium which will yield a fair degree before it breaks. Such materials are called ductile (as opposed to brittle). The flattened curve to the right could be representative of chewing-gum. If we pull at a piece of gum, it seems elastic at first but begins to yield if we pull harder. Pull harder still, and it tears apart. Materials – notably metals – contract and expand with changes in temperature. If a structural member is attached at both ends, it is appropriate to consider thermal strains and stresses, and the effect they may have on the integrity of the surrounding structures.

hn

In the case of a strut or beam, the thermal deformation ∆L is related to the change in temperature ∆T according to: ∆L = α L ∆T

(9.5)

fT

ec

where α is the thermal expansion coefficient, L is the strut’s initial length, and ∆T is measured in Kelvin. Having calculated ∆L, and knowing the geometry L and A and the elasticity modulus E for the strut, we are able to compute the strain ε from Eq 9.4, the stress σ from Eq 9.2, and the corresponding force F from Eq 9.3.

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Equations 9.2 through 9.5 apply equally to tension and compression while the material is in the elastic phase. Buckling under compression usually occurs at the elastic stage.

9.2.3. Quasi-Static and Dynamic Loads

The structural engineer is concerned about buckling, yield and failure in the context of quasi-static loads, such as those intermittent static loads experienced during handling on the ground and during the launch phase (Chapter 12). Ground handling includes suspending the satellite from cranes and other lifting devices, and tensile stress becomes an issue. Launch accelerations inflict significant compressive forces, with load factors of 4 g and up. A load factor is the multiple of the satellite’s weight on the ground, so if a satellite weighs 1000 kg before lift-off, it may experience loads corresponding to a weight of 4000 kg and beyond due to launch acceleration. (Fighter pilots experience considerable load factors when banking sharply at high speeds, and wear so-called g-suits to mitigate the discomfort.) While quasi-static loads are fairly easy to quantify analytically, dynamic loads are more complex and therefore more problematic for the structural engineer. The different kinds of dynamic launch loads are discussed extensively in Section 12.2 and will not be repeated here. We will therefore focus on one of the most hazardous phenomena in a spacecraft’s lifetime, namely vibration resonance between the launch vehicle and the satellite.

All objects manifest natural frequencies. We are all familiar with the natural frequencies of guitar strings that give rise to musical tones, but there are more subtle examples in our everyday life. For example, if we place a portion of jelly on a plate and shake it gently sideways, the substance will wobble. At a given frequency of shaking, the substance appears to wobble more violently than when shaken more quickly or more slowly, to the point where it risks flying off the plate and landing on the floor. 313

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The Tacoma Narrows suspension bridge in the United States collapsed on November 7, 1940 (you can watch it on YouTube). One theory suggests that the collapse was caused by structural fatigue after the wind that blew through the strait induced so-called Kármán vortex shedding. The shedding phenomenon is periodic, and on that particular day the vortex shedding frequency happened to coincide with the bridge’s lowest natural frequency.

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An object’s lowest natural frequency is known as the fundamental frequency and is usually the most severe in terms of structural loading. In the case of complex constructions such as a satellite, each element has its own fundamental frequency. To make matters worse, these frequencies are altered by the manner in which the element is attached, and by the mounting platform’s own dynamic behaviour.

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In the above examples involving the jelly and the suspension bridge, disaster strikes when the induced frequency coincides with the fundamental frequency. This is what is known as resonance. If left undamped, the vibration amplitude of a resonating object will reach infinity.

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Resonance is easy to illustrate mathematically. Lay a toy car with mass m on a table and attach it to the end of a wall-mounted spring with the spring constant k. Pull the object and let go. If there is no damping (either in the spring itself or due to friction), the body will oscillate forever at constant amplitude, and with a frequency corresponding to the system’s fundamental frequency. We have: m
x
+ kx = 0

(9.6)

Eq 9.6 describes a balance of forces, namely between the inertial force m
x
due to the body’s cyclic acceleration and deceleration, and the spring force whose magnitude and direction are proportional to the cyclic extension and compression of the spring. k 2 x = 0 , or
x
+ ω n x = 0 , where ωn is the m angular velocity of the natural frequency fn, such that ωn = 2πfn and Eq 9.6 may be rewritten as follows:
x
+

fn =

1 2π

k m

(9.7)

The solution to Eq 9.6 for the fundamental frequency is: x = A sin(ωnt + φ)

(9.8)

where A and φ are integration constants (A may be set to 1 and φ to 0 when studying the general case). Eq 9.8 is illustrated in Figure 9-7.

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1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0

0

1

2

3

4

5

6

7

8

9

10

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Time

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Amplitude

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Figure 9-7 Undamped oscillation (fn = 1 Hz).

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In mechanical systems there is usually an element of damping due to internal or external friction. The damping force Fdamp is, in the idealized viscous case, proportional to the translation velocity x
, and Eq 9.6. now reads as follows:

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m
x
+ cx
+ kx = 0

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where c is called the damping constant.

(9.9)

2 Eq 9.9 may be rewritten as
x
+ 2ςω n x
+ ω n x = 0 , where 2ζωn = c/m, and therefore the damping ratio ζ is found to be:

ς=

and:

c

2 km

(

x = Be − ςωnt sin ω n 1 − ς 2 t + φ

)

(9.10)

(9.11)

If the damping coefficient c is zero, then ζ = 0, and Eq 9.11 reverts to the undamped solution in Eq 9.8. Eq 9.11 is plotted in Figure 9-8, with ζ = 0.05 and the integration constants B = 1 and φ = 0.

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1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0 0

1

2

3

4

5

6

7

8

9

10

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Time

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Amplitude

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Figure 9-8 Damped oscillation with fn = 1 Hz and ζ = 0.05.

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So far we have studied the oscillating systems in equilibrium, i.e. without any external influences. This is paramount to pulling out the object m, letting it go, and watching it oscillate in a damped or undamped manner. Making a more drastic analogy, it would be like kicking a satellite hard and then listening to its fading death rattle. A scenario of greater interest to the structure engineer is the one where the launch vehicle generates vibrations, propagates them through the interface with the satellite, and causes the satellite to resonate. The question is whether the resonance will cause the satellite to break up during launch. To illustrate this point, we make the assumption that the vibration emanating from the launch vehicle is a simple sinusoidal force FLV(t) = K sin ωt. How will the satellite with its (simplified) ωn respond to ω?

We may modify Eq 9.9 as follows:

m
x
+ cx
+ kx = K sin(ω t )

(9.12)

The analytical solution is laborious, but the resonant amplitude Ar is shown to be:

K

Ar =

(ω

2

n

−ω

)

K

m

2 2

+ (2ςω n ω)

2

=

mω n

2

(1 − q ) + (2ς q ) 2 2

(9.13)

2

where q = ω/ωn. Eq 9.13 is plotted in Figure 9-9 with K/mωn2 normalized to 1.

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5

ζ=

4

0 0.2

3

0.4

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0.6

2

0.8

1

1

0 0.0

0.5

1.0

1.5

2.0

2.5

hn

Ratio z = ω/ω n

3.0

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Figure 9-9 Resonant amplitude response to a forced sinusoidal stimulus.

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Figure 9-9 illustrates how the resonant amplitude in an undamped system moves towards infinity as the frequency ω of the rocket’s forcing function approaches the fundamental frequency ωn of the satellite (i.e. ω/ωn ≈ 1). To prevent the satellite from being destroyed during launch, we must either achieve substantial vibration damping ζ or remove the satellite’s fundamental frequency ωn a safe distance from the launcher’s induced frequency ω. Introducing a vibration damper at the satellite/launcher interface has been tried on Ariane, but the preferred solution is to separate ωn from ω. It is usually easier to find design measures that raise ωn rather than lower it, namely by stiffening the satellite’s structure. However, introducing stiffeners will make the satellite heavier, so there is a strong incentive to select a structural layout that inherently possesses the required stiffness. Typical values of vibration frequencies f induced by the launch vehicle are shown in Table 9-1, remembering that ω = 2πf. The preferred satellite fundamental resonance frequencies fn are also tabulated for comparison, whereby ωn = 2πfn.

Launcher Satellite induced f (Hz) fundamental fn (Hz) Axial 30 >40 Lateral 10 >15

Table 9-1 Typical launcher induced vibration and preferred satellite fundamental frequency. Axial vibrations occur along the thrust axis. Lateral vibrations are induced by load asymmetries due small differences in the timing of strap-on booster ignition, and by wind gusts during the ascent.

Here is some terminology used in satellite structural engineering, especially in the context of testing: •

The limit load is the predicted maximum load the structure will encounter at any stage during its lifetime within a probability of 99%.

•

The allowable load is the highest load a structure can survive without suffering failure in the form of buckling, yield or rupture. 317

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•

The design load is the limit load plus a factor of safety (FS) against buckling, yield or rupture. The FS is typically 10 - 25% for yield and 50 – 100% for rupture. The design load must obviously be less than the allowable load; the ratio serves to define – allowable load − 1 . For a valid structural design, the The margin of safety (MS) = design load MS must be greater than 0. Compared to the FS, the MS may be viewed as an additional reserve of structural strength.

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•

Figure 9-10 offers a graphical comparison between the different concepts.

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Permanent Elastic phase or buckling Allowable load

MS

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Design load

FS

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Limit load

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damage

Figure 9-10 Relationship between load types and load margins.

9.2.4. Mathematical Modelling

So far we have only scratched the surface of structural analysis. In addition to struts, there are panels, shells and closed contours of every conceivable geometry and manner of attachment. Apart from tensile and compression forces acting axially, there are lateral forces, shear forces, torques and bending moments. The forcing functions from the launch vehicle to the satellite are not pure sinusoids but a composite of more or less random vibrations and shocks covering a wide band of frequencies and amplitudes. Moreover, connected objects interact with each other. Structural modelling by analytical means is possible for only the very simplest geometries, such as beams, enclosed contours, or structures with few members. The actual layout of a satellite is obviously far more complex. Analytical solutions become too cumbersome, and it is necessary to resort to numerical methods. The advantage of analytical modelling is that the solutions are valid at any point along or within the structure being analyzed. Analytical modelling also facilitates the understanding of the physics involved. However, as the geometric complexity increases, the analytical equations become very difficult and time-consuming to solve. Moreover, when dynamics are involved, the equations are predominantly differential rather than algebraic, in which case analytical solutions are even more elusive. Numerical solutions overcome these difficulties in part by turning differential equations back to algebraic, which are solved in an iterative process. The first step is to approximate the structure by a multitude of nodes, e.g. distinct elements with simple geometries such 318

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as flat plates or cylindrical, spherical and conical contours. Each node is attributed a set of plausible physical properties (e.g. mass, moment of inertia, elasticity, degrees of freedom of displacement) which are made to interact with adjacent nodes in a realistic manner. A set of parallel algebraic equations results, which are formed into matrices and are readily solved by computer. Two of the best-known numerical techniques are the finite element method and the finite difference method.

Figure 9-11 Example of finite element modelling.

These methods have the added advantage that the structural behaviour of the launch vehicle and the satellite may be modelled independently and then combined in a coupled loads analysis which reveals the static and dynamic load patterns in the satellite due to the launcher, and vice versa. The main difficulty with numerical methods is that the number of nodes must be neither too small nor too large for the problem at hand. In the former case, the modelling density and accuracy may be unsatisfactory, and in the latter case the processing time increases exponentially with no corresponding gain in realism. Also the accuracy is only as good as the attribution of properties to the nodes. Mastering the attribution task well requires a good understanding of the physics involved; hence the need for engineers to be familiar with basic analytical theory.

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Is testing of the real article not more

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Q: Why bother with mathematical modelling? realistic?

A: There are three answers. Firstly, we cannot test an object until we have built it. Mathematical modelling allows us to make an intelligent guess as to what the object should look like to survive the expected load conditions.

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Secondly, testing is very expensive, especially when it comes to large structures like satellites and rockets. Rather than testing for every conceivable load condition, it is more cost-effective to test a few cases, compare with equivalent runs of the math model, adjust the model where necessary, and then use it to simulate all the other cases of interest. The philosophy here is to calibrate the model against a few real tests, and then assume that the model is highly representative of the physical article. The same philosophy is used in aircraft design, where wind-tunnel tests serve as reality checks for mathematical models used to analyze aerodynamic flow as well as structural integrity.

Mechanisms

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9.3.

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Thirdly, many load conditions are impracticable or downright impossible to test on earth. One example is the load coupling between the satellite and the launch vehicle; another example is weightlessness.

As mentioned in the Introduction, moving parts are inherently unreliable, especially in the severe environment of outer space. Most of them are also mission critical, i.e. if any one of them fails, the entire mission may be jeopardized. Many spacecraft designers therefore see mechanisms as a liability to be avoided at almost any cost. That said, they provide as fascinating an engineering challenge as any other part of the satellite. Figure 9-12 illustrates the types of mechanism commonly found onboard a satellite.

Thruster latch valves

Reaction wheels

Deployment mechanism

Hinge

BAPTA

Figure 9-12 Examples of satellite mechanisms.

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In the above example, only two solar subpanels are shown, but the solar array wings on GEO satellites often feature between four and six hinged subpanels each, and these constitute flimsy and delicate assemblies. The solar array deployment mechanism may consist of redundant pyro cutters to initiate deployment, spring-loaded hinges to unfold the subpanels, and an elaborate set of wires, pulleys and brakes to ensure that the affirmative deployment ends with a gentle lock-up without backlash.

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The bearing and power transfer assembly (BAPTA) is an electric step motor with redundant slip rings and brushes for rotating the solar panels and transferring electric energy from the cells to the power management subsystem. In addition to power, the BAPTA may also transfer signals, e.g. from wing-mounted sun sensors used for autonomous panel sun-pointing. The BAPTA is sometimes known as a solar array drive mechanism (SADM).

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The antenna deployment mechanism in Figure 9-12 is a simpler version of the solar panel ditto, given that the antenna shown is not articulated. A few communications satellites in GEO (notably Thuraya and Alphasat) feature 12 – 15 m trussed parabolic antennas whose deployment mechanisms represent cutting edge technology.

Figure 9-13 Alphasat. (Credit Inmarsat)

Mechanisms may be divided into three classes, each with its own technological challenges: a) Once-only devices, such as deployment mechanisms, separation cutters and pyro valves. b) Intermittently operating mechanisms, for example antenna pointing devices, retractable gravity gradient booms, and latch valves. c) Continuously operating motors, including BAPTAs, antenna despin motors, gyroscopes and momentum wheels. 321

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One would have thought that the design of once-only devices should be trivial, but a depressing number of category (a) deployment mechanisms have failed in the past, leading to partial or total loss of the mission shortly after a successful launch. Many of these failures have involved solar panels. The difficulty has often been to estimate the deployment forces necessary to overcome friction, keeping in mind that cold welding and evaporation of lubricants may occur in the vacuum of outer space (see Chapter 12). Blindly oversizing the forces is not a solution, since latch-up backlash might damage the structure and upset the attitude control. Large structures are also prone to warp due to thermal strains. Getting the design right is further complicated by the notorious difficulty to simulate weightlessness while testing large, flimsy structures on the ground.

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Category (b) devices present many of the same challenges as those in category (a) but may be less mission critical. For example, the failure of an antenna pointing mechanism might be partially overcome by changing the satellite’s attitude, or by using another antenna with lower gain.

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In category (c), the problem of wear-out is the predominant concern. Examples are ball bearings, which become pitted over time, or lubricants that change their chemical properties under the influence of temperature, space radiation or vacuum. R&D efforts continue to build friction-free bearings based on magnetic suspension. The key to successful implementation of mechanisms is imaginative design, careful materials selection, rigorous quality assurance during manufacturing, and exhaustive testing within realistic limits. These subjects are dealt with at some length by Fortescue & Stark [6].

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9.4.

Solved Problems

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See Appendix C for solutions.

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9.4.1 A circular, solid aluminium beam is variously cooled and heated in relation to its equilibrium temperature as the satellite moves in and out of eclipse. The temperature excursions amount to ∆T = ± 30°C = ± 30 K. The strut is bolted at both ends to walls with negligible relative movement. Its length is L = 1 m, and its diameter D = 10 cm. Calculate the strain ε, the stress σ, the resulting compression and tensile forces Fc and Ft, and establish whether the beam will yield, break or buckle.

9.4.2

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L=1m

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D

What happens if we choose a thinner strut with D = 1 cm instead of 10 cm?

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9.4.3 A satellite weighing m = 100 kg is attached to a rocket via an adapter with a spring constant k = 104 N/m. At the moment of rocket engine burn-out, the satellite experiences a sudden shock. What is the natural frequency fn of the satellite/adapter assembly if the rocket’s mass is much greater than that of the satellite, and if there is no damping? 9.4.4 Now introduce a damping constant c = 5200 Ns/m. What is the value of the damping ratio ζ?

9.4.5 During lift-off, the rocket exposes the above satellite to a constant acceleration a = 4g, where g = 9.81 m/s2. What is the response of the satellite/adapter assembly in terms of x(t)? Outline the response in a diagram.

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10. Thermal Management

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10.1. Introduction

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Most of the materials and components used in a satellite have been developed for applications on earth, and their resilience to temperatures not found on earth is limited. The thermal engineer’s primary task is therefore to recreate an earth-like temperature regime inside the satellite. This is no trivial task, considering the extreme cold in space and the brutal, unfiltered illumination of the sun. Matters are made worse by temperature gradients as the sun moves within the satellite’s reference frame, or as the satellite travels in and out of the earth’s shadow.

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Onboard propellants and batteries are examples of equipment that are particularly temperature sensitive. Hydrazine freezes at around +2°C, and subsequent thawing can cause pipes to burst. Battery performance and lifetime are also severely affected by temperatures beyond the recommended operating range. For most other equipments, an out-of-limit temperature will affect the performance in the first instance (e.g. infrared sensors), and to a lesser degree the unit’s survival.

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Thermal monitoring via telemetry is therefore an essential tool for spacecraft controllers to ascertain the health of a satellite, the same way a physician relies on a thermometer to assess a patient’s health. Thermal control is performed automatically by passive means as far as possible, but active control is sometimes necessary. On unmanned spacecraft, active control usually involves heating rather than cooling, and is triggered by onboard logic or via telecommand. Figure 10-1 shows some typical thermal boundaries for the good operation of satellite equipment. Hydrazine tanks & pipes Travelling wave tubes Batteries General electronics Electric motors Solar cells IR sensors

-200 -150 -100

-50

0

50

100 deg C

Figure 10-1 Typical equipment temperature operating limits.

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10.2. Thermodynamic Principles

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Back on earth, thermal energy is transported in three ways: through radiation, conduction and convection. For example, the warming of an object by the sun is the result of energy transfer by radiation. Heat conduction takes place inside solid substances, e.g. from the inside of a wall to the outside. Convection occurs in moving liquids and gases. The movement is the result of either thermal gradients within the medium, or of the medium being pumped.

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In a typical household, thermal energy is transported through convection in water from the boiler to the radiators. The energy is then conducted through the radiators’ metal manifold, and is finally radiated into the rooms.

Figure 10-2 Examples of convection, conduction and radiation.

Out in space, there is no convection in the absence of an atmosphere, so the temperature of a satellite is primarily the result of external radiation and conduction (though, as we shall see, convection exists in so-called heatpipes and is sometimes introduced artificially). The primary source of radiation energy is the sun, closely followed by albedo and earthshine. But heat sources within the spacecraft, such as radio transmitters, also contribute to its temperature balance. Conduction is used primarily to dissipate excess heat from the inside of the spacecraft to the outside, i.e. into space. One of the challenges of spacecraft design is to ensure that the satellite's thermal radiator areas are large enough to dissipate the waste heat from internal equipment into space, or else parts of the satellite might overheat. The size of the radiators also contributes to the overall size of the satellite, so it is essential that they are neither under- nor overdimensioned. As a rule of thumb, a space-facing wall of the satellite body can dissipate up to 350 W of waste heat energy per square meter. This value holds true if the radiator is constantly in shadow, as is (more or less) the case of the north and south panels of a GEO satellite. If the radiator is illuminated by the sun, the figure will be lower, since.the dissipation is curtailed by simultaneous energy absorption.

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10.3. Radiation

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10.3.1. Thermal Equilibrium

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A spacecraft in earth orbit is subjected to heat-generating radiation from three different sources: • direct solar radiation (“sunshine”) primarily in the visible spectrum; • solar radiation reflected by the earth’s surface in the visible spectrum (“albedo”); and • thermal radiation from the earth (“earthshine”) in the infrared (IR) spectrum.

S ≈ 1367 W/m2

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Max albedo ≈ 0.25 S

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Max IR ≈ 0.15 S

Figure 10-3 Radiation contributions from sunshine, earthshine and albedo.

Only the IR earthshine is relatively constant throughout the day, although it is height dependent. The direct solar radiation power density S ranges from 1321 W/m2 in early July to 1413 W/m2 in early January, with an annual average at 1367 W/m2. From a satellite’s viewpoint, S goes to zero in the earth’s shadow, although there are orbits (such as GEO and certain SSO) where eclipse transits are rare or nonexistent. Albedo radiation originates from the daytime side of the earth and depends on the orbital height as well as on the degree of light diffusion occurring over land, oceans and the atmosphere. In the following we will concentrate on the predominant source of radiation, namely the sun. The thermal prediction techniques are also applicable to the other sources of radiation. The present Section deals with the equilibrium situation where the illumination of the satellite is presumed constant. Most of the sun’s radiation is confined to the visible part of the spectrum (0.40 – 0.76 µm), i.e. the rainbow part that we can see with our eyes. A proportion of this energy is absorbed by the satellite, while the remainder is reflected back into space. A small portion of the sunlight extends into the near-infrared region (λ > 0.76 µm) and the near-ultraviolet region (λ < 0.40 µm).

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VIS

IR

0.25 0.20

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Solar Radiation I (W/cm2/µ µm)

UV

0.15 0.10 0.05 0.00 1

10

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0.1

Wavelength λ (µ µ m)

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Figure 10-4 Spectral distribution of solar radiation.

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Incoming light is either absorbed or reflected by the illuminated surface, or else transmitted if the surface is transparent. The corresponding fractions are denoted α for absorption, ρ for reflection and τ for transmission. It follows that α + ρ + τ = 1.

ρS

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S

αS

τS

Figure 10-5 Absorption, reflexion and transmission of incoming sunlight.

Transparent surfaces are unusual on satellites; we shall therefore assume in the following that τ = 0, such that α + ρ = 1. In the case of a perfectly black body, α = 1 and ρ = 0. A perfect mirror has the opposite properties: α = 0 and ρ = 1. The proportion of S absorbed by the satellite depends on the surface colour and treatment, and is characterized by the absorption constant α mentioned above, also known as the absorbivity. The absorbed energy per second is: qa = αAaS

(W)

(10.1)

where Aa is the absorbing surface area, calculated as described at the end of Section 3.2.3. As we have seen, 0 ≤ α ≤ 1. The absorbed energy has the effect of heating up the satellite. Some of this heat energy is emitted into space in the infrared spectrum according to the Stefan-Boltzmann law of radiation: 328

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qe = εAeσT4

(W)

(10.2)

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where ε is the emissivity constant, Ae is the emitting surface area, T is the body temperature in Kelvin, and σ is the Stefan-Boltzmann constant = 5.67·10-8 W/m2K4. Here again, 0 ≤ ε ≤ 1, the actual value being dependent on the satellite’s colour and surface treatment. Ae is the total emitting surface area, not just the illuminated area.

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Note that the values of α and ε are the same for any given wavelength. For example, in the case of earthshine, the radiation into and out of the spacecraft occurs at approximately the same IR wavelengths, in which case α ≈ ε. However, the sun shines mainly in the visible spectrum while the satellite emits in IR, so α is usually different from ε in the corresponding equations.

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By equating Eq. 10.1 and 10.2, we obtain the equilibrium temperature of the satellite, assuming that there are no internal heat sources. (Kelvin)

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 α Aa S   T =  ε σ A e  

(10.3)

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Table 10-1 gives approximate values for α and ε for commonly used paints and polishes. The true values depend on the surface treatment and can be found in numerous textbooks and handbooks.

White paint Black paint Aluminium (unpolished) Aluminium (polished) Gold Graphite epoxy Glass fibre Aluminized kapton Optical solar reflector (OSR) Second surface mirror (SSM) Solar cells, Si, filtered

α (VIS)

ε (IR)

α/ε

0.20 0.95

0.90 0.90

0.22 1.05

0.25

0.25

1.00

0.20

0.05

4.00

0.25 0.95 0.90 0.50

0.05 0.75 0.90 0.60

5.00 1.25 1.00 0.83

0.08

0.80

0.10

0.15

0.80

0.19

0.80

0.90

0.90

Table 10-1 Typical α and ε values.

It is worth repeating that α and ε assume the same value for any given radiation wavelength. Therefore, whenever Eq 10.1 is used to evaluate the effect of incoming IR radiation from the earth, α is often replaced by the surface’s ε.

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The α/ε ratio alone does not fully characterize the thermal behaviour of the surface in question. For example, an ideal black body has α = ε = 1, and hence α/ε = 1. In the case of unpolished aluminium, α = ε = 0.25, and also in this case α/ε = 1. But whereas the black body absorbs and emits radiation without delay, aluminium with its low α and ε is more reluctant to absorb and emit, and therefore offers greater thermal inertia. So even though the equilibrium temperature is the same in both cases, the latter is more resistant to rapid thermal variations. See Figure 10-6 and Eq 10.6 below. Black body

1

Low inertia

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COLD

0.8

0.4

α/ε > 1 High inertia

0.2

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0.6

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Emissivity ε

α/ε < 1

WARM

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0

0

0.2

0.4

0.6

0.8

1

Absorbivity α

Figure 10-6 Thermal balance based on the α/ε α/ε ratio.

To illustrate the calculation of Aa and Ae, we will take the simple case of a sphere with radius r. Here Aa = πr2 is the projected area in the direction of the sun (i.e. the “shadow”), while Ae = 4πr2 is the total area. The ratio Aa/Ae in Eq. 10.3 is then = 1/4. The greater the Aa/Ae ratio, the higher the equilibrium temperature – which is hardly surprising when the heat-absorbing area is large in relation to the heat-emitting area. Similarly, the equilibrium temperature is proportional to the α/ε ratio (Figure 10-7). Eq. 10.3 is plotted in Figure 10-7 for a sphere with the α/ε ratio as the variable.

Figure 10-7 Equilibrium temperature of a sphere as a function of α/ε. α/ε

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As another example, let us take a cylinder having radius r and height h. If the cylindrical wall is turned towards the sun, then Aa = 2rh. The emitting area Ae is made up of the two end surfaces plus the area of the cylindrical mantle, i.e. Ae = 2πr2 + 2πrh = 2πr(r + h). The ratio Aa/Ae in Eq. 10.3 now becomes

Aa h = Ae π (r + h)

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If, on the other hand, the cylinder is turned such that the top or the bottom faces the sun, then Aa = πr2, while Ae = 2πr(r + h) as before. The new ratio is:

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Aa r = Ae 2(r + h)

qa = αAaS + qs

(W)

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and Eq. 10.3 now reads:

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If the satellite contains a heat energy source qs of its own (e.g. a power amplifier), Eq. 10.1 ismodified as follows:

1/ 4

 αA S + qs   T =  a  εAeσ 

(Kelvin)

(10.4)

10.3.2. Thermal Gradients

Note that the equilibrium temperature depends solely on the satellite’s shape, surface treatment and internal heat dissipators. It is independent of the satellite’s mass and construction material. The situation changes if we now look at thermal gradients, e.g. those occurring as the satellite enters and exits the earth’s shadow. The rate of heating or cooling is clearly a function of the satellite’s thermal inertia, which is defined as the product of the satellite’s mass m and its specific heat c. The value of c depends on the materials used and can be found in standard handbooks. For example, for the most common aluminium alloys, c ≈ 900 J/kg K.

For a satellite experiencing a change in illumination, the following relationship applies: mc

dT = αAa S + q s − εAeσT 4 dt

(10.5)

We may rewrite Eq 10.5 as follows:

qs dT ε α 4 =  Aa S − AeσT  + dt mc  ε  mc

(10.6)

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from which we deduce that the thermal inertia (i.e. the magnitude of dT/dt) is influenced by m·c as well as ε. This makes intuitive sense: The greater the spacecraft mass, or the greater the specific heat, the smaller the magnitude of dT/dt, and the longer it takes for the body to heat up or cool down. Similarly, low emissivity ε slows down the ejection of body heat, thereby increasing the thermal inertia.

In a steady-state thermal situation (e.g. in the absence of eclipse), dT/dt = 0, and we are back to Eq. 10.4. If the body enters the earth’s shadow, then S = 0 in Eq 10.5, and the temperature drops in accordance with the differential equation dT = q s − εAeσT 4 dt

(10.7)

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mc

(10.8)

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1 1 3εAeσ = 3+ t 3 mc T T0

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If also qs = 0, we have the solution

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where T0 is the initial temperature at the start of eclipse. For example, a 30 kg satellite with a body shell of unpolished aluminium alloy (c = 900 J/kg K, ε = 0.25, Ae = 24 m2) will experience the temperature drop shown in Figure 10-8 for various initial temperatures T0 (recall that T is in Kelvin in the above equations).

Figure 10-8 Satellite temperature drop in eclipse at different initial temperatures.

The temperature curves converge as they approach the equilibrium temperature in eclipse (3 Kelvin, or –270°C). If the mass is increased tenfold from 30 kg to 300 kg, the thermal inertia increases, as one would expect (Figure 10-9):

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T0 ºC

70 60 50 40 30 20 10 0 -10 -20 -30

0 30

0

10

20

30

40

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60

50

Time from eclipse entry t (min)

60

70

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Temperature T (°C)

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Figure 10-9 Same situation as in Figure 10-8 for a heavier satellite.

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T = 1 − e −0.4( t / κ +C ) Teq

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Returning to the general case in Eq. 10.5, this differential equation is more difficult to solve. Moreover, it can only be solved for the elapse time t as a function of the temperature T, whereas in real life it is usually the inverse relationship that is of interest, i.e. T = f(t). To overcome this problem, we have resorted to the solution of fitting curves. It transpires that radiative heating can be modelled by the equation (10.9)

where Teq is the equilibrium temperature in sunlight as per Eq. 10.4, κ is a time constant, and C is an integration constant. The time constant is a measure of how quickly the temperature changes and is found to be:

κ=

mc

4εAeσTeq

3

(10.10)

The integration constant C is obtained by setting t = 0 and T = T0, where T0 is the initial temperature at the exit of each eclipse cycle::  Teq   C = 2.5 ln T −T  0   eq

(10.11)

The equivalent curve-fit equation for radiative cooling is:

T = 1 + 1.3e − 2(t / κ +C ) Teq

(10.12)

Here, Teq is the equilibrium temperature in eclipse – a value >> 0 K if the satellite contains heat-dissipating equipment. As before, the value of the integration constant C is obtained by solving for T = T0 at t = 0, where T0 is the initial temperature at the start of each eclipse cycle:

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 Teq   C = 0.1312 + 0.5 ln T −T   0 eq 

(10.13)

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1.2 1.0

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0.8 0.6 0.4 0.2 0.0 1

2 3 4 Relative time t/κ + C

5

6

fT

0

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Relative temperature T/Tq

Eq. 10.9 and 10.12 are plotted in Figure 10-10 and Figure 10-11, respectively. The curvefit is accurate to within 5% of the analytical solutions. For the precise solution, refer to Gordon & Morgan [7].

Relative temperature T/Tq

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Figure 10-10 Radiative heating in sunlight.

1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1.0

0.2

0.4

0.6

0.8

1.0

1.2

Relative time t/κ κ+C

Figure 10-11 Radiative cooling in eclipse.

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Numerical example: A spherical satellite is launched into a 600 km sun-synchronous noonmidnight orbit. It has a radius r = 1 m, weighs 100 kg and is made of unpolished aluminium (α = 0.25, ε = 0.25, c = 900 J/kg K). An amplifier inside the spacecraft radiates heat at qs = 200 W. Calculate the temperature of the spacecraft as it enters eclipse, returns to sunlight, and goes through eclipse one more time. The initial temperature at the start of the sequence -8 2 4 T0 = 10°C (283 K). The Stefan-Boltzmann constant σ = 5.67·10 W/m K . The earth’s mean radius R = 6371 km.

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Solution: First we calculate the orbital period from Eq 2.12: τ = 2πa√(a/µ) = 5792 s. The eclipse duration is obtained from the figure as tecl = 2θ/360°· τ = 2126 s, since -1 θ = sin [R/(R+h)] = 66°. Consequently the time in sunlight tsun = τ - tecl = 3666 s.

h

2

2

R

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R θ R

h

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R

2

2

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Moving on: Aa = πr = 3.14 m ; Ae = 4πr = 12.57 m .

 αAa S + q s Teq =   εAe σ

 qs   Teq =  ε A σ  e  κ=

(a) Cooling (Eq 10.12):

1/ 4

= 290 K in sunlight

1/ 4

mc

4εAe σTeq

  

3

T

Teq

= 183 K in eclipse (since S = 0)

= 5171 s in sunlight, κ = 20593 s in eclipse.

= 1 + 1.3e

− 2 (t / κ+ C )

.

Replacing T with T0 = 283 K at t = 0 yields the integration constant C = 0.433. With t = tecl in Eq 10.10 we have Texit = 264.7 K, which now becomes the new T0. (b) Heating (Eq 10.9):

T

Teq

= 1− e

− 0.4 ( t / κ + C )

.

Proceeding as before, the new integration constant C = 6.082. After tsun = 3666 seconds we reach the temperature T = 270.9 K, which becomes the T0 for the subsequent eclipse transit – and so forth. The temperature history is plotted below (converted to minutes and Celsius for ease of interpretation).

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10 5 0 -5 -10 -15 -20 0

10

20

30

40

50

60

70

80

90

100 110

120

130

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Time t (minutes)

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Temperature T (°C)

15

10.3.3. Thermal Interaction

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So far we have studied the thermal behavior of a single irradiated body, namely a satellite exposed to sunlight in space. But recall from Figure 10-1 that different types of equipment manifest different degrees of tolerance to temperature extremes, and that most prefer room temperature. Spacecraft equipment designers are therefore less interested in the thermal behaviour of the spacecraft as a whole, and are more concerned with the temperatures that their individual units will experience. These temperatures are the result not only of external radiation, if any, but also of thermal radiation among spacecraft equipments.

Let us illustrate this point by examining the interaction between two parallel surfaces having the same area A (Figure 10-12). The surfaces are assumed to be opaque (τ = 0), have the absorbivity coefficients α1 and α2, and the reflectivity coefficients ρ1 and ρ2. It follows that ρ = 1 – α, α2q1 +

α2ρ1ρ2q1

+

α2ρ12ρ22q1 + .....

Plane 2

ρ1ρ2q1

q1

ρ2q1

ρ12ρ22q1 ρ1n ρ2n q1 ρ1ρ22q1

Plane 1

Figure 10-12 Thermal reflexion between two parallel surfaces.

The total heat energy absorbed by Plane 2 is: ∞

q1→ 2 = α 2 q1 ∑ (ρ1ρ 2 ) = n= 0

n

α 2 q1 , since 1 − ρ1ρ 2

Inserting ρ = 1 – α and q1 = ε1AσT14 gives us:

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∞

∑ (ρ ρ )

n

1

n= 0

2

=

1 for ρ < 1. 1 − ρ1ρ 2

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q1→ 2

ε1α 2 AσT1 = α1 + α 2 − α1α 2

4

q1→ 2 =

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Since the radiation between the two surfaces occurs on the same IR wavelength, we have α = ε, and therefore: 4

ε1ε 2 AσT1 AσT1 4 = ε e AσT1 = 1 1 ε1 + ε 2 − ε1ε 2 + −1 ε1 ε 2

4

and the net energy flow from Panel 1 to Panel 2 is: 4

4

)

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(

q12 = q1→ 2 − q 2→1 = ε e Aσ T1 − T2

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q 2→1 = ε e AσT2

hn

where εe is called the effective emissivity. Similarly, the radiation absorbed by Panel 1 from Panel 2 is:

(10.14)

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Consequently, the effective emissivity between two parallel planes is: 1 1 1 −1 = + ε e ε1 ε 2

(10.15)

It can be shown that the effective emissivity for two concentric spheres with radii r1 (inner) and r2 (outer) is:

 r  1 1  1 = +  − 1 1  ε e ε1  ε 2  r2 

2

(10.16)

and for two long, concentric cylinders:

 r  1 1  1 = +  − 1 1  ε e ε1  ε 2  r2 

r2

(10.17)

r2

r1

r1

Figure 10-13 Concentric spheres and concentric cylinders.

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Note, however, that Eq 10.16 and 10.17 are only applicable for radiation from the inner core to the outer shell, where all the emitted radiation is absorbed. We cannot use Eq 10.14 to calculate q21, since part of the radiation from surface 2 misses 1 and is reabsorbed by 2 on the opposite side. The next Section deals with that particular case. Other expressions for εe may be found in the literature for additional shapes, such as concentric cubes, pyramids, etc.

10.3.4. Local Spacecraft Thermal Modelling

View Factors

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10.3.4.1.

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In the general case, two parameters are needed to analyze the flow of thermal energy between two arbitrary surfaces, namely the view factor F and the radiation coupling factor R.

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The view factor Fi→j, or Fij for short, quantifies the fraction of radiation from surface i that is intercepted by surface j. Similarly, Fji symbolises the fraction in the opposite direction. It follows that 0 ≤ F ≤ 1. For example, in the case of a radiating sphere “1” contained inside an outer sphere “2”, F12 = 1, because all the radiation from the inner sphere impinges on the outer sphere (Figure 10-14). On the other hand, F21 ≠ 1 because some of the radiation from the inside wall of sphere 2 misses sphere 1 and falls on the opposite wall of sphere 2.

Generally speaking, if a concave surface is divided into two halves labelled “1” and “2”, then F12 adopts some value between 0 and 1, because part 1 can “see” part 2. A similarly divided flat or convex surface has F12 = 0, because no radiation flows from one part to the other.

1

2

1

F12 = 0

2

F12 = 0

1

2 1 2

0 < F12 1) with their golden appearance. Photographs of satellites often show blanket-covered satellites with cut-outs for mirror surfaces. MLI blankets are also used for local thermal shielding. The blankets consist of layers of aluminized Mylar or Kapton film. The layers are either separated 351

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by a Dacron mesh to avoid conduction through contact, or else they are crinkled such that contact only occurs sparingly. The outermost layer is coated with indium tin oxide (ITO) to make it electrically conductive; this is to prevent electrostatic discharge and associated EMI problems. To avoid hot or cold spots inside the spacecraft, most electronic equipment has black surfaces (high α and ε), since black facilitates the interchange of thermal energy. The rear sides of solar panels are painted or treated with low α/ε agents to keep the solar cells cool for maximum efficiency.

MLI White paint

White paint

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MLI

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OSR

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Figure 10-21 Examples of MLI, OSR and white paint cover.

Black paint

MLI

Black paint

Figure 10-22 Examples of MLI and black paint cover.

When using paints on surfaces exposed to sunlight, care must be taken to select products that do not change colour over time, particularly under the influence of the sun’s ultraviolet radiation. It is not uncommon for white surfaces on earth to turn yellow and even brown. The same discoloration is even more likely to happen in space where the UV radiation is unfiltered, causing an undesirable change in α and ε. Radio receivers onboard satellites must be kept at very low temperatures to prevent selfgenerated thermal noise from drowning out the weak incoming radio signal. Radio transmitters, on the other hand, generate a considerable amount of heat and must be installed away from the receivers. They are usually mounted inside an external wall whose space-facing surface is covered with mirrors. Heat pipes (Figure 10-23) are mobilized to spread concentrated equipment heat sources evenly across the inside of the radiator to improve the dissipation efficiency.

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Condensate

Satellite wall

olo gy .

Vapour

Heat source

Heat sink Figure 10-23 Heat pipe.

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Heat pipes work on the same principle as convection refrigerators. A medium (e.g. water or ammonia) is sealed inside a pipe and is kept near saturation pressure, allowing it to evaporate as well as condense within the intended temperature range. The heat source (e.g. a radio amplifier) is mounted at one end of the pipe and causes the medium to evaporate. The evaporation process removes thermal energy from the amplifier, thereby lowering its temperature. The internal vapour pressure forces the medium to travel to the other end of the pipe which terminates in a space radiator and therefore constitutes a heat sink. There the gas condenses into liquid form, giving off heat energy. Capillary action in a wick or in grooves along the inside wall send the liquid back to the heat source end of the tube, where it evaporates once more – and so forth. The heat pipe is an excellent dissipator of excess heat energy from inside the spacecraft to the outside. Figure 10-24 shows the range of temperatures that can be dissipated using different media. Lithium Sodium Cesium Mercury Water Ammonia Methane Nitrogen Neon Hydrogen Helium

-600 -400 -200

0

200

400 600 800 1000 1200 1400 1600 Temperature (°C)

Figure 10-24 Energy transport media and corresponding temperature ranges for heat pipes.

In practice, heat pipes are more often used to distribute heat from several dissipating units across the inside of a radiator panel (Figure 10-25), thereby improving the amount of thermal energy that can be dissipated per m2 of panel surrface. Sometimes, heat pipes also connect opposing radiator panels so as to equalize their temperatures.

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Figure 10-25 Interlinked heat pipes for spreading thermal energy.

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Heat pipes are either surface mounted or embedded in the radiator panel. The left picture in Figure 10-26 shows a heat-dissipating equipment mounted on a surface-mounted heat pipe, while the right figure illustrates the embedded approach. Surface mounting is less expensive in manufacturing and allows the routing of heat pipes to be changed in the course of satellite integration. Embedded heatpipes offer a shorter heat conduction path from the equipment to the radiator and is therefore more efficient.

Dissipating equipment

Dissipating equipment

Heat pipes

Honeycomb panel

Heat radiation

Heat pipes embedded in honeycomb panel

Heat radiation

Figure 10-26 Surface-mounted and embedded heat pipes.

Conduction is sometimes used to passively dissipate heat. The equipment to be cooled is mounted on a spreader, also known as a doubler, which is typically a flat, bevelled aluminimum plate (Figure 10-27) mounted on the inside of a radiator. With its wide footprint, the spreader ensures improved redistribution and eventual dissipation of the heat energy.

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Figure 10-27 Equipment mounted on a thermal spreader.

10.5.2. Active Thermal Control

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The most straightforward method of active heating is to attach small electric heaters to relevant units, notably hydrazine tanks and pipes to prevent the propellant from freezing solid. Adjacent radio amplifiers keep each other within acceptable operating temperatures as long as they are all operating. But if one of the amplifiers is idle (e.g. because it is a spare unit), it may be necessary to activate an electric substitution heater in its place.

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Active cooling devices include pumped-loop systems, cryogenic coolers and thermoelectric coolers, although these are more common onboard large manned spacecraft than on unmanned satellites. Thermistor

Heater

Heater

Figure 10-28 Propellant tank equipped with thermistor and electric heaters.

Another method of active heating as well as cooling is to use louvres whose external α or ε differs from that of the cavity floor (Figure 10-29). The equipment to be temperature controlled is attached to the underside of the cavity. The louvres are opened and closed automatically either by a bimetal spring arrangement, or by a thermostat acting in a feedback loop, to ensure that the correct equipment temperature is maintained at all times. When used to regulate energy dissipation, the effective emissivity εe = f(ε1, ε2, θ). Louvres are then used on the shaded side of the satellite, since direct solar illumination would make the thermal performance erratic due to partial shadowing inside the cavity. Conversely, louvres may be used to regulate the amount of solar absorption. The main 355

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advantage of louvres over electric heaters is that they draw little or no current, but a price must be paid in terms of added mass.

olo gy .

ε1 θ

ε2

hn

Equipment

ec

Figure 10-29 Louvres for regulating thermal dissipation.

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We mentioned in the beginning of this chapter that convection is rarely used in spacecraft applications, with the exception of heat pipes. Convection is actually a powerful means of heat removal on a large scale, and is therefore widely used onboard large structures such as the Space Shuttle and the International Space Station. Energy from a heat-dissipating unit is conferred to a liquid, which is subsequently pumped to a radiator, where it is ejected into space. In the former Soviet Union, many communications satellites used a similar technique. The dissipating unit was placed inside a gas-filled chamber. An electric fan made the gas circulate past the unit and across a heat exchanger, from where the heat was transported to a suitable space radiator. Spherical or cylindrical pressure vessels were needed to contain the pressurized gas; hence the lumpy appearance of many Soviet spacecraft.

10.5.3. Layout Summary

Figure 10-30 shows a cross-section of a typical geostationary satellite body with various structural and thermal elements outlined. OSR

Aluminium honeycomb

Spreader

Heat pipe

CFRP

Black surfaces for internal radiative heat transfer

CFRP

White paint for minimum thermo-elastic deformation

Heater

MLI

Aluminium honeycomb

OSR

Figure 10-30 Typical satellite cross-section.

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White paint for minimum thermo-elastic deformation

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10.6. Solved Problems See Appendix C for solutions.

olo gy .

10.6.1 A cylindrical satellite with no internal heat dissipation has a radius r = 1 m and a height h = 1 m. At the beginning of the mission it is orientated such that one end surface faces the sun, while at the end of the mission the cylindrical wall faces the sun. All the surfaces have an absorbivity coefficient α = 0.6 and an emissivity coefficient ε = 0.4. Calculate the satellite’s equilibrium temperature TBOL in sunlight at the beginning of life and TEOL at the end of life. The solar power density S = 1367 W/m2, and the StefanBoltzmann constant σ = 5.67 ·10-8 W/m2K4.

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BOL:

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EOL:

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10.6.2 Returning to Problem 10.6.1, how much internal energy dissipation qs is required at EOL to bring the satellite’s temperature up to +20°C?

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10.6.3 A spherical satellite with radius r = 1 m, mass m = 100 kg and specific heat c = 900 J/kg K is launched into a noon-midnight orbit at 200 km altitude. The satellite’s surface coefficients α = 0.8 and ε = 0.3. A transmitter inside the satellite dissipates qs = 100 W. Calculate its temperature at each eclipse entry and exit during four successive orbits. The initial temperature at the first eclipse entry T0 = 10°C.

10.6.4 A satellite made of aluminium has reached a temperature T0 = 20°C when it enters the earth’s shadow. The eclipse duration te = 35 minutes. The satellite is a sphere with the radius r = 50 cm, the mass m = 100 kg, and the emissivity ε = 0,3. Its only internal heat source is a radio transmitter whose RF output power is 300 W, and whose efficiency η = 66,7%. Calculate the satellite’s temperature when exiting the earth’s shadow.

10.6.5 The six sides of a hollow, rectangular box are sized and numbered as shown in the figure. Find the view factor F15, i.e. from the bottom panel to the top. (Note: Panel 4 is at the front; panel 6 is at the rear.) 6

5

b=3m

2

4

3

1

a=1m

c=6m

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11. Launch Vehicle Selection

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Payload section

3rd stage

Oxidizer

2nd stage

Fuel

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The complexities of detailed launch vehicle design do not fall within the scope of the present book. However, the design of satellites is intimately linked to that of launchers in terms of physical compatibility and orbital parameters, so it is worth spending a chapter on the general layout and performance of launch vehicles. In order for the satellite design work to proceed smoothly, it is important that the satellite manufacturer and his customer agree at an early stage about the preferred vehicle and one or maybe two back-ups. The agreement will be based on the evaluation of certain criteria which are explained in the following.

Tandem adapter & heatshield

fT

Rockets seem to possess a romantic aura that satellites lack, even though the satellite is the sole reason for building and launching the rocket in the first place. Several well-heeled private individuals in the West have entered the costly and risky business of building launch vehicles, and few have succeeded. Most successful launchers are designed and built by large corporations with decades of experience in missile technology.

olo gy .

11.1. Introduction

Strap-on boosters

1st stage

Most expendable launch vehicles are constructed along similar lines. The core vehicle consists of 2 – 4 rocket stages. The first stage is always the most powerful to overcome the earth’s gravity, and is frequently assisted by a set of 2 – 9 strap-on boosters. These boosters are jettisoned after burnout while the first stage is still operating. The last stage is the most efficient in terms of propellant consumption, so as to maximize the kinetic energy of the orbit. In line with this philosophy, the lower stages typically burn UDMH or kerosene because of their high thrust yield. The thrust is 1.5 – 3 times the lift-off weight of the overall vehicle. The upper stage is frequently designed around cryogenic H2 / O2, whose thrust is insufficient to even lift the stage itself off the ground, but whose high Isp (≈ 450 s) ensures adequate powered flight efficiency in weightlessness. The total propellant mass represents around 80% of the launch vehicle’s lift-off mass. Launcher people refer to the satellite as the launch vehicle's “payload,” since it is literally the load that pays for the launch (not to be confused with the satellite's own payload). The 359

11 / LaunchVehicle Selection

olo gy .

payload is housed inside a protective heatshield, sometimes called a fairing or a shroud. By injecting clean, dry and temperate air inside the heatshield, the satellite is maintained in a clean-room environment up until launch. During the ascent, the shield protects the satellite from damage due to excessive aerodynamic heating and dynamic pressure. It is made up of two bullet-shaped half-shells which split open and separate from the rocket as soon as the outside conditions permit, usually during the second stage burn.

Co p by yri Lu ght leå © Un 201 ive 4, rsi ty o

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From the satellite’s point of view, one of the most important rocket elements is the separation system (Figure 11-1). The system forms part of the interface adapter between the vehicle and the satellite. The adapter consists of a hollow, truncated cone structure with flanges at both ends. The bottom flange is bolted to the upper stage of the launch vehicle, while the upper flange is mated to a similar flange on the satellite. These two flanges are held together by means of a Marman clamp band. (In everyday plumbing, we are familiar with Marman clamps as those metal screw collars that tighten a hose around a pipe with the help of a screwdriver.) At the moment of satellite injection, the clamp band is released by means of explosive bolts, and the satellite is pushed into space by compressed springs.

Release

Satellite flange

}

Interface adapter

Clamp band

Launcher flange

Figure 11-1 Conical interface adapter with Marman separation clamp.

The conical adapter also serves as a conduit for power and signal cables between the rocket and the satellite. The signal cables carry telemetry and command signals necessary to monitor the satellite’s health during launch, and to configure the satellite correctly. A few large launch vehicles come equipped with tandem adapters (“tandem” meaning “one behind the other”) or other adapter arrangements for accommodating several satellites at once. The satellites are separated in a timed sequence to avoid collision.

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11.2. Rocket Engine Architecture

Fuel

Turbine

olo gy .

Figure 11-2 shows the typical layout of a chemical bipropellant rocket engine.

Gas generator Oxidizer

Pump

ec

Pump

hn

Igniter

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Nozzle

fT

Combustion chamber

Figure 11-2 Bipropellant rocket engine.

In the example shown, the primary elements are the two propellant tanks and the thruster. One tank holds the fuel, the other contains the oxidizer. Pumps withdraw propellant from the tanks and feed it under pressure to the combustion chamber. Before entering the chamber, the propellant passes through injectors which diffuse the liquids to a fine mist. If combustion occurs spontaneously, the propellant is said to be hypergolic. Nonhypergolic propellants have to be lit by an igniter. The fuel and oxidizer pumps are driven by a turbine, which runs on the same propellant as the rocket. Some engines have both pumps mounted on a common shaft which is either coupled directly to the turbine drive shaft or connected via a gearbox. Other engines employ independent pump drives for the fuel and the oxidizer. Combustion takes place in a gas generator which is a miniature version of the main combustion chamber. A charge of solid propellant inside the gas generator is ignited to start up the turbine. Exhaust from the turbine is either released through a duct into the atmosphere (open cycle) or fed to the thrust chamber where it participates in the main combustion (closed cycle). Closed-cycle operation improves the efficiency of the engine but complicates the design. Whereas liquid-propellant engines can be stopped and re-started in flight, solid-propellant motors burn continuously until all the propellant is consumed. The latter are used primarily where a high thrust level and simplicity are more important factors than operational flexibility, as in certain strap-on boosters.

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11.3. Ascent and Injection

olo gy .

A typical launch vehicle’s journey from the launch pad to the point of satellite injection is illustrated in Figure 11-3. 3rd stage burnout & satellite injection into GTO.

4000 km

300 km 200 km 100 km 6000 km 0 km

hn

2000 km

0 km

ec

2nd stage burnout Heatshield jettison 1st stage burnout

Figure 11-3 Launch vehicle ascent trajectory.

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During the 1st stage burn, the launch vehicle rises vertically to pass through the dense atmosphere as quickly as possible. Thereafter, the vehicle gradually pitches over to a horizontal trajectory through a combination of active guidance and passive gravity turn. The latter results naturally due to the earth’s gravitational pull. It is a trajectory optimization method that uses gravity to steer the vehicle onto its desired trajectory with minimal angle of attack, so as to conserve propellant and minimize aerodynamic stress on the structure. Figure 11-4 shows a typical acceleration profile experienced by the satellite during the ascent. 6

Acceleration (g)

5 4

3 2 1

1

2

3

0

0

200

400

600

800

1000

1200

1400

Flight duration (sek)

Figure 11-4 Accelerations during ascent.

Given that rocket engines are designed for constant propellant mass expulsion, and hence constant thrust, one would expect the acceleration to be constant as well according to Newton’s law F = m·a. There are two reasons why the acceleration actually increases 362

11 / Launch Vehicle Selection

F = Vedm/dt + Ae (pe – pa)

olo gy .

during the burn of each stage. The main reason is that the rocket mass m decreases with time as propellant is consumed, so if F is to remain constant, the acceleration must increase. A secondary reason is that F is not truly constant despite the constant propellant mass flow dm/dt. The equation for rocket engine thrust is: (11.1)

where Ve is the velocity of propellant exhaust at the nozzle exit;

hn

Ae is the surface area of the nozzle exit; pe is the combustion pressure at the nozzle exit; and pa is the ambient pressure.

pa

Co p by yri Lu ght leå © Un 201 ive 4, rsi ty o

pe

fT

Ve

ec

Eq 11.1 states that the pressure difference between the inside and the outside of the nozzle exit contributes to the thrust – see Figure 11-5. Since the ambient pressure pa decreases with altitude (i.e. the pressure differential increases), the thrust F experiences a corresponding increase.

Figure 11-5 Engine exit velocity and pressure differential.

It also follows from the definition of specific impulse (Eq 6.1) that Isp improves with altitude, since: I sp =

F dt g dm

(11.2)

For a large launch vehicle, the ascent from lift-off until satellite separation lasts anywhere from 20 minutes to 9 hours, depending on whether the injection is direct, or the upper stage performs multiple orbit tailoring manoeuvres to achieve a particular set of COE. Injection of the satellite into earth orbit usually occurs within seconds or minutes after the final burn of the upper stage. The relative separation velocity between the satellite and the rocket is typically 0.5 m/s. Any force asymmetry between the separation springs will cause the satellite to tumble randomly in orbit. This may be acceptable to some satellites whose attitude control subsystems are designed for automatic attitude recovery. Other satellites prefer a clearer definition of injection attitude by being temporarily spinstabilized. Certain launch vehicles therefore come equipped with a spin-up facility which is activated after upper stage burnout but before satellite separation.

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11.4. Choice of Launch Sites

olo gy .

Satellite builders and operators do not have a choice of launch sites, because the site is predefined by the chosen launch service. The present Section therefore deals with how the launch sites were chosen by the launch service providers. Figure 11-6 shows the locations of major launch sites in the world. Some of the most active sites are Cape Canaveral and Vandenberg in the USA, Kourou in French Guiana, Baikonur in Kazakhstan, and Plesetsk in Russia.

hn

Plesetsk

Svobodny

Kourou

Baikonur

Xichang

Sriharikota

Co p by yri Lu ght leå © Un 201 ive 4, rsi ty o

Sea Launch

Cape Canaveral

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Vandenberg

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Jiuquan Taiyuan Tanegashima

Figure 11-6 Geographical locations of major launch sites.

The sites have been chosen on the basis of several criteria, notably: a) Proximity to the equator in order to achieve full orbital inclination flexibility, and to take maximum advantage of the earth’s eastward surface velocity. b) Absence of populated areas down-range from the launch site, for safety reasons. c) Political stability and autonomy to mitigate the consequences of popular unrest or injunctions by foreign jurisdictions. d) A stable physical environment devoid of earthquakes, hurricanes, flooding, etc.

These criteria can seldom be met in their entirety. For example, Cape Canaveral may be situated in the southernmost corner of the USA, but at latitude 28.5°N it is still a long way from the equator. Vandenberg is further north still, which is one of the reasons why that site is used exclusively for high-inclination launches. A similar situation exists in Russia in regard to Baikonur and Plesetsk. Baikonur has the added disadvantage of being located in another country, and rockets launched from either site overfly (sparsely) populated areas. Only Kourou satisfies all the criteria (a) through (d), recalling that politically and economically, French Guiana forms an integral part of France.

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Other solutions have been contemplated to satisfy all four criteria. For example, Sea Launch has succeeded by launching from a converted oil platform in the middle of the Pacific Ocean.

olo gy .

Let us examine criterion (a) more closely. Looking at Figure 11-7, it is evident that the inclination of the satellite orbit can never be lower than the latitude of the site from which the launch takes place – at least not without the rocket or the satellite performing complex and propellant-consuming dog-leg manoeuvres (see Section 5.4.5.1).

ec

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For example, the minimum achievable inclination following a direct easterly injection from Cape Canaveral is 28.5°. GEO satellites launched on Atlas or Delta vehicles must therefore use their apogee engines to not only circularize the GTO, but also remove a substantial inclination. The latter manoeuvre costs propellant which could otherwise have been better used to extend the satellite’s lifetime, or else be traded for a higher dry mass. A launch by Ariane from Kourou or by Sea Launch from their equatorial ocean platform avoids this problem.

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N

λ

i3 =λ i2

i1

180°-i3

180°-i1

180°-i2

Figure 11-7 Orbital inclination i as a function of launch site latitude λ in the case of a direct launch injection.

In practice, population clusters down-range prevent most launch sites from exploiting the full range of direct injection inclination possibilities. For example, Kourou is restricted to an inclination interval of 5.2 – 100.5 degrees. The corresponding sector for Cape Canaveral is 29 – 57 degrees, and for Vandenberg it is 56 – 104 degrees (Figure 11-8).

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57° 100.5° 90°

5.2°

39°

104°

olo gy .

28.5°

56°

90°

Cape Canaveral

Vandenberg

hn

Kourou

Figure 11-8 Allowable launch sectors and their corresponding orbital inclination limits.

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11.5. Launch Vehicle Profiles

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Launch vehicles come in every size and shape, ranging from modified ICBMs and winged “cruise missiles” to eardrum-shattering ballistic behemoths. Some are manufactured in small numbers, while others are mass-produced in modular configurations to best suit each particular satellite customer. The following inventory is not exhaustive, but has been selected to illustrate the multitude of existing design concepts, some of which are more imaginative than others. The profiles highlight vehicles available on the commercial market. The lift-off mass (in metric tons) is given for each vehicle to offer an impression of its size. The inventory is limited to vehicles that are fully operational at the time of writing. Half a dozen others are still in development; the most promising of these are included in the performance tables in Figure 11-12. More detailed launch vehicle and launch site information can be found via the Internet links contained in Annex F. 11.5.1. Small Vehicles

Launch vehicles designed to lift satellites weighing less than 2,500 kg to LEO are classified as small. Several of these are former ICBMs, which have been converted to space launchers as part of arms limitation treaties. During the 1990s, when half a dozen satellite constellations were being planned to serve the mobile communications market from LEO, business analysts foresaw a bright future for small vehicles. Meanwhile, most of these constellations are either faltering or have disappeared altogether. Those that entered service (notably Iridium and Globalstar) were usually clustered onboard medium-sized launch vehicles, leaving the multitude of small launchers to compete for less lucrative payloads. Cosmos 65M (Ukraine): This two-stage Ukrainian vehicle, also known as Cosmos 3M, dates back to 1967. The second stage is restartable and steerable for improved orbital

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targetting. Both stages are propelled by UDMH in combination with nitric acid. Lift-off mass: 109 tons.

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Pegasus XL (USA): Pegasus is one of the few purpose-built space launchers in the small vehicle class, most of the others having military missile origins. It is also the only winged vehicle. It is attached to the belly of a modified passenger jet and is lifted up to cruising altitude before being jettisoned and ignited. The three rocket stages use solid propellants. Mass at ignition: 24 tons.

Rockot (Russia): A three-stage vehicle derived from the ICBM code-named SS-19 in the West. All three stages use UDMH for fuel and N2O4 for oxidizer. Lift-off mass: 107 tons.

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Start-1 (Russia): This modified ICBM (SL-19) with its 4 solid-propellant stages offers a real fireworks display when launched from its transporter on wheels. Lift-off mass: 47 tons.

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Vega (Europe): Together with Ariane 5 and Soyuz, the new 4-stage Vega vehicle completes Europe's family of launchers for satellites weighing 1 - 10 tons at lift-off from Kourou. Vega typically launches satellites weighing 1.5 tons to a 700 km orbit. The first three stages employ solid propellant. The 4th stage uses liquid propellant and is restartable, so that multiple satellites can be jettisoned into different orbits. Lift-off mass: 137 tons. 11.5.2. Medium Vehicles

Mid-sized vehicles are those with a lift capacity between 2,500 kg and 10,000 kg to LEO. They are also capable of placing GEO satellites weighing less than 4,000 kg in GTO. Antares (USA): Launched from Wallops Island on the U.S. East Coast, this new vehicle is highly modular, featuring a liquid propellant first stage, a choice of various solidpropellant second stages, and an optional bi-propellant third stage. It is capable of launching satellites weighing around 5,000 kg into LEO or 1,800 kg into GTO, depending on the chosen stage mix. Lift-off mass: 240 tons. Delta II – 7900 series (USA): Like the Atlas vehicle, the Delta II has been a workhorse in the U.S. launcher stable for many years and has undergone many modifications. It is unclear whether production of this vehicle will continue. The 7900 series has three core stages and up to 9 strap-on boosters, 6 of which are fired and jettisoned before the remaining three. The first stage operates on a kerosene / liquid O2 mix, while the second stage uses Aerozine-50 in conjunction with N2O4. The third stage as well as the boosters burn solid propellants. Lift-off mass with 9 boosters: 232 tons. Dnepr (Russia): This is yet another Russian former ICBM (SS-18) which has been converted to a civilian space launcher. The three rocket stages use UDMH / N2O4. Lift-off mass: 268 tons. Soyuz ST (Russia): The history of the Soyuz vehicle goes back to the legendary ICBM called R7 “Semyorka”, which launched Sputnik in 1957. Since then, approximately 1700 Soyuz rockets have been launched in a multitude of configurations, albeit not without accidents. The vehicle remains Russia’s workhorse for transporting cosmonauts and

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provisions to the International Space Station. A Russian-French company called Starsem markets it commercially in the West. The two core stages as well as the four strap-on boosters use kerosene in combination with liquid O2. Lift-off mass: 310 tons. The European equatorial launch site at Kourou has been prepared for launching an almost identical version of Soyuz, thereby enhancing its payload performance significantly compared to a Baikonur or Plesetsk launch.

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Tsiklon-4 (Ukraine): The maiden flight of this modernized version of the earlier successful Tsiklon series of vehicles is planned for 2015 from the Alcantara launch site in Northern Brazil, taking advantage of the site's proximity to the equator. Launches are also planned from Baikonur and Plesetsk. All three stages use UDMH / N2O4. The vehicle is capable of carrying satellites weighing 5,500 kg to a 500 km LEO and 1,700 kg to GTO. Lift-off mass: 193 tons.

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11.5.3. Heavy Vehicles

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Heavy vehicles are capable of injecting satellites weighing more than 10,000 kg into LEO, or more than 4,000 kg into GTO.

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Angara-A3, A5 (Russia): A highly modular design that has been under development for many years and has yet to fly. It is intended as a future replacement for several existing vehicles, notably Cosmos-3M, Rockot, Zenith and Proton. Ariane 5 (Europe): In 2003, the last of the world-leading Ariane 4 vehicles was launched, following a policy decision in Europe to henceforth rely entirely on the more powerful Ariane 5. This drastic departure from the proven to the unproven is reminiscent of NASA’s premature decision in the mid-1980s to forgo the highly reliable Atlas and Delta vehicles in favour of the Space Shuttle. Ariane 5 is a two-stage vehicle flanked by two solid-propellant strap-on boosters. The first core stage uses cryogenic H2 / O2, while the second stage employs MMH together with N2O4. Lift-off mass: 746 tons. Atlas V-500 series (USA): Version 551 is the most powerful of the new Atlas V-500 series vehicles and features 5 solid-propellant strap-on boosters. The first digit shows the diameter of the fairing in meters; the second digit indicates the number of boosters; and the third digit is the number of engines of the Centaur upper stage. The core vehicle consists of two stages, the first of which is propelled by a Russian kerosene / O2 engine. The second stage (Centaur) uses cryogenic H2 / O2. Lift-off mass of Version 551: 547 tons. Delta IV Heavy (USA): The development of Boeing’s Delta IV series has run in parallel with that of Lockheed Martin’s Atlas V, and has been the latter’s most important competitor for US Government launch business. Paradoxically, or perhaps naturally, the two manufacturers have joined forces to form United Launch Alliance. Delta IV Heavy is the most powerful version of the Delta IV series. The vehicle features two core stages and two strap-on boosters, all of which use cryogenic H2 / O2. Lift-off mass: 733 tons. Falcon 9 (USA): If this new, two-stage, privately financed vehicle meets its stated performance, reliability and price targets, it is likely to revolutionize the commercial launcher market, mainly because its stated price is at least 30 percent below that of its closest competitors. The economies are achieved, in part, by performing the design,

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development and manufacturing in-house at SpaceX, rather than farming it out to dozens of subcontractors. The first stage is made up of a cluster of Merlin engines, while the upper stage uses a single engine of the same type. All the engines use kerosene and liquid oxygen. Lift-off mass: 333 tons (version v1.0) or 505 tons (v1.1). Version v1.1 is for launching satellites into GTO and is designed to allow soft ocean landing of the first stage for recovery and re-use. Land Launch (Russia): This vehicle is a land-based version of Sea Launch (see below) and is launched from Baikonur.

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Long March 3B (China): This member the extended Long March family takes its first two UDMH / N2O4 stages from the earlier Long March 2C vehicle, and comes with an added hi-tech cryogenic H2 / O2 third stage. Lift-off mass: 204 tons.

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Proton (Russia): The four-stage Proton is the only operational launch vehicle that routinely takes payloads (< 3 tons) directly into GEO, thereby obviating the need for the satellites to carry apogee engines. The two and three stage versions are also used for lifting heavy space station elements into LEO. The vehicle exists in two basic versions – Proton K with a Block DM fourth stage and the more powerful Proton M with a Breeze fourth stage. All four rocket stages use a combination of UDMH and N2O4. Lift-off mass: 713 tons.

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Zenit 2 (Ukraine): This two-stage launch vehicle was the first attempt by the USSR to develop a dedicated space launcher, in contrast to the ICBM heritage of the other rockets in the country’s stable. In the event, the Zenit development became long and difficult, and matters were not helped by the slow disintegration of the USSR itself. Finally, in 1985, the maiden flight took place from the Baikonur Cosmodrome. The Zenith vehicle constitutes the core of Sea Launch and Land Launch. Both rocket stages use kerosene / liquid O2. Lift-off mass: 459 tons. Sea Launch (Misc.): During the 1980s, several launch service providers came up with creative solutions to finding launch sites near the equator. One of these consisted of a modified supertanker that would serve as both rocket transporter and equatorial launch pad. Sea Launch is an evolution of that idea, and a model of international cooperation. The Ukraine provides the two core stages of the Zenit 2 launcher, Russia adds the Block DM third stage, and Norway has supplied the launch pad (a self-propelled, converted oil exploration platform) as well as the command vessel. A launch campaign begins at the Sea Launch home port in San Diego, where the satellite is fuelled and integrated with the launch vehicle onboard the command ship. The assembly is stored horizontally in a hangar onboard the platform which travels to a point on the equator in the Pacific Ocean. Upon arrival, the rocket is erected, fuelled and launched. All three rocket stages operate on a kerosene / liquid O2 combination. Lift-off mass: 471 tons.

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Figure 11-9 Sea Launch being readied for launch from its floating platform.

Historical note: The launch of the last American Space Shuttle took place in 2011. During the heyday of the space race between the superpowers, the USSR decided to compete with the USA by building its own space shuttle. The Soviet version was called Buran and flew only once, unmanned, by remote control. It was subsequently mothballed for cost reasons as the Soviet Union disintegrated in 1991. The stored orbiter was finally destroyed in 2002 when the hangar that housed it in Baikonur collapsed.

Though the Buran orbiter was almost identical to that of the Space Shuttle, the two launch systems featured some conceptual differences. While the American designers chose to install the main engines in the orbiter, the Russians opted to mount them at the bottom of the external tank. In fact, that tank with its integral engines constituted the immensely powerful Energia launch vehicle.

Space Shuttle and Buran.

The advantage of the Soviet solution was that Energia could be used independently as a heavy launch vehicle, for which Buran was only one of many possible payloads. The disadvantage was that the very costly engines could not be recovered after launch. By contrast, the American approach allowed the engines to be reused during follow-on missions, but the Space Shuttle configuration lacked the versatility of the Buran/Energia combination.

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11.6. Launch Vehicle Performance 11.6.1. Overview

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The choice of launch vehicle is of course a major preoccupation among satellite builders, since the spacecraft design has to fit the rocket in terms of physical accommodation, launch environment and orbital performance, not to mention the budget.

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A launch vehicle’s performance is measured by how big a satellite (in terms of mass and volume) it is able to lift into a given orbit. The performance depends on a number of parameters, notably the propulsive power of the rocket engines, the rocket’s lift-off mass, the number of stages, the chosen orbital parameters, and the latitude of the launch site. We will quantify these parameters in the following, but let us first take an overview of the key performance parameters of all major launch vehicles currently in operation or in the final stages of development.

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Most launch service providers publish the performance parameters in user manuals which are readily available on their Internet sites (see Appendix F). The format of their performance diagrams varies but, one way or another, they convey the information illustrated in Figure 11-10. The family of curves indicates the perigee altitude in ascending order, as per the arrow. The higher the desired perigee, the more effort is demanded of the launcher, and a price must be paid by lowering the admissible satellite payload mass. The same argument applies to the apogee altitude. Not shown in Figure 11-10 is the performance loss if the rocket is also required to perform a dog-leg manoeuvre to alter the orbital inclination (e.g. to go from an inclined GTO to a GEO with zero inclination). In fact, most launch service providers break down their performance diagrams into several sub-diagrams: one each for LEO, SSO, GTO, HEO, another for deep space trajectories, and yet another for direct injection into GEO, with altitude and inclination shown as independent parameters. Payload mass (kg)

Perigee altitude (km)

Circular orbit

Standard GTO

Apogee altitude (km)

Figure 11-10 Typical launch vehicle performance diagram.

Providers of launch services to GTO often present payload lift capacities using a perigee in the 200 – 400 km range and the apogee at GEO altitude, with i = λ as before. These may be the optimum conditions for the launch vehicle’s performance, but not for the satellite, which must perform the necessary dog-leg manoeuvre from i = λ to i = 0 deg 371

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either on its own or with the help of the rocket. In the first scenario, the extra apogee engine propellant to be carried for the purpose raises the satellite’s lift-off mass, necessitating a corresponding reduction in its useful mass. In the second scenario, the rocket must carry the extra propellant, and its payload lift capacity is reduced accordingly.

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Figure 11-12 summarizes the payload lift performance of all GTO and direct-to-GEO launchers available today or in the near future. To make their GTO performance data comparable from the satellite builder’s viewpoint, the performance has been normalized to the “∆V = 1500 m/s point.” This happens to be the ∆V required of an apogee engine to eliminate the satellite’s inclination of 6 deg and to circularize the orbit when launched by Ariane 5. Clearly, if a satellite is to be compatible with several launch vehicles, and if one of these is Ariane 5, then it will want to use the same apogee engine on all of them. Therefore, the satellite owner is likely to request those other launchers to achieve the ∆V = 1500 m/s point before ejecting the satellite into GTO. This they can usually do, but since their launch site latitudes λ – and hence their favoured GTO inclination – is nearly always higher than 6 deg, they will have to perform a dog-leg manoeuvre to reach the ∆V = 1500 m/s point for the satellite, losing lift performance in the process. This is why the actual payload lift capacity may be lower than reported in their user manuals, and it is this normalized value that is shown in Figure 11-12.

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Obviously, other ∆V points may be more optimal for satellites whose inventory of compatible GTO launchers does not include Ariane 5. In fact, since the achievable ∆V is a function of the mass carried, according to Eq 6.3, it can be shown that less propellant is needed for the satellite to perform the dog-leg manoeuvre on its own, rather than having the rocket’s upper stage do it. This is so because the apogee engine only has to carry the satellite’s mass, whereas the upper stage must carry both the satellite and its own mass. Available volume under the heatshield is another important parameter to be considered when selecting a launch vehicle. Many vehicles come with a choice of heatshield lengths and diameters. The satellite must fit within the usable volume (dotted contour in Figure 11-11). The margin between the physical and usable volumes allows the satellite and the heatshield to flex under the influence of launch vibrations. Launch service user manuals contain detailed drawings of applicable heatshields and the usable volumes

Figure 11-11 Usable volume within a heatshield.

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Atlas V 551 Atlas V 541 Atlas V 531

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Atlas V 521 Atlas V 511 Atlas V 501 Atlas V 431 Atlas V 421 Delta IV Medium (5,4)

Atlas V 411

Delta IV Medium (5,2)

Atlas V 401

Delta IV Medium (4,2)

0

1,000

2,000

3,000

4,000

5,000

6,000

7,000

hn

Payload mass (kg) to ∆ V=1500 m/s point

Delta IV Medium Delta IV Heavy Delta II 7925 Delta II 7920

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Proton-M; KVRB

Delta II 7420

Proton-M; Breeze M

Delta II 7320

Angara A5; KVRB

2,000

4,000

6,000

8,000

10,000 12,000 Angara A5; Breeze-M

Payload mass (kg) to ∆ V=1500 m/s point

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0

Angara A3; Breeze-M

0

Soyuz ST; Fregat

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Kourou launch

Soyuz ST; Fregat

1,000 2,000 3,000 4,000 5,000 6,000 7,000

Payload mass (kg) to ∆ V=1500 m/s point

Baikonur launch

Land Launch 3SLB

Falcon 9 S9

Sea Launch 3SL

Falcon 9 S5

Ariane 5; EPS-V

Falcon 9

Ariane 5; ECA

Falcon 5

0

2,000

4,000

6,000

8,000

10,000

Payload mass (kg) to ∆ V=1500 m/s point

0

2,000

4,000

6,000

8,000

10,000

Payload m ass (kg) to ∆ V=1500 m /s point

PSLV

Long March CZ-3C Long March CZ-3B Long March CZ-3A

Long March CZ-3 H-II A202

GSLV Mk III

0

1,000

2,000

3,000

4,000

5,000

Payload mass (kg) to ∆ V=1500 m/s point

Figure 11-12 Lift capacity of heavy launch vehicles to GTO (lower bar) and directly to GEO (upper bar).

11.6.2. GTO or Direct Injection into GEO

Proton is capable of injecting satellites directly into GEO. As indicated in Figure 11-12, several other launch vehicles offer the same capability, at least in their user manuals. In theory, this is a great service, because the satellite can dispense with the mass, cost and

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risk associated with apogee engines. However, there are three disadvantages with direct injection:

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Without an apogee engine, the satellite cannot fly on other launch vehicles that do not provide direct injection capability, thereby depriving itself of some back-up launchers. It is true that Atlas V, Delta IV, Sea Launch and Long March CZ-3B also offer direct injection, but they have little or no track record in doing so. Soyuz and Land Launch also purport to offer direct injection, but these vehicles have neither the lift capability nor the track record expected by Proton-class satellites (i.e. 2000 kg and above). As for Angara, the development has been very slow, and it remains to be seen when it becomes operational.

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As mentioned earlier, it is less efficient to have the rocket’s upper stage perform the necessary dog-leg manoeuvre than using the apogee engine. The attendant loss of launch vehicle performance limits the satellite’s useful mass rather severely, even if the apogee engine has been removed.

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For a given satellite design, flying with an apogee engine on one rocket and removing the engine when using a different rocket is far from trivial. The tanks and the plumbing must be removed or re-installed. The volume that is freed up after the tanks are removed is probably wasted, because it is difficult to mount equipment on the cylindrical walls of the central tube (Figure 9-2). The OBDH computer must be reprogrammed to account for the altered dynamic behaviour of the satellite due to the changed mass distribution. Lastly, the satellite will have been qualification-tested (Chapter 14) based on a certain mass distribution. If the distribution changes dramatically due to the presence or absence of the apogee engine, the satellite will need to be re-qualified – a very expensive and timeconsuming proposition. For these reasons, the choice between GTO and direct-to-GEO injection requires careful analysis at the earliest design stages of a geostationary satellite. 11.6.3. Computation of Performance

To understand how launch vehicle manufacturers arrive at the above performance curves, let us analyze the velocities involved (Figure 11-13).

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11 / Launch Vehicle Selection Vkin

∆V

Az v (North)

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η

w

v Vpot Vearth

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u

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Figure 11-13 Definition of ascent velocity components in a topocentric uvw coordinate system centered on the launch site.

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Before going into detail, we notice immediately that the following vector addition applies: Vearth + ∆V = Vpot + Vkin

or:

∆V = Vpot + Vkin -Vearth

(11.3)

Here, ∆V is the total velocity increase that the launch vehicle has to produce from lift-off until burnout of the last stage. It remains to define the three vector components Vpot , Vkin and Vearth. The launch vehicle has to perform two separate tasks: overcome earth’s gravity, and acquire the requisite orbital velocity. The first task amounts to building up potential energy, while the second task involves acquisition of kinetic energy. From classical mechanics we have the expression for potential energy: Epot = mgh

(11.4)

where m is the mass of the object, g is the gravitational acceleration, and h is the height above the earth’s surface. In our case the matter is complicated by the fact that the height is so substantial that the decrease of g must be taken into account. According to Eq 2.16: E pot = mg (r )r =

and therefore: g (r ) =

µ r2

mµ r

(11.5)

(11.6)

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11 / LaunchVehicle Selection where µ is the earth’s gravitational parameter = 398,601 km3/s2, and r is the radial distance from the earth’s centre.

mV pot

2

2

=

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A rocket may gain potential energy by lifting itself against gravity to a height h. To do so, it must generate kinetic energy, which is converted to potential energy. The launch vehicle therefore has to produce enough kinetic energy to bridge the gap in potential energy between the earth’s surface (r = R) and the orbital height (r = R + h):

mµ mµ − R R+h

2µh R ( R + h)

(11.7)

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V pot =

hn

or:

2µh w R ( R + h)

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Vpot =

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Since the force of gravity acts solely along the w-axis, we may write the velocity as a vector:

or:

Vpot

 0 = V pot 0 1

(11.8)

with w being the unit vector towards zenith, as per Figure 11-13. Having built up the potential energy necessary to overcome earth’s gravity, the launch vehicle must now acquire kinetic energy to maintain an orbit: Ekin = ½ mVkin2

(11.9)

According to Eq 2.13:

1 2 1  2 Vkin = µ  −  = µ −  r a R+h a

(11.10)

with h and the semimajor axis a being elements of the specified orbit. In order for us to write Vkin in vector form, the mission analysts also have to specify the orbital inclination i and the flight path angle η at the burnout point. The inclination gives us the azimuth Az (Figure 11-14) of Vkin from the cosine law of spherical triangles:

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Launch

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Az site

λ

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i

Figure 11-14 Relationship between launch azimuth Az and inclination i.

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 cos i  Az = sin −1    cos λ 

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cos i = -cos 90° cos Az + sin 90° sin Az cos λ

(11.11)

where λ is the latitude of the launch site. Note that Az = 0 deg when injecting the satellite into a polar orbit (since i = 90°), and Az = 90° deg when injecting straight East (since i = λ according to Section 11.4 and Figure 11-7). Eq 11.11 applies to a scenario where the launch site is near the ascending node, as shown in Figure 11-14. If the launch site is near the descending node, Eq 11.11 is modified as follow:  cos i  Az = 180° − sin −1    cos λ 

(11.12)

If also the flight path angle η is specified, we may now express Vkin as a vector (Figure 11-13):  sinη sin Az  Vkin = Vkin  sinη cos Az   cosη 

(11.13)

The Vkin vector is the same as the satellite injection vector. The flight path angle η is defined in Section 2.3.4.3 as the angle between V and the radius vector r. Thus, η = 90° on injection at perigee or into a circular orbit. It remains to determine the eastward velocity contribution Vearth from the earth’s rotation. Knowing that the earth’s angular velocity ω is 360° in 23 hr 56 min 4 s, or 2π/86164 rad/s, we have:

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N ω

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ρ R λ

hn

R

2 π ⋅ 6371 cos λ = 0.464 cos λ (km/s) 86164

1  Vearth = Vearth 0 0

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(11.14)

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Vearth = ωρ = ωR cos λ =

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Figure 11-15 Computing the earth’s eastward surface rotation velocity.

(11.15)

Returning to Eq 11.3, we arrive at a complete vector equation for V:

Vkin sinη sin Az − Vearth   ∆Vu    ∆V =  Vkin sinη cos Az  =  ∆Vv   Vkin cosη + V pot  ∆Vw   

and

2

2

∆V = ∆Vu + ∆Vv + ∆Vw

(11.16)

2

To this ∆V it is necessary to add a ∆Vloss to account for various losses during the ascent. These losses are due to atmospheric drag, and to a less than optimum ascent trajectory as the guidance system makes corrections en route. The atmospheric backpressure into the engine nozzle (Eq 11.1) also contributes to the losses. Hence: 2

2

2

∆Vtot = ∆Vu + ∆Vv + ∆Vw + ∆Vloss

(11.17)

where ∆Vloss typically amounts to 0.7 – 1.5 km/s.

As we shall see in the following, ∆Vtot is the guiding light for the design of launch vehicles, and in particular for the dimensioning of their individual rocket stages.

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Numerical examples: What ∆V does a launch vehicle have to generate when launching from Cape Canaveral (λ = 28.5°) and injecting a satellite in a 185 km circular orbit at an inclination i = λ = 28.5°? km/s 6371

Vpot =

1.879

µ (km /s )=

398601

Vkin =

7.797

Ap (km) = Pe (km) =

185 185

Vearth =

0.408

ra (km) =

6556

∆Vu =

rp (km) =

6556

∆Vv =

a (km) = i (deg) =

6556 28.5

∆Vw =

hn

R (km) =

λ (deg) = η (deg) =

28.5 90

∆Vloss =

Az (deg) =

90

7.389

0.000

1.879

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3

∆Vtot =

1.500 9.124

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The answer is ∆Vtot = 9.124 km/s, assuming a Vloss = 1.5 km/s. Note that the flight path angle η = 90° and the launch azimuth Az = 90°, as one would expect when injecting straight eastward into a circular orbit. What would happen to ∆Vtot if instead we launched into a polar orbit (i = 90°)? In this scenario, the launch vehicle no longer receives the benefit of the earth’s eastward velocity component Vearth, so we should expect ∆Vtot to increase.

R (km) =

km/s

6371

Vpot =

1.879

µ (km /s )=

398601

Vkin =

7.797

Ap (km) = Pe (km) =

185 185

Vearth =

0.408

ra (km) =

6556

∆Vu =

-0.408

rp (km) =

6556

∆Vv =

7.797

a (km) = i (deg) =

6556 90

∆Vw =

1.879

λ (deg) = η (deg) =

28.5 90

∆Vloss =

1.500

Az (deg) =

0

∆Vtot =

9.531

3

2

And that is precisely what happens. Note also that the launch azimuth has gone to zero, since we are launching straight north.

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11.7. Designing a Launch Vehicle 11.7.1. Vehicles Without Strap-On Boosters

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Let us say that NASA or ESA has issued a request for proposals (RFP) to several competing launch vehicle manufacturers. The RFP specifies a vehicle that can lift a satellite weighing 2000 kg to a 185 km circular orbit with an inclination of 28.5°. The manufacturer who offers the best product for the best price will win the contract.

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We are one of those competing manufacturers. In Section 11.6 we established that the rocket has to produce a total ∆V of 9.124 km/s for the satellite to reach the specified orbit. To make our proposal as cost-effective as possible, we decide to procure existing, off-theshelf rocket motors rather than developing our own designs. The question is therefore what motors are available on the open market, and how should we stack them to create an optimized multi-stage design.

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∆V −   g Isp   ∆m = m0 1 − e    

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Maybe the very first question we should ask ourselves is whether the rocket can be built as a single stage. Eq 5.43 gave us the propellant mass ∆m as a function of lift-off mass m0 and ∆V: (11.18)

A tried and tested chemical bipropellant such as UDMH + N2O4 might produce an Isp of 300 s. Eq 11.18 gives us, for a single-stage vehicle: ∆m/m0 = 0.956

which means that 95.6% of the lift-off mass would have to consist of propellant. But recall from Section 11.1 above that a more realistic ratio would be 80%, and that figure does not even include the mass of the satellite payload. Consequently, a single stage is not feasible in our case. Accepting that we will need to build a multistage vehicle, we begin by re-arranging Eq 11.18 as follows:

m  m0   = g I sp ln before ∆V = g I sp ln mafter  m0 − ∆m 

(11.19)

Eq 11.19 states that ∆V is a function of the mass of the rocket before ignition and after burnout. We also know that, in our case ∆Vtotal = ∆V1 + ∆V2 + ... + ∆Vn = 9.124 km/s

(11.20)

where ∆V1 is the velocity increment contributed by the first stage, etc. The rocket motor manufacturers we contact will declare the gross mass m0 of their stage, along with the Isp, the amount of propellant ∆m it can carry, and the burn time ∆t (we will use ∆t later).

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11 / Launch Vehicle Selection Armed with this information, we may use Eq 11.19 to calculate ∆V for that stage. The challenge is therefore to find a set of motors that satisfy Eq 11.20.

olo gy .

Let us now make another working assumption, namely that our launch vehicle requires three stages. If the first stage has the lift-off mass m1 and carries an amount ∆m1 of propellant, then:

  m1 + m2 + m3 + m sat  ∆V1 = g I sp1 ln m + m + m + m − ∆ m 2 3 sat 1   1

(11.21)

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  m3 + msat  ∆V3 = g I sp 3 ln  m3 + m sat − ∆m3 

(11.22)

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Similarly, for the third stage:

  

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 m2 + m3 + m sat ∆V2 = g I sp 2 ln  m2 + m3 + msat − ∆m2

hn

since the propellant of the first stage is needed to lift not only the first stage, but all three stages plus the satellite. After the first stage has burnt out and been jettisoned, we find for the second stage:

(11.23)

A survey of rocket motors has yielded the following shortlist:

Stage No. 1 2 3

Make

Propellant

Castor 120 Solid ESBM Solid OAM Hydrazine

Stage gross mass (kg) 53,000 11,000 700

Stage propellant mass (kg) 48,800 10,000 340

Isp (s)

Burn time (s)

270 290 400

80 150 1,500

Table 11-1 Shortlist of rocket motors for a three-stage launch vehicle.

Let the Castor 120 be the first stage, the ESBM the 2nd stage and the OAM the 3rd stage. From Eq 11.21 – 11.23: 53,000 + 11,000 + 700 + 2,000   ∆V1 = 9.81 ⋅ 270 ln  = 3,484 m/s  53,000 + 11,000 + 700 + 2,000 − 48,800  11,000 + 700 + 2,000   ∆V2 = 9.81 ⋅ 290 ln  = 3,724 m/s  11,000 + 700 + 2,000 − 10,000 

 700 + 2,000  ∆V3 = 9.81 ⋅ 400 ln  = 528 m/s  700 + 2,000 − 340 

and consequently ∆Vtot = ∆V1 + ∆V2 + ∆V3 = 7,736 m/s. But the requirement was ∆Vtot = 9,124 m/s, so we are obviously on the wrong track. Or are we? What if we were to add 381

11 / LaunchVehicle Selection

another Castor 120 and make it the first stage in a 4-stage configuration? Then Eq 11.21 would read as follows:

olo gy .

  m1 + m2 + m3 + m4 + msat  ∆V1 = g I sp1 ln  m1 + m2 + m3 + m4 + msat − ∆m1 

and so forth. Repeating the above calculations, we arrive at the following expanded table:

Castor 120 Solid Castor 120 Solid ESBM Solid OAM Hydrazine

Stage gross mass (kg) 53,000 53,000 11,000 700 2,000 119,700

Stage propellant mass (kg) 48,800 48,800 10,000 340 107,940

Isp Burn ∆m/∆t Thrust (N) (s) time (s) (kg/s)

∆V (m/s)

270 270 290 400

1,389 3,495 3,781 550

80 80 150 1,500

1,615,707 1,615,707 189,660 889

610 610 66.67 0.23

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Propellant

1,810

9,215

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1 2 3 4 Payload Total

Make

ec

Stage No.

Table 11-2 Shortlist of rocket motors for a four-stage launch vehicle.

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With ∆Vtot = ∆V1 + ∆V2 + ∆V3 + ∆V4 = 9,215 m/s, we meet the requirement in Eq 11.20 with some margin (disregarding the fact that we have not accounted for the mass of interface adapters, navigation electronics and other overheads). By a remarkable coincidence, we have actually re-invented the old Athena 2 launch vehicle. Note the additional columns containing the thrust F and the propellant consumption ∆m/∆t. If the manufacturers have not already provided these performance parameters in their data sheets, we can calculate F from Eq 6.1: I sp =

F ∆t g ∆m

where ∆t is the burn duration and ∆m is the amount of propellant carried by the stage in question, as per the table.

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11 / Launch Vehicle Selection

Q: Will adding more stages give the rocket a proportional increase in lift capacity?

olo gy .

A: Let us examine Eq 11.19 again. By adding stages, we increase the total mass m0, which the propellant of the first stage ∆m must lift. Eventually m0 will dominate, such that the argument of the natural logarithm goes towards 1. Since ln (1) = 0, we end up getting no ∆V from the first stage. This tallies with our intuition: If the overall mass of the vehicle becomes greater than the thrust of the first stage, the rocket will not lift off at all, and there will be no ∆V. In other words, adding more stages leads to diminishing returns in terms of ∆V performance.

hn

Q: What happens to ∆V if the rocket’s “dry” mass (i.e. its structure) is infinitely small, such that the vehicle consists entirely of propellant? (This is of course nonsense from an engineering point of view, but let us allow it as an experiment of thought.)

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11.7.2. Vehicles With Strap-On Boosters

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A: In this case m0 = ∆m, and Eq 11.19 gives us ∆V = ∞. How is this possible? Obviously this very “wet” rocket needs a lot of propellant initially to lift its own weight. As propellant is consumed, so its weight diminishes, and less propellant is needed to accomplish a given ∆V. Towards the end, when only a milligram of propellant remains, we still obtain a considerable ∆V from burning even half a milligram ... and so forth, ad infinitum. So ∆V does indeed go towards infinity in this hypothetical case.

Computing ∆V for a launch vehicle with strap-on boosters follows the same logic as in the tandem case. To illustrate the point, let us choose a launch vehicle design similar to the one on the cover of this book, and simplified in Figure 11-16.

In most cases, the main (first) stage ignites at the same time as the boosters, and continues to fire after the boosters have burnt out and been ejected. Let ∆V1A denote the velocity increase of the first stage during the burn time ∆tS of the strap-on boosters. A quantity ∆m1A of propellant is consumed during this time. Using Eq 6.1: ∆m1 A =

dm1 F ∆t S = 1 ∆t s dt gI sp1

where F1 is the thrust of the main stage. If our sample rocket has two identical boosters, each with lift-off mass mS, we may visualize the vehicle as having, in effect, four stages. • •

• •

Stage 1A consists of the two boosters and the main 1st stage. Stage 1B consists of the 1st stage, less the propellant ∆m1A consumed up to the point where the boosters are jettisoned. Stage 2: As shown in Figure 11-16. Stage 3: As shown in Figure 11-16.

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11 / LaunchVehicle Selection

S S

Returning to Eq 11.21:

(11.24)

ec

  (m1 − ∆m1 A ) + m2 + m3 + m sat  ∆V1B = g I sp1 ln  (m1 − ∆m1 A ) + m2 + m3 + msat − ∆m1B ) 

hn

  2mS + m1 + m2 + m3 + msat  ∆V1 A = g I sp1 ln  2mS + m1 + m2 + m3 + msat − (2∆mS + ∆m1 A ) 

olo gy .

Sat

3

2

1B

1A

Figure 11-16 Launcher with strap-on boosters

fT

(11.25)

whereby ∆m1A + ∆m1B = ∆m1, i.e. the total propellant mass of the main 1st stage.

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The ∆V for the remaining stages is computed as before, using Eq 11.22 and 11.23. From Eq 11.20: ∆Vtotal = ∆V1A + ∆V1B + ∆V2 + ∆V3

When calculating ∆V1A from Eq 11.24 above, we assumed that the boosters and the main 1st stage share the same Isp. This may hold true if the boosters use the same propellant as the main 1st stage. If they do not, we must calculate an equivalent specific impulse IspE. For estimation purposes, IspE may be taken as the average value of the participating specific impulses. Hence, for the above sample vehicle: I spE =

2 I spS + I sp1 3

(11.26)

For example, if the main 1st stage consumes H2 + O2 (Isp1 = 400 s) and the two boosters burn solid propellant (IspS = 270 s), the equivalent specific impulse IspE = 313 s. If the vehicle were to have N identical strap-on boosters rather than 2, the coefficient 2 in Eq 11.24 must of course be replaced by N. The equivalent IspE is then: I spE =

384

N ⋅ I spS + I sp1 N +1

(11.27)

11 / Launch Vehicle Selection

11.8. Launch Vehicle Reliability

olo gy .

Having spent typically 2 – 5 years developing, building and testing a new satellite, the teams involved dread the thought of their creation being destroyed in a launch failure. Selecting a suitable launch vehicle is therefore a major preoccupation throughout the project.

hn

Statistically, most of the launch vehicles profiled in Section 11.5 above show a reliability of 70 – 100% (to the extent that there has been a statistically significant number of launches). A reliability of, say, 80% implies that one in five launches fails, which is unacceptable to most satellite customers. Some customers find even a 95% reliability worrisome, since it means that one in 20 launches fails. However, reliability figures should be taken with a grain of salt for the following reasons: A new vehicle design, which has only flown a few times, does not offer a statistically meaningful database.

•

Despite their long heritage, well-established vehicles are subject to regular modifications which contaminate the reliability statistics.

•

Even in cases where a launch vehicle design has remained unmodified for a long time, a different reliability figure is obtained depending on whether the last 5, 10, 20 or 50 launches are counted (Figure 11-17). In their publicity material, launch service providers often pick the number that gives the most favourable statistic.

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ec

•

5 attempts, 3 failures, R = 40%

10 attempts, 5failures, R = 50%

20 attempts, 6 failures, R = 75%

Figure 11-17 Example of how a launch service provider might “improve” the claimed reliability R by quoting the last 20 launches rather than the last 5 or 10.

For all their flaws, reliability statistics are nonetheless the only tool available to satellite customers to assess their chances of launch success. Insurance underwriters charge premiums based partly on the overall reliability statistics for a particular vehicle, and partly on their loss or profit during the previous year. It is therefore useful to examine the reliability of individual launch vehicles from the date of their maiden flight until today's date. If the launch frequency is low compared to some other vehicles, one has to ask oneself why. One reason could be poor reliability, with long down-times to remedy the cause of failures. Another reason could be high price, making the vehicle unattractive at least for commercial satellite customers. Moreover, the manner in which launch failures are distributed over time could tell us something about whether the failures were caused by faulty design or poor workmanship. 385

11 / LaunchVehicle Selection

olo gy .

Design flaws are usually concentrated to the beginning of the launch programme, while workmanship problems are typically spread out more evenly over time. In other words, design flaws tend to recur until they are remedied once and for all, while workmanship issues are a sign of poor quality management in the rocket manufacturer's organisation and should be of particular concern for the satellite customer.

hn

The failure history of the European Ariane 5, the Russian Proton and the Indian GSLV provide interesting examples of the above, as illustrated in Figure 11-18 through Figure 11-20. The diagrams held true at the time of writing (January 2014) and were generated according to the formula (No.of launch successes) / (No. of launch attempts) x 100 at any one time. The arrows highlight where in the launch sequence each failure took place.

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100 90 80 70 60 50 40 30 20 10 0

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Reliability %

Ariane 5

0

5

10

15

20

25

30

35

40

45

50

55

60

65

70

Launch attempt No.

Figure 11-18 Launch failure history of Ariane 5.

The failures are concentrated to the beginning of the Ariane 5 programme and were indeed caused by fundamental design problems.

Reliability %

Proton/Breeze

100 90 80 70 60 50 40 30 20 10 0

0

10

20

30

40

50

60

70

Launch attempt No.

Figure 11-19 Launch failure history of Proton-K & -M/Breeze-M.

Proton suffered several failures at an early stage, suggesting a mixture of design and workmanship issues. The more recent failures were the result of workmanship problems.

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11 / Launch Vehicle Selection

GSLV Mk.1 & 2 100 90

olo gy .

Reliability %

80 70 60 50 40 30 20 10 0

1

2

3

4

5

6

7

8

9

10

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Launch attempt No.

hn

0

Figure 11-20 Launch failure history of GSLV.

Flight 1: Stage 3 shut down 10 seconds too early. Flight 4: Failure in one of the strap-on boosters, rocket diverted from course. Flight 5: Underperformance of upper stage, lower than planned orbit Flight 6: Upper stage failed. Flight 7: Control lost after 47 seconds, disintegrated after 53 seconds.

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• • • • •

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The GSLV programme has suffered more failures than successes due to deficiencies that appear to be primarily of a design nature:

The most recent launch attempt was successful, and it remains to be seen whether the design problems have now been resolved. One has to keep in mind that the vehicles in the above statistics may have undergone relatively minor modifications during their lifetime. Purists might argue that one should only include identical designs, but this would reduce the size of the statistical population, thereby adding to the uncertainty. Some launch service provider might also argue that not all the shown anomalies were complete failures, since the satellite(s) achieved at least some kind of orbit, even if it was not the specified one. In a few instances involving dual satellite launches, one of the satellites did achieve the correct orbit, but the other satellite did not. The reliability statistics are summarized in Table 11-3 in the context of average launch frequency per year. Notice how Proton, Long March 3 and Ariane are currently in the lead as concerns launch frequency, suggesting that these two vehicles offer optimum ratios in terms of proven reliability, availability and price.

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11 / LaunchVehicle Selection

Launch vehicle

04/06/1996 21/08/2002 20/11/2002 04/06/2010 20/04/2001 29/08/2001 28/03/1999 29/01/1984 05/07/1999 08/02/2000

29/08/2013 24/01/2014 28/08/2013 06/01/2014 05/01/2014 27/01/2013 31/08/2013 20/12/2013 26/12/2013 19/12/2013

71 43 24 8 8 29 43 72 75 38

Average Current success Number of number of rate failures launches per (January 2014) % year 4 4.1 94.4 1 3.8 97.7 1 2.2 95.8 1 2.2 87.5 5 0.6 37.5 1 2.5 96.6 4 3.0 90.7 5 2.4 93.1 8 5.2 89.3 2 2.7 94.7

hn

Ariane 5 Atlas V Delta IV Falcon 9 GSLV Mk 1 & 2 H-2A Land & Sea Launch Long March 3 Proton-K & -M/Breeze-M Soyuz/Fregat

Number of launches

olo gy .

Most recent First launch launch

Table 11-3 Summary of launch vehicle reliability statistics.

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Gunter's Space Page (www.skyrocket.de) provides a convenient and up-to-date overview of launch vehicle flight histories.

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Q: If there is a 1-in-5 chance that a particular launch vehicle will fail, and if four launches have already taken place successfully, is the failure of the fifth launch a foregone conclusion? A: No, a launch has no memory of the previous launches. All one can say is that each launch has a 1-in-5 chance of failing, no more, no less. It is possible (though unlikely) that ten such launches will succeed in succession, and it is equally (un)likely that ten launches in succession will fail.

11.9. Launch Vehicle Availabilisty

From a satellite customer's point of view, launch vehicle availability is measured against the following criteria: a) Number of free slots in the launch service provider's manifest (i.e. launch queue) during the time frame of interest. b) Stability of the rocket manufacturer's production line (i.e. whether the launch vehicle production is likely to proceed at the necessary rate). c) Launch delays caused by reliability issues. d) Political obstacles (e.g. ITAR, by which the U.S. government prohibits any satellite containing U.S. components to be sent to China for launch).

Criterion (a) is assessed by direct contacts with the launch service providers of interest. The most popular vehicles have full manifests during the coming 2 - 3 years, although free slots may become available on a stand-by basis. Criteria (b) and (c) are evaluated by reading the trade press such as Space News. Criterion (d), as it relates to ITAR, is currently a fact of life, although there have been rumours that the U.S. government may relax the prohibition at some future date.

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11 / Launch Vehicle Selection

11.10. Launch Costs

olo gy .

The supply and demand of launch services, and hence the price, vary from year to year. The price swings are also influenced by the advent of new entrants on the launcher market, and by the success or otherwise of the launches themselves.

hn

Though most launch service providers publish a “sticker price” for their services, some satellite customers are able to negotiate significant discounts through quantity purchases, or by making their satellites compatible with several different launch vehicles and having the corresponding providers compete for the launch contract. Significantly lower prices are also available to satellite customers who are prepared to accept the risk of launching on the maiden flight of a new rocket.

fT

ec

In any event, the launch price must be seen in the context of the launch vehicle’s lift capacity. Using a rocket’s capacity to the full is obviously more cost-effective than flying half-empty. In many cases, two or more satellite owners get together to fill the capacity of a heavy vehicle, thereby saving money as compared to flying separately on smaller rockets.

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In addition to the cost of building a satellite and procuring a launch service, there is the significant albeit optional expense of buying launch insurance. Insurance premiums range from 10 to 20 percent of the insured assets (the launcher alone, the satellite alone, or the two combined), depending on the vehicle’s track record. Especially commercial customers are likely to pay extra for a very reliable launcher, knowing that some of the extra expense will be offset by the cheaper insurance premium. Most importantly, the ultimate aim of launching a commercial satellite is not to recover the insurance money, but to have a satellite in orbit that, over time, earns its owner 5 – 10 times the original investment.

11.11. Making the Final Selection

The easiestway of justifying a launch vehicle selection is to establish a matrix as shown in Table 11-4, and to assign a figure of merit to each combination of criterion and vehicle. Score "1" could signify that the vehicle is ruled out, while score "5" might represent the most preferred. By adding up the figures, one arrives at the most preferred vehicle, followed by potential back-ups. Launch vehicle

Launcher A Launcher B Launcher C Launcher D Launcher E Launcher F Launcher G

Optimized performance 5 1 3 4 4 2 1

Reliability

Availability

Cost

Total

Ranking

2 4 5 4 5 4 3

4 3 1 5 5 3 5

4 3 2 5 3 4 5

15 11 11 18 17 13 14

3 6 6 1 2 5 4

Table 11-4 Sample order of preference when selecting launch vehicles.

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11 / LaunchVehicle Selection

olo gy .

In the above example, Launcher D with its total count of 18 comes out as the most preferable, followed by Launchers E and A as potential backup vehicles. Some analysts might "weight" one criterion - e.g. cost - more heavily than the others, which means that the numbers in that column are multiplied by a "weighting factor" of, say, 2 or 3. This will obviously change the relative ranking.

hn

Case study: A satellite customer has paid $300m for a GEO satellite weighing 3,300 kg at lift-off. The most suitable launch vehicles appear to be the hypothetical rockets A, B and C, since the satellite fills their lift capacity reasonably well. Any spare capacity can be used up by filling additional propellant into the satellite, for the benefit of its orbital lifetime. How much will the customer have to pay to lift his satellite into GTO? The price per kg for each of the three vehicles is:

Failure rate

510 420 70

2 out of 3 0 out of 25 2 out of 7

$45,000 $26,000 $15,000

ec

Vehicle A: Vehicle B: Vehicle C:

Spare capacity (kg)

fT

Price per kg

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From a pricing point of view, Vehicle A is clearly unattractive, and the rocket’s reliability history to date would rule it out once and for all. The temptation would be to sign up a Vehicle C with its low price per kg, but here again the customer might balk at the poor reliability statistics. Vehicle B gives the customer a reliable launch as well as sizeable spare capacity for extra propellant. His capital expenditures would be: Satellite cost: Launch price: $26k x (3,300 + 420 kg) Insurance: 12% of ($300m + $97m) Total:

$300m $97m $48m $445m

By comparison, costs for the Vehicle C alternative would have been as follows: Satellite cost: Launch price: $15k x (3,300 + 70 kg) Insurance: 20% of ($300m + $50m) Total:

$300m $50m $70m $420m

From these two examples we conclude that the higher insurance premium for the riskier Vehicle C makes this alternative only marginally more economical than Vehicle B. If getting the satellite into orbit is the customer’s first priority, there is no doubt that he will spend the extra $25m ($445m - $420m) for the added reliability of Vehicle B.

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11 / Launch Vehicle Selection

11.12. Solved Problems See Appendix C for solutions.

olo gy .

11.12.1 A launch vehicle is to inject a satellite from a site located at latitude 60°N into a 600 km circular, sun-synchronous orbit with an inclination i = 98°. With what azimuth Az must the vehicle be launched? 11.12.2 A satellite is to be inserted into a 600 km circular orbit. How much propellant mass will the launch vehicle consume if its lift-off mass is 50 tons (including the payload) and its Isp = 450 s? Assume that ∆Vloss = 1.0 km/s. The earth’s radius R = 6371 km.

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hn

11.12.3 A secondary launch site for the Soyuz ST launch vehicle at Kourou (latitude 5°N) has been arranged to give the vehicle greater inclination flexibility and lift capacity compared to the primary launch facility at Baikonur (latitude 46°N). How much ∆V will the vehicle gain from Kourou compared to Baikonur when launching straight east into a 400 km circular orbit, and how will this gain translate into added payload lift performance? Assume that the rocket engines have an average Isp = 300 s, and that orbital injection occurs horizontally.

391

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12 / Launch & Space Environment

12. Launch and Space Environment

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12.1. Introduction

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hn

A satellite has to survive in a harsher environment than is usually found in engineering. During the launch and ascent phase, static and dynamic loads threaten the integrity of the satellite’s structure and equipment. Following separation from the launch vehicle, the spacecraft experiences an extreme thermal regime, as described in Chapter 10. In addition, it has to cope with unshielded radiation and charged particle bombardment from the Van Allen belts, the sun, and the deeper realms of space. Low-flying satellites are subjected to drag friction in the upper layers of the atmosphere. The drag makes the orbit shrink and may upset the spacecraft’s attitude. Lastly, the vacuum of outer space poses its own problems.

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fT

All these environmental influences have the effect of curtailing the satellite’s operational lifetime in the short, medium and long term. The spacecraft designer’s challenge is to implement counter-measures using a wide range of available technologies, and to verify the adequacy of these measures through rigorous testing. The present chapter describes the environmental challenges. The countermeasures are discussed in Chapter 13.

12.2. Launch Environment

The complex vibration environment during launch may be divided into static loads and several kinds of dynamic loads. It is useful to do so, because the various structural elements of a satellite react differently to these loads. The spacecraft designer must be aware of these differences in order to arrive at an optimum construction in terms of strength, rigidity and mass. The static and dynamic loads have axial as well as lateral components. In the case of ballistic launch vehicles (“no wings”), the axial loads predominate under the influence of engine operation, while lateral loads are caused by wind shear and by small but measurable ignition timing discrepancies between strap-on boosters. Winged vehicles, such as the Pegasus, manifest axial and lateral loads of almost equal magnitude due to aerodynamic effects and the horizontal mounting of payloads. The load profiles illustrated in the following pertain to heavy launch vehicles of the Ariane-Atlas-Delta class. For more accurate information, refer to the relevant User Manual (see Appendix F). 12.2.1. Static Loads

As the name implies, static loads manifest no cyclic variation during the ascent from the moment of lift-off until orbital injection. Their effect on the satellite structure is to cause buckling and yield phenomena, as opposed to fatigue breakage typical of dynamic loads (Chapter 9).

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12 / Lauinch & Space Environment

olo gy .

In actual fact, the “static” loads vary considerably, but their frequency of change is too slow to cause fatigue problems. They arise as a result of the rocket stages accelerating during motor burn, and are interspersed by periods of relative weightlessness in the lull between the burnout and ignition of successive rocket stages. The acceleration of each rocket stage is not constant. Newton’s law states that the instantaneous thrust force F is a function of the mass m and the velocity V as follows: F = d(mV)/dt = m dV/dt + (-Ve dm/dt) = ma - Ve dm/dt

(12.1)

ec

hn

Ve is the exit velocity of the expended propellant mass. From Newtonian mechanics we immediately recognize the component F = ma, but that expression assumes a constant mass, as in the case of applying a force to accelerate a solid object. In rocketry it is necessary to account for the change in mass as well; hence the additional term -Ve dm/dt. The minus sign implies that the mass is diminishing over time. Rearranging Eq 12.1: 1  dm   F + Ve  dt  m(t ) 

fT

a (t ) =

(12.2)

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Since rocket motors are designed to produce relatively constant thrust F, propellant consumption dm/dt and exit velocity Ve, it follows that the acceleration a will increase over time as the net mass m decreases. This process is illustrated in Figure 12-1 for a typical three-stage launch vehicle assisted by strap-on boosters. The acceleration is given as multiples of the gravitational acceleration g = 9.81 m/s2. The digits indicate the rocket stage number. S is the moment of burnout of solid propellant strap-on boosters, while L coincides with liquid propellant booster burnout. Acceleration (g) 6 5 4 3

L

2

S

1

1

2

3

0

0

200

400

600

800

1000

1200

1400

Flight duration (seconds)

Figure 12-1 Typical acceleration profile for an Ariane 4SL launch vehicle.

In Eq 11.1 we saw that the acceleration is assisted by the thrust F∆p occasioned by the pressure difference between the inside of the nozzle pe and the ambient pressure pa:

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12 / Launch & Space Environment

F ∆p = A e ( p e - p a )

(12.3)

Ve

pa

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Ae pe

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Figure 12-2 Thrust force occasioned by combustion and ambient pressure differentials.

F ∆t g ∆m

(12.4)

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I sp =

ec

where Ae is the area of the nozzle’s mouth. F∆p increases as the rocket gains in altitude, since pa goes towards zero. This is the reason why the specific impulse Isp of a rocket motor improves with altitude, recalling from Eq 11.2 that

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The maximum acceleration remains in the vicinity of 4 g for heavy rockets, while payloads onboard small sounding rockets may experience up to 12 g (e.g. Skylark).

12.2.2. Dynamic Loads

The examples of dynamic load profiles given in the following are those contained in a typical launch vehicle User Manual. They are not true values as measured in flight, since the levels change over time, and since each flight is different depending on payload configuration and trajectory. Instead the levels represent generic design envelopes with which the spacecraft builder is expected to comply. 12.2.2.1.

Sinusoidal Vibration

The low-frequency sinusoidal component of the vibration spectrum is potentially hazardous. If it coincides with a natural frequency of the spacecraft, the vibration may be amplified to the point of structural disintegration. Sinusoidal vibrations have their origins in the launch vehicle structure’s response to aerodynamic buffeting and to the operation of rocket engines. They propagate to the satellite through the interface adapter. The acceleration amplitude rarely exceeds 2 g in the axial sense and 0.5 g laterally. The pogo effect refers to a particularly violent flow oscillation in the launch vehicle’s propellant feed lines. There are always small variations in the propellant feed rate to the rocket engines, and hence in the engine thrust. If the frequency of thrust variations comes anywhere near the launcher’s natural frequency, the thrust-induced acceleration cycles are magnified through a vicious closed loop between body accelerations, feed irregularities and thrust instabilities. The resulting vibrations enter the satellite through the interface adapter. 395

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Another propellant instability phenomenon in rockets is called water-hammering. It is reminiscent of the loud humming noise that is sometimes heard when slowly turning off a water tap connected to inferior plumbing. Launch vehicle manufacturers go to great lengths to eliminate both pogo and water-hammering effects in their design.

Random Vibration

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Destructive resonance between the launcher and the satellite can be avoided by installing a suitable vibration damper at the spacecraft/launcher attachment interface. A more common approach is to stiffen the spacecraft’s structure such that the fundamental frequencies land above the range of propagated vibrations. This range is typically 15 – 30 Hz in the axial direction, and 10 – 15 Hz in the lateral sense. The actual values depend on the choice of launch vehicle. Consequently, a satellite designed to be vibrationcompatible with a particular vehicle cannot be launched on a different vehicle without prior analysis and test.

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The combustion process in the rocket motor chambers gives rise to random vibrations, which reach the satellite through the attachment interface adapter. These vibrations are typically in the range 20 – 2000 Hz, depending on the flight phase and the launch vehicle. Figure 12-3 shows the vibration envelope for a heavy vehicle. It is customary to measure this type of vibration in terms of energy spectral density (g2/Hz), because the r.m.s. value quantifies the equivalent acceleration experienced by the satellite.

-3 dB

3 dB

Figure 12-3 Random vibration profile.

To facilitate testing, the slopes of the curve are labelled in decibel per octave, where an octave is the interval over which the frequency is doubled (25 – 50 Hz, 50 – 100 Hz, 100 – 200 Hz, etc.). For example, in Figure 12-3 the positive slope goes from 0.015 to 0.030 g2/Hz over the octave interval 100 – 200 Hz, i.e. 10 log10(0.030/0.015) = 3 dB per octave. Low-frequency random vibration influences the design of core elements of the satellite’s primary structure.

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Q: Why is the word “octave” used for the doubling of frequency intervals? It sounds like something to do with “eight.” A: The term octave comes from music, where eight notes are used in the major (“happy”) or minor (“sad”) diatonic scale to move from one frequency to its doubled value. For example, Middle A is at 440 Hz, the next A down is at 220 Hz, and the next A up is at 880 Hz.

Acoustic Vibration

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12.2.2.3.

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Acoustic noise is a form of random vibration. While the variety described in Section 12.2.2.2 has internal sources and affects the satellite’s core structure, acoustic noise arrives from the outside and rattles flimsy external structures such as stowed solar panels and antenna reflectors. The main origin of acoustic noise is ignition-related sound waves reflected off the launch pad during lift-off. Another important cause is the disintegration of the aerodynamic boundary layer on the outside of the heatshield at transonic speeds. The level of noise affecting the spacecraft can be reduced by cladding the inside of the heatshield with acoustic blankets. Figure 12-4 shows a typical acoustic noise profile. Here the acoustic pressure level I is measured in decibels with reference to a pressure level of 2 × 10-5 Pascal (or N/m2). Expressed mathematically: I = 20 log10

F (dB) 2 ⋅ 10 −5

(12.5)

where F is the linear value of the acoustic field intensity. The levels are represented as constant for each octave to facilitate testing.

Acoustic pressure I (dB)

140 135 130 125 120 115 110

105 100

10

100

1000

10000

Frequency (Hz)

Figure 12-4 Acoustic noise profile.

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Q: Why make the distinction between “random” and “acoustic” vibrations? If both are truly random, don’t they cover the whole audible spectrum? A: Noise can be random within a limited frequency spectrum. Compare the roar of a jet taking off (low random frequency) with the hiss of compressed air streaming out of a bottle (high random frequency).

Shock

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The terminology “random” and “acoustic” is used to distinguish between the sources of vibrations (internal or external), its effect on different structural members (sturdy core or flimsy appendages), and the corresponding test method (using an electromechanical vibration table or blasting through horns).

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The satellite experiences mechanical shock in connection with various launch transition and separation events, e.g. rocket motor ignition or cut-off, rocket stage separation, heatshield jettison, satellite separation from the rocket, and release of units onboard the spacecraft itself (perigee or apogee motors; protective covers; solar panels and antenna dishes). Some of these shocks can be particularly violent, notably those involving pyrotechnic devices.

Acceleration response (g)

Though the shocks are of short duration (milliseconds), their acceleration amplitude can be in the thousands of g’s and must be accounted for in the design and testing of satellites. Launch vehicle manufacturers specify shock load envelopes as exemplified in Figure 12-5. 10000

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Frequency (Hz)

Figure 12-5 Typical shock spectrum

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12.3. Space Environment

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The two environmental parameters that first come to mind are weightlessness and vacuum. There are, however, two other categories that are of equal or greater concern to spacecraft designers, namely charged particle bombardment and electromagnetic radiation. Particle bombardment involves protons and electrons trapped in the Van Allen belts, along with a variety of charged particles emanating from our galaxy and beyond. The electromagnetic radiation of greatest interest in satellite design is ultraviolet light (UV) and solar pressure.

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12.3.1. Weightlessness

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Frequently, charged particle bombardment is also referred to as “radiation” but, in the present context, it is useful to define radiation as a transfer of massless energy quanta, as opposed to particles which clearly have mass.

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The near-total absence of gravity in orbit is a blessing for most moving parts onboard the satellite, since it allows for a much lighter construction than on earth. The best examples are deployment mechanisms for solar panels and antenna reflectors. However, the difficulty of simulating weightlessness during ground verification has sometimes led to deployment mechanisms being inadequately tested, with dire consequences after launch. Weightlessness causes propellants to float around haphazardly inside their tanks. An undesirable side effect is dynamic instability of the entire spacecraft due to sloshing at the start of attitude manoeuvering. An associated difficulty arises during thruster activation, since propellant may be absent at the drain end of the tank. Solutions include creating artificial “gravity” by spinning the satellite, or else by constraining the propellant inside elastic bladders within the tank volume (Figure 12-6). See also Chapter 6.

Bladder

Propellant

Figure 12-6 Harnessing propellant using centrifugal force or elastic bladder.

12.3.2. Vacuum

One would have thought that vacuum constitutes an ideal conservation medium for a spacecraft, in the absence of humidity, pollution and other corroding influences of the atmosphere. In reality, life in vacuum presents its own unique problems.

One such problem is outgassing, defined as the escape of volatile gases and compounds inside the satellite. We know from experience on the earth that liquids boil and begin to 399

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evaporate more readily at higher altitudes (e.g. on top of mountains) than at sea level. The lower the air pressure, the lower the boiling temperature. In the early days of space exploration, it happened that lubricants between moving satellite parts turned dry or gritty as the agent responsible for the viscous consistency evaporated into space. As a result, momentum wheels and tape recorders seized up. During the first few hours, days and even weeks after launch, a certain amount of air remains trapped inside the satellite. The air contains water vapour (moisture). As the air seeps out, the vapour is attracted to cold surfaces, where it condenses as a thin layer of ice. The condensate can blur optical lenses and change the emissivity of heat-radiating surfaces.

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Another problem related to vacuum is cold welding, whereby metal surfaces designed for unlubricated movement fuse. For example, solar panels that deployed repeatedly during ground testing would fail to do so a few days later in space. We often overlook the fact that the atmosphere inserts itself as a thin cushion between moving parts. In outer space, the absence of this natural and ubiquitous lubricant can give cause to cold welding.

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Historical note: Soon after the first Meteosat satellite was launched in 1977, meteorologists noticed that the infrared images of the earth were getting pale, i.e. the photo contrast was fading to the point of rendering the images unusable. Inspection of housekeeping telemetry revealed that the camera’s infrared sensor was too warm, thereby masking the thermal pictures with its own temperature. To keep the sensor as cool as possible, it had been mounted at the apex of a conical radiator that opened up into space. Because trapped water vapour had condensed on the radiator surface and changed its emissivity, the radiator was no longer able to do its job. Fortunately, the spacecraft designers had foreseen this eventuality, and had therefore installed switchable heater coils inside the radiator wall. Infrared imagery was suspended for a few days to allow the heaters to burn off the ice, much like the rear-window defroster in a car. Eventually the heaters were disconnected, and infrared imagery resumed with perfect picture quality.

For all its apparent emptiness, the vacuum in space is not total. It is estimated that, on a typical day in outer space, approximately one million particles occupy every cm3. Most of these particles are charged and constitute the plasma in which satellites “swim.” Plasma is like superheated vapour, in which the neutral molecules have been stripped apart to form an energetic brew of ions, protons and electrons. If the satellite were electrically conductive throughout, there would be no difference in electric potential between the satellite and the surrounding plasma. In reality it is impractical to create full conductivity, since the requirements on strength, thermal behaviour and functionality of the spacecraft call for different materials to be employed. The interaction of the satellite’s surfaces with the surrounding plasma is complex, but the net result is a build-up of voltage differences between some surfaces and the plasma, as well as between dissimilar surfaces. Whenever these differences exceed the breakdown voltage, discharge occurs through an arc, similar to a microscopic lightning storm. This phenomenon is known as electrostatic discharge (ESD). In the worst case, the arc can destroy electronic components and perforate insulating blankets. 400

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Satellites in high earth orbits, such as GEO, are particularly prone to ESD during intermittent magnetic substorms. These storms occur in the magnetotail on the night side of the earth and have the effect of propelling highly energetic charged particles towards the earth. The resulting hot plasma exacerbates the charge/discharge phenomena described above.

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12.3.3. Aerodynamic Drag

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Historical note: The launch of satellites into GEO accelerated in the 1970s. Most of the satellites carried telecommunications and meteorological payloads. Spacecraft controllers on the ground were frequently confounded by random switchings onboard the satellites, many of which led to service interruptions for several hours while the anomalies were being investigated. The controllers were particularly annoyed by the fact that most of the random switchings consistently occurred between midnight and 4 a.m. when the control centres had minimum staffing. It was only in the early 1980s that the physics of ESD became fully understood, such that corrective action could be taken at the spacecraft design level.

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The earth’s atmosphere extends far into space, though its density decreases exponentially with altitude. A satellite travelling in an orbit with a semimajor axis a ≈ 7000 km or less (or < 600 km circular orbit height) will experience orbital decay at a rate which matches typical operational lifetimes (5 – 10 years) of LEO satellites.

The most obvious effect of orbital decay is to slow down the satellite, i.e. making the orbit lose kinetic energy through atmospheric friction. During each satellite passage through the atmosphere, it is the apogee height that loses height the most. Eventually the orbit becomes circular, and the satellite begins to spiral its way deeper and deeper into the atmosphere. By the time it has descended to a circular height of 130 km, re-entry is imminent.

Figure 12-7 Orbital decay.

Predicting the orbital lifetime, i.e. the rate of decay, is relatively straightforward in principle but difficult in practice. If the density variation with height were always the same, and if the satellite’s attitude with respect to the velocity vector were fixed, the 401

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Lifetime (days) 900 1100 1300 500 600 700 800 1000 1200 400

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1100

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1200 300

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1000 900 800 700 220

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Apogee height Ap (km)

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aerodynamic drag coefficient CD would be entirely predictable, as would the decelerating force. The progression of orbital decay could then be computed by numerical integration. The satellite would disintegrate when the orbital height fell below 130 km. Figure 12-8 shows theoretical lifetimes of a cylindrical satellite weighing 140 kg as a function of the chosen initial apogee and perigee height values.

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Perigee height Pe (km)

Figure 12-8 Theoretical orbital lifetime as a function of apogee and perigee height for a launch in 2021 (approaching solar minimum).

In reality, the lifetime is influenced by a number of factors, some of which are more predictable than others: • •

•

Changing spacecraft mass (varies over time if propellant mass is expended) Changing spacecraft attitude relative to the velocity vector (causes CD to vary over time) Variations in atmospheric density at given orbital heights (differs between the day-side and the night-side of the earth, and is also a function of the amplitude and phase of the 11-year solar cycle).

The last of these is the most significant and the least predictable. As a consequence, lifetime predictions can easily go wrong by a factor of 5.

To give an idea about the quantities involved, the smoothed solar cycle profile is illustrated in Figure 12-9, while the corresponding atmospheric density is shown in Figure 12-10. The drag coefficient CD varies between 2 and 4 for most satellite shapes, and the instantaneous drag force equals D = ½ρAV2CD

(12.6)

where ρ is the air density, A is the surface area confronting the oncoming air molecules, and V is the orbital velocity.

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Solar flux (x 10-22 W/m2 Hz)

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Figure 12-9 Solar cycle flux profile.

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1.E-09

Density (kg/m 3)

1.E-10

Solar maximum

1.E-11 1.E-12 1.E-13

Solar minimum

1.E-14

1.E-15 200

300

400

500

600

700

Altitude h (km )

Figure 12-10 Atmospheric density as a function of altitude.

Figure 12-11 gives an impression of how orbital decay progresses. The thin curve shows the ideal case with constant mass, attitude and density. The staircase shape of the fat curves is the result of density modulation by the 11-year solar cycle. The upper fat curve shows the progression as a result of an optimistic prediction regarding the amplitude of the solar cycle. The lower fat curve represents a pessimistic prediction.

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7100 7000 6900

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Semimajor axis a (km)

7200

6800 6700 6600 6500 6400 1

1.5

2

2.5

3

3.5

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Solar cycle No.

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Figure 12-11 Possible orbital decay of a satellite launched into a 700-km circular orbit.

12.3.4. Atomic Oxygen

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In both predictions, the detrimental effect of peak solar activity on orbital lifetime is clearly in evidence.

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The elements that combine to make up the atmosphere become dissociated by the sun’s ultraviolet radiation at altitudes above 100 km. One of the constituents is atomic oxygen, whose concentration peaks at a height of approximately 170 km, and whose nuclei are heavy enough to “sand-blast” the surfaces of a satellite. Kapton film used in thermal blankets, as well as silver junction leads between solar cells, are prone to oxygen erosion, as are optical lenses. The “sand-blasting” is referred to as sputtering.

Oxygen bombardment also has the effect of oxidizing certain materials, the more so since atomic oxygen (O) is far more corrosive than oxygen molecules (O2). One well-known effect is for the α and ε values of exposed surfaces to undergo a change, such that their thermal control effectiveness is compromised. Certain lubricants have been known to become gritty for similar reasons. 12.3.5. Van Allen Belts

Protons and electrons emanating from the sun are trapped in the earth’s magnetic field to form doughnut-shaped concentration rings along the geomagnetic equator. Figures 12.11 and 12.12 illustrate these so-called Van Allen belts, with the areas of maximum energy concentration shown as shaded areas. The concentration is measured in terms of particle flux above a given energy threshold. The boundaries for the shaded areas in Figure 12-12 are 1 proton > 100 MeV per cm2 and second, and 106 electrons > 0.5 MeV per cm2 and second. These concentration ranges are clearly harmful to satellites. To facilitate a comparison, the proton and electron belts have been drawn to the left and the right of the vertical dash-dot line, respectively. In reality, of course, each belt goes all around the earth. Note that there are two concentric electron belts of peak concentration.

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Altitude 10,000

Electrons > 0.5 MeV Earth LEO

10,000 MEO

20,000

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Protons > 100 MeV

30,000 km GEO

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Figure 12-12 Van Allen belts of high-flux, high-energy protons and electrons.

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The Van Allen belts are often referred to as radiation belts. This is a misnomer, since here we are dealing with charged particles rather than electromagnetic radiation.

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Trapped protons and electrons create a variety of problems in electronics, notably bit flips in digital circuits, as well as performance degradation of electronic components and solar cells. In the former case, the remedy consists in making the walls of electronic boxes thick enough to withstand penetration of high-energy particles. Minimizing damage to solar cells is a primary concern when deciding the orbit for the satellite. From Figure 12-12 one can conclude that low earth orbits (LEO) below 1500 km, medium earth orbits (MEO), and geostationary orbits (GEO) offer relatively lenient environments. Inspection of satellite catalogues reveals that few low-inclination spacecraft travel in the zones of maximum concentration, i.e. in the height ranges of 1,500 – 9,000 km and 20,000 – 30,000 km. Satellites travelling at high inclinations - notably the navigation satellites GPS, Glonass and Galileo - are less affected by the Van Allen belts, since they only encounter the high-energy belts at low latitudes. The highly elliptic but mostly low-inclination geostationary transfer orbit (GTO), with its perigee height at around 300 km and apogee height at 36,000 km, offers an interesting operational challenge, since the orbit sweeps through the proton belt as well as both electron belts. It has been suggested that the solar arrays lose 0.5 - 1% of their performance for each round in GTO. Consequently there is an incentive to fire the apogee engine at the first apogee. This leaves only 5 hours from injection to perform the necessary attitude determination and attitude manoeuvres. The challenge is compounded by the fact that TT&C contact with ground stations may be intermittent during this critical time.

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High-energy electrons

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Orbit

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Earth’s magnetic field

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High-energy protons & electrons

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Figure 12-13 Artist’s impression of the proton and outer electron belts.

12.3.6. Meteorids and Man-Made Debris

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Let us first clear up the confusion between meteorids, meteors and meteorites. A meteorid is a piece of rock or metal of planetary origin that travels through the solar system. Most meteorids of interest for earth-orbiting satellites have their origins on the moon, on Mars or in the asteroid belt. A meteor is a meteorid in the process of burning up in the atmosphere. A meteorite is a meteorid that survives the journey through the atmosphere and strikes the ground. The prefix micro- means that the fragment is particularly small.

Cumulative Flux Φ (Q/km 2 yr)

Clearly, meteorid is the proper term to use in the present context. The occurrence of meteorids is fairly evenly distributed across the LEO – GEO range of altitudes. The probability of a spacecraft being struck by a meteorid can be surmised from Figure 12-14, in which the flux Φ is plotted against mass. Here, Q is the quantity of meteorids. 1.E+16 1.E+12 1.E+08 1.E+04 1.E+00 1.E-04 1.E-08 1.E-12

1.E-16 1.E-16

1.E-12

1.E-08

1.E-04

1.E+00

1.E+04

1.E+08

Meteorid Mass m (g)

Figure 12-14 Meteorid flux versus mass.

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As a statistical example, one meteorid with a mass m = 1 gram will traverse an area of 1 km2 once a year. This may not sound like a big threat, but the velocities involved paint a different picture. Take a 1-gram meteoroid approaching an earth-orbiting satellite at 10 km/s. The unelastic impact energy would be similar to a 100-ton object striking the satellite at 1 m/s, since V 2   10,000 2  2 2  = 100,000 kg ½ m1V1 = ½ m2V2 ; m2 = m1  1 2  = 0.001 1 V    2 

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The equivalent statistics for a meteorid with a mass m = 1 milligram is 1000 such meteorids per km2 per year, corresponding to an equivalent impact of 100 kg at 1 m/s. Now the probabilities become more worrisome for the satellite designer, and especially for the designers of large orbiting space stations, the more so since meteoroid velocities of up to 45 km/s have been measured.

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The density of meteorids in earth orbits varies during the year, as well as from year to year. The Leonides make up a cluster of meteorids through which the earth travels in early November each year. The predicted intensity of the meteorid shower has prompted some satellite operators to temporarily suspend services in order to feather the solar panels away from the direction of onslaught, thereby reducing the probability of damage to solar cells. Man-made debris, consisting mainly of expended solid rocket propellant dust, exploded rocket stages and jettisoned satellite hardware, is another source of concern for spacecraft designers and operators. With average velocities of 11 km/s relative to orbiting spacecraft, the above comparative impact equation yields impressive results. In contrast to meteorids, man-made debris is mostly confined to specific orbital bands. Impacting meteorids and debris are of primary concern in manned space flight where loss of hermeticity can be life-threatening. A common solution is to encapsulate the module with a double-walled hull, such that an incoming particle is vaporized or fragmented by the first wall before being deflected by the second.

Historical notes: Two GEO satellites – Westar 6 and Palapa B2 – were launched in August 1984 but finished up in LEO when their perigee kick motors failed to ignite. They were retrieved by the Space Shuttle in April 1985. After 8 months in low earth orbit, they showed clearly visible meteorid impact craters in the cover glass of several solar cells. Examples of collisions in space between man-made objects are the 0.2 mm paint flake that caused a 4-mm crater in the Space Shuttle’s wind screen in 1983, the fragment of an Ariane rocket that knocked off the gravity gradient mast of the Cérise spacecraft in July 1996, and the head-on collision between an expired Cosmos spacecraft and a still active Iridium satellite in February 2009.

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Many other satellite anomalies (e.g. attitude disturbances, solar panel failures and optical instrument defects) have been attributed to impacting particles. There is a natural temptation to reach for this explanation whenever onboard failures seem to defy logic.

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12.3.7. Galactic Cosmic Radiation

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Outer space is criss-crossed by a motley collection of charged particles originating from stars other than our own sun, and possibly from the Big Bang itself. These particles consist of hydrogen, oxygen, helium and various other nuclei. Though their density is extremely low, their energy is sufficient to cause upsets in digital circuits onboard satellites. They are also blamed for generating background noise in attitude sensors, causing the sensors’ output signals to be corrupted.

12.3.8. Solar Wind and Particle Showers

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The sun emits a steady breeze of charged particles, primarily protons and electrons, known as the solar wind. Many of these become trapped in the Van Allen belts discussed earlier. Eruptions on the sun occasionally turn the breeze into veritable storms which, like cosmic radiation, have been known to upset and even cripple satellites.

Single Event Phenomena: High-energy charged particles are capable of penetrating equipment walls and corrupting digital states. A single particle can cause a bit-flip which is either temporary (single event upset, or SEU) or more permanent (single event latch-up, SEL). While a SEU is rectified by subsequent bit traffic and does no permanent damage to the component in question, the operational consequences can be significant. Examples are inadvertent thruster activation or mission interruption. SEL is potentially more serious since the anomaly can cause both functional outage and component damage. The problem is solved by removing and then restoring power in the entire circuit.

12.3.9. Solar Pressure

While the solar wind is made up of charged particles, solar pressure is caused by pure electromagnetic radiation. The two concepts are sometimes confused, perhaps because the radiation acts like a wind. Any orbiting surface facing the sun experiences a pressure force. If the surface is flat, perpendicular to the sun vector, and absorbs all the solar energy, then the corresponding solar force F is obtained as: F=A

S c

(12.7)

where A is the area of the surface, S = 1367 W/m2 is the average solar energy flux in earth orbit, and c ≈ 3 × 108 m/s is the speed of light. (The equation is easier to remember if we recall that Watts is the same as Newton-meter and make a dimensional analysis, yielding F in Newton.) Even though the force is miniscule, it is a contributing factor to orbital perturbations. Moreover, a satellite that manifests geometrical asymmetry, as seen from 408

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the sun, is subject to attitude disturbances (Figure 12-15). Both types of disturbance are explored further in Chapters 4 and 5.

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Figure 12-15 Attitude and orbit perturbations due to solar pressure.

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In practice, the illuminated surface may be neither flat nor perpendicular, nor indeed totally absorbing. We must therefore: a) calculate the surface that is effectively facing the sun; and

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b) estimate to what extent the surface offers specular reflexion, diffuse reflexion, and absorption, and compute the force components resulting from each.

To achieve (a), we return to the method described at the end of Section 3.2.3, i.e. we measure the size of the shadow cast by the surface on an imaginary screen placed perpendicular to the sun vector: d

h

r

Figure 12-16 Estimating effective illuminated areas.

For example, the area of the shadow behind a sphere A = πr2, while behind a cylinder or a flat panel the area A = dh cos φ if illuminated at an angle φ to the normal. The area behind a tilted box requires a little more skill in trigonometry, as shown in the sketch to the right. As for (b), let us examine Figure 12-17.

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φ

Absorption:

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Fa = S/c * A cos φ

φ Specular reflection:

φ

s

φ φ

Diffuse reflection:

s

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Fs = 2 S/c A cos2φ

Fd = 2/3 S/c A cos φ

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Figure 12-17 Main types of solar illumination.

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The net solar pressure force vector F is the vector sum of the above three components, suitably weighted with an absorption coefficient α, a specular coefficient β and a diffuse coefficient γ:

F = αFa + βFs + γFd

with

(12.8)

α+β+γ=1

A⋅ S A⋅ S 2 A⋅ S cos φ s; Fs = 2 cos2φ n; Fd = cos φ n c c 3 c s is the solar unit vector; n is the panel normal unit vector.

Fa =

and

For example, the solar panel in Figure 12-15 may have α = 0.7, β = 0.2 and γ = 0.1 (recalling that the main purpose of a solar panel is to absorb sunlight so as to generate the maximum amount of electric energy). If we are interested in the torque that disturbs the satellite’s attitude, we should look at the component of F that is parallel with the sun vector s: F = F·s = αFa·s + βFs·s + γFd·s = A⋅ S = cos φ (α + 2β cos2φ + ⅔γ cos φ) c

(12.9)

Let us assume that φ = 0, as in Figure 12-15, and that the panel area A = 1 m2. Then: F=

1 ⋅ 1353 (0.7 + 2·0.2 + ⅔·0.1) = 5.26 ·10-6 N 8 3 ⋅ 10

i.e. a very weak force indeed, but sufficient to cause unacceptable attitude disturbances over time.

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Numerical example: If a solar sail measuring A = 1 km is deployed 1 A.U. from the sun and is turned to face it, and if the sail absorbs all the sunlight, it will find itself pushed into deep space with a force F = 4.5 N. This may not seem like much. But suppose the sail weighs m = 450 kg and starts its journey with near-zero velocity. It will then experience an 2 acceleration a = F/m = 0.001 m/s . After one year (t ≈ 30,000,000 seconds) it will have 2 12 covered a linear distance s = ½ a·t = 450 × 10 km and be travelling at a speed of V = a·t = 30 km/s.

Ultraviolet Radiation

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A solar sail is therefore an excellent solution for interplanetary exploration, at least conceptually. Practical hurdles include building, launching and deploying the hyper-thin sail, keeping it on a linear trajectory, maintaining the attitude, and coping with the everweakening sunlight as the distance from the sun grows.

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Ultraviolet light from the sun accelerates the oxidation of oils and resins traditionally found in plastics, lubricants and paints, causing a deterioration over time in their functionality. Even on earth, where the atmosphere constitutes a fairly effective UV filter, we sometimes notice how white-painted surfaces turn yellow and plastic objects become brittle (e.g. household appliances stored near a window). A satellite in space enjoys no protection whatsoever from UV radiation, and the designer therefore has to take extra care in his selection of UV-resistant materials. The adhesive employed to attach cover glasses to solar cells has been known to become opaque under the influence of UV radiation; hence the practice of applying a UV filter on the inner surface of cover glasses (the outer surface is occupied by an anti-reflective coating). Anti-reflective coating

Cover glass

UV filter

Adhesive

Solar cell

Figure 12-18 Solar cell coatings.

12.4. Internal Environment

The environments described above have their origins external to the satellite. It is also meaningful to talk about internal environments, notably thermal and electromagnetic. The thermal environment has already been described in Chapter 10, so here we will deal with electromagnetic interference (EMI) and electromagnetic compatibility (EMC) Any electronic circuit that conducts rapidly changing currents is prone to emit an electromagnetic wave, and this wave may cause currents in adjacent circuits through 411

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conduction and radiation. Air travellers are familiar with the cabin crew’s announcements before take-off and landing that all electronic equipment must be switched off, be it laptop computers, mobile phones or electronic games. The reason is that the internal circuitry conducts modulated carriers, and these seep out of the equipment in the form of radio waves. In the absence on an antenna, the signals are usually weak, but they are still capable of causing interference with the aircraft avionics. Activating a power switch might also give rise to an interfering electromagnetic pulse.

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Within the confines of a satellite, electromagnetic interference is a major headache for the designers. EMI is conveyed in two ways. Low-frequency signals (below 10 MHz) travel mostly by conduction via the power distribution wiring, and sometimes through the mounting platform. Higher-frequency signals are predominantly radiated through the vacuum inside the satellite.

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One refers to conducted and radiated emission to quantify an equipment’s propensity to emit EMI. At the receiving end, units are classified by their conducted and radiated susceptibility. For the designer, the remedy against EMI is to accomplish electromagnetic compatibility (EMC) between the units. Methods include internal frequency planning, relocating or reorientating units, introducing bypass capacitors, shielding cables, and twisting wires.

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These days, when nearly all electronics is digital, the most common effect of EMI is bit flips in digital registers. The inverted bits might corrupt transmitted images and other data. In the worst case, the satellite may suffer irreparable harm. Attempts have been made to model EMI mathematically for use in simulators, but the results are not always accurate. A more reliable method is to inject test currents judiciously in an actual satellite. For the results to be trustworthy, the design of the tested satellite must be as authentic as possible. It is therefore tempting to use the flight model itself, even if there is some risk of damaging it, and even if any modifications that arise become prohibitively expensive at such a late stage. Note that EMI does not always originate internally, but may sometimes have external causes. The satellite itself may also be the cause of EMI elsewhere. For example, launch service providers impose strict EMI limitations on satellite designers to prevent interference with the launcher’s destruct system. (The destruct system is basically a detonator strapped to the rocket. It is ignited by ground command to destroy the rocket if it veers off course during launch and threatens to impact on the ground.)

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12.5. Solved Problems See Appendic C for solutions.

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12.5.1 A geostationary satellite offsets its solar wings by an angle φ = 10 deg from normal in a windmill fashion. The aim is to achieve attitude control around the roll and yaw axes through quarter-orbit coupling. The satellite body is a 2 x 2 x 2 m cube. Each of the two solar wings is 6 m long and 2 m wide, and is mounted on a 1 m yoke. How long will it take for the satellite to turn an angle Ω = 1 deg?

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Assume that α = 0.7, β = 0.2 and γ = 0.1, and that the moments of inertia Ix = Iz = Ixz = 6,000 kg m2.

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13. Product Assurance

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13.1. Introduction

Once launched, a satellite is beyond reach for repairs, so it has to be designed and built with the idea in mind that it should last on its own for 5 – 15 years in a very hostile environment. The quality requirements are therefore exceedingly strict and account for a significant portion of a satellite’s cost. Product assurance (PA) is a somewhat wider concept than quality assurance, and it is quite specific.

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Under PA, satellite components undergo rigorous analysis and test to ascertain that they will survive in orbit.

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Materials such as metals, composites, paints and lubricants are similarly scrutinized, as are the processes used to combine them.

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A good PA programme leads to high product reliability. As we shall see in the following, reliable systems may be achieved by combining even unreliable functional in a certain manner. Quality control is a subset of PA and involves a number of activities to ascertain that manufacturing is performed by qualified personnel in a suitable environment. As designs evolve, it is essential that their mechanical and electrical interfaces remain compatible – a responsibility of configuration management.

Many activities during the manufacturing, testing and launching of satellites are hazardous. Therefore, elaborate safety measures must be taken to prevent mishaps, and to protect human beings and facilities if they do occur.

13.2. Parts Engineering

A piece part, also known as a component, is an electrical, electronic or mechanical device that performs a single function and is made of elements which cannot be separated without destroying its functionality. Examples are semiconductors, transformers, crystals and filters. The design of most electronic components used onboard satellites stem from commercial product lines, as used in ground-based hi-tech instruments and even in consumer goods. What makes them different is the environment in which they are manufactured, and their rigorous testing programme. Manufacturing takes place under strictly controlled environmental conditions, with cleanliness and humidity being primary parameters. Once produced, all of the parts are subjected to non-destructive screening tests aimed at weeding out components prone to

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“infant mortality.” The test programme may include acceleration, vibration, thermal cycling, visual and X-ray inspection, and hermeticity.

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Space-qualified parts are identified in qualified parts lists (QPL) maintained by the PA departments in major space organisations. Within the QPL there is a preferred parts list (PPL) containing components with a particularly favourable quality record. Manufacturers that use PPL components are ensured a smooth ride through the PA parts scrutiny.

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However, a particular PPL part may not always be the component of choice, either because it is no longer being produced (obsolescence), or because a newer and ostensibly better component has come on the market but has not yet made it into the QPL and PPL. The unavailability of obsolescent components may be problematic e.g. when a manufacturer has been asked to build a follow-up satellite based on the original design. If he opts to swap the components for new designs, he runs the risk of having to re-qualify the entire unit in which the components are used. Alternatively he may go shopping among other manufacturers to see if they still have the old items in stock. If he prefers to use a new component design, he must have someone qualify it, or else do it himself, in either case under strict PA surveillance

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Mechanical and electromechanical components include relays, bearings, hinges and cutters. These usually form part of mechanisms. As discussed in Section 9.3, mechanisms are inherently unreliable, and PA involvement is sought at an early stage of design. In contrast to electronic components, mechanical parts often have no design heritage in the commercial sector and are therefore purpose-built. As a result, there may not be a meaningful statistical sample for reliability analysis, so a more holistic approach is necessary. The approach consists in ensuring that enough design margins exist within the predicted flight envelope, and that these margins are demonstrated through test. Ideally a prototype should be built of the entire mechanism and be subjected to destructive testing, in order to establish which part constitutes the most critical link. However, most manufacturers hesitate to proceed in this manner due to the time and cost implications.

13.3. Materials and Processes

We are all familiar with the tendency of certain painted surfaces to change colour if exposed to daylight. For example, the top of a white fax machine may turn yellow, or a piece of coloured fabric may fade. If these materials were used for passive thermal control onboard a satellite, the colour changes would alter their absorbivity and emissivity coefficients, with potentially harmful consequences for the units thus covered. Since the beginning of space exploration, the behaviour of materials under the influence of vacuum, radiation and extreme temperatures has yielded many surprises, most of them unpleasant. Lubricants have been known to turn gritty, paints have flaked off, clear adhesives have become opaque, and synthetic compounds have turned brittle. One of the most persistent problems has been so-called outgassing, a form of “sweating” where certain materials decompose in vacuum and the most volatile components condense on adjacent surfaces, causing a change in their thermal or optical properties. Outgassing is discussed further in Section 12.3.2. 416

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Over the years, the above anomalies have been catalogued, and many products have been banned from use in space. In analogy with approved parts lists, there exist approved materials lists for easy reference by spacecraft designers. New materials with attractive properties appear regularly on the market. To make it into the list of approved materials, they are first subjected to extensive testing to ascertain their immunity against outgassing, discoloration, embrittlement, etc.

13.4. Reliability

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PA is concerned not only with the materials themselves, but also in their manner of application. Taking one more example from daily life, we know that a wooden window frame needs to be dry, scraped clean, primed and undercoated before it receives two coats of gloss paint. This is known as the process of applying materials and must follow established procedures. Other process examples are welding, soldering, bonding, metal finishing, encapsulation, and crimping of wires.

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The reliability R of a functional unit is the probability P that it will continue to function for a given duration t. At that time the unit is assumed to fail, or at least become unusable. We have 0 ≤ P(t) ≤ 1 , i.e. P lies in the range 0% – 100%. R and P are used interchangeably in many texts, so in the present Section we will adhere to P throughout.

13.4.1. Mathematical Modelling

As a first approach, we may postulate that the probability takes the following form: P (t ) = e − λt

(13.1)

where λ is a constant failure rate (i.e. λ failures every so many hours, on average) and t is the time from the beginning of the unit’s activation. Graphically, Eq 13.1 would appear as in Figure 13-1: Probability P(t) of satisfactory performance 1.0

0.0

Time

Figure 13-1 Probability of satisfactory performance over time.

The curve in Figure 13-1 agrees with our experience from daily life, namely that the longer we use an appliance or an instrument, the higher the probability that it will fail. 417

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Stated differently, the probability that it will continue to function in a satisfactory manner diminishes over time.

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In Eq 13.1 we made the assumption that the failure rate λ is constant. In real life, this may not be true. Figure 13-2 shows typical failure rates for electronic and mechanical equipment onboard satellites in space. Failure rate λ

Infant mortality

Random failures

Wear-out

Wear-out

Mechanical equipment

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Electrical equipment

Infant Random mortality failures

Time

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Failure rate λ

Figure 13-2 Variable failure rates over the lifetime of an equipment (not to scale).

Here again, the diagrams in Figure 13-2 tally with our everyday experience. For example, lightbulbs usually work after initial installation but will eventually fail after a few months or even years. On the other hand, a new automobile may require some warranty repairs after the first few weeks of purchase and needs ever more frequent and expensive repairs after 150,000 km. This is to say that most electric appliances seem to enjoy a long period of constant and low failure rate, whereas complex mechanical equipment have a more protracted failure incidence at the beginning and end of life. Satellites, with their complex mix of electrical and mechanical equipment, manifest both bathtub curves in Figure 13-2. It is not uncommon for satellites to suffer “infant mortality” shortly after a successful launch – e.g. mechanisms that refuse to deploy or electronics that cannot be activated. The electronics tend to settle down faster than continuously running mechanisms, such as electromechanical motors. Statistically, the latter also show earlier signs of wear-out. For example, increased friction in the bearings manifests itself in the telemetry as an increase in the amount of current drawn by the motor. The most cost-effective way of reducing the risks of infant mortality and wear-out is to implement a rigorous product assurance programme, notably in the selection of parts and materials, and in the quality oversight. A well-chosen testing strategy will reduce the risks further, particularly during the infant mortality and wear-out stages. However, when it comes to preventing random failures, the randomness itself limits the effectiveness of quality assurance and testing. A better approach is to either operate the components in a derated mode, or else to create functional redundancy. Most satellite designers prefer a combination of both. 418

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13.4.2. Component Derating

140 120

Loading (%)

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Failure rate λ

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Most component manufacturers indicate within which physical limits a component may be operated to achieve full performance. These are the rated limits. For example, the rated limits for a transistor may be maximum 5 volts at a maximum temperature of 80°C. Some manufacturers also specify the component’s random failure rate λ when operated at full rating as well as in a backed-off (derated) mode. The specification may look like this:

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80 60 40 20

0

20

40

60

80

100

Temperature °C

Figure 13-3 Component rating.

Figure 13-3 illustrates how a dramatic improvement in the random failure rate may be achieved by operating the component at a lower than operational temperature, or else by imposing a lower load in terms of volts, amps, watts, Newtons, or whatever. It is not unusual for certain satellite components to be thus “derated” to 50% or less of their rated values. Derating improves the failure rate not only during the random failure phase, but also during the infant mortality and wear-out stages.

13.4.3. Redundancy

Redundancy is synonymous with duplication, i.e. we create redundancy by duplicating (or triplicating, etc.) equipment. Functionally, redundancy is achieved by connecting units in parallel rather than in series. Looking at averages, the assumption that random failures occur at a constant rate has proven to be fairly accurate, as long as the causes of failure are truly independent of each other. Eq. 13.1 with its constant failure rate λ is therefore a valid starting point for designing optimum redundancy into the satellite’s subsystems. The random failure stage deserves a thorough analysis in the present text, the more so since it lasts typically 90% of the satellite’s lifetime. 419

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A satellite owner’s functional specification will stipulate that the satellite shall perform adequately in orbit for a duration of, say, 10 years with a probability of 0.7, and he expects the manufacturer to demonstrate through analysis that this requirement will be met. By “adequately” he means that the mission objectives are to be met in full, allowance being made for redundant equipment replacing malfunctioning equipment if necessary.

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Working from the top down, we assume that each of the payload and platform subsystems is mission critical, i.e. the satellite cannot perform adequately if a subsystem fails. We must therefore demonstrate that, at the end of the 10-year lifetime, the following identity applies: Psat = PpayloadPpowerPattitudePpropulsionPOBDHPS&MPthermal ≥ 0.7

(13.2)

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In our example, the OBDH includes TT&C, and the subscript S&M stands for Structures & Mechanisms.

Q: Why are the subsystem probabilities in Eq 13.2 multiplied instead of added together?

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A: In probability theory the word “and” corresponds to multiplication, while “or” is represented by the plus sign. Eq 13.2 therefore states that Ppayload and Ppower and Pattitude and ... must all work. If the requirement had been for exactly one of the subsystems to work, we would have said Ppayload or Ppower or Pattitude or ..., and Eq 13.2 would have been written: Psat = Ppayload + Ppower + Pattitude + Ppropulsion + POBDH + PS&M + Pthermal ≥ 0.7

Where do these probability figures come from? They are not arbitrary, but have been compiled and tabulated over the years from ground testing and flight experience. Actually it is not the probabilities that have been compiled, since they are time-dependent, but rather the failure rates λ. For these rates to be trustworthy, they must refer to relatively simple items that have been built in large quantities – in other words, components such as diodes, transistors, capacitors, inductors, resistors, bearings, etc. The sequence of events in reliability calculations is therefore to begin with the satellite owner’s lifetime requirement, apportion the probability among the subsystems, re-apportion them among the units within the subsystems, and then use component failure rates to design compliant units.

The failure rates of components are usually given in FITs, which means “Failures in 109 hours.” Hence, λ = FITs · 10-9, and the probability P of a unit surviving during t hours equals e-λt, as per Eq 13.1. (The reason for preferring FITs to λ is that the former quantities are more manageable.) FIT values for basic components lie in the range 1 – 50. In Table 13-1, FIT values have been tabulated for some typical assembled units, along with their derived probabilities for a 10-year survival. Thus we have – P(10 years) = exp(-FIT · 10-9 · 10 · 365 · 24) = e-0.0000876 FIT

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0.991 0.991 0.957 0.991 0.983 0.983 0.996 0.983 0.974 0.974 0.996 0.991 0.987 0.949

Propellant tank Thruster Thruster latch valve Pyrotechnic valve Propellant filter Telecommand antenna Telecommand receiver Telecommand decoder Telemetry formatter Telemetry transmitter Telemetry antenna Central processor Mass memory Remote terminal unit

100 400 600 50 0 0 700 800 500 600 0 300 100 25

0.991 0.966 0.949 0.996 1.000 1.000 0.941 0.932 0.957 0.949 1.000 0.974 0.991 0.998

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100 100 500 100 200 200 50 200 300 300 50 100 150 600

P FITs 10 yrs

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Solar arrays Array switching regulator BAPTA Battery Battery charge regulator Battery discharge regulator Power distribution Sun sensor Earth sensor Star tracker Gyro Sensor data converter Attitude control actuator Reaction wheel

P FITs 10 yrs

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Lifetime (years): 10

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Table 13-1 Typical failure rates for satellite units.

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In Eq 13.2 we have 7 subsystems. For the equation to be satisfied, it is clear that each subsystem probability should preferably be (Psubsystem)7 > 0.7, or Psubsystem > 0.95. This is an average, so if one subsystem falls short, one or more of the others will have to make up for it with a higher probability. Our top-down approach leads us to the units of each subsystem. If two active units A and B are connected in series (Figure 13-4), we know that: PAB = PA PB

A

(13.3)

B

Figure 13-4 Two functional units connected in series.

Let us see what happens to their combined probabilities if they are connected in parallel (Figure 13-5).

A B

Figure 13-5 Two functional units connected in parallel.

In this case we have:

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QAB = QA QB

(13.4)

where Q is the “unreliability” of the unit in question, such that Q = 1 – P. Therefore, (13.5)

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PAB = 1 – QAB = 1 – (1 – PA)(1 – PB) = PA + PB - PAPB

all this under the assumption that units A and B are in hot standby (i.e. both are activated) and that the switches (shown as diamonds) are perfectly reliable (P = 1).

Serial: Parallel:

PAB = 0.9 x 0.8 = 0.72 PAB = 1 – (1 – 0.9)(1 – 0.8) = 0.98

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Which of the two configurations is more reliable? Let us take a numerical example with PA = 0.9 and PB = 0.8. Then:

PAB = 0.7 x 0.5 = 0.35 PAB = 1 – (1 – 0.7)(1 – 0.5) = 0.85

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Serial: Parallel:

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The parallel connection is clearly much more reliable. The improvement is even more spectacular, the lower the reliability of the individual units. For example, with PA = 0.7 and PB = 0.5 we have:

These two examples illustrate the fact that two unreliable units are best connected in parallel, while there is little benefit in doing so with two highly reliable units. The above discussion is artificial insofar as we are unlikely to want to connect two dissimilar units in parallel. For instance, it makes no sense to connect the solar panel and the BAPTA in parallel, because the BAPTA cannot make up for a failed solar panel, and vice versa. However, we could improve the reliability of the assembly by letting the BAPTA drive two separate but identical solar panels. If the panels are called A and the BAPTA B, then Figure 13-5 is modified as follows:

A

B

A

Figure 13-6 Two functional units connected in parallel.

Here again, let us assume that PA = 0.7 and PB = 0.5. Then:

Unit A non-redundant:PAB = PA PB = 0.7 x 0.5 = 0.35 Unit A redundant: PAB = [1–(1–PA)(1–PA)]PB = [1–(1–0.7)(1–0.7)] x 0.5 = 0.45

An even better result would have been achieved if we had duplicated the least reliable unit, namely the BAPTA, instead of the more reliable solar panels:

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PAB = PA [1–(1–PB)(1–PB)] = 0.7 x [1–(1–0.5)(1–0.5)] = 0.53

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In practice it is of course engineering nonsense to try to duplicate the solar panels or the BAPTAs in this manner, so the above examples merely serve to quantify the advantage of duplicating unreliable units.

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So far we have discussed redundant units in hot standby, whereby the spare unit remains powered on and is immediately available if the primary unit fails. It is therefore assumed that the lifetime of the spare is being consumed at the same rate as the operational unit. In cold standby, the spare unit is switched off until such time that it is needed, such that its “body clock” (i.e. time t) does not start ticking until it is switched on. The following identities apply: Two units A in hot redundancy: PAA(t) = 1 – [1 – PA(t)]2 (as per Eq 13.5)

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(13.6)

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Two units A in cold redundancy: PAA(t) = PA(t) ·(1 + λt)

(13.7)

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The relative reliability of two units in hot and cold redundancy is illustrated in Figure 13-7. 1.000

Reliability P AA

0.998 0.996

Cold

0.994 0.992 0.990

Hot

0.988 0.986 0.984

0

5

10

15

Lifetime t (years)

Figure 13-7 Reliability comparison between hot and cold redundancy using two identical units A with a failure rate of 1000 FITs each.

Eq 13.7 assumes that the “cold” unit will never fail while dormant, and therefore cold redundancy is obviously preferable to hot redundancy. However, the assumption itself is flawed. Let us take a familiar analogy with two automobiles, one of which is in daily use while the other is left parked in the street for many months. The active vehicle may exhibit failures as the months go by, but so may the dormant vehicle due to a variety of aging processes, like lubricants being contaminated by moisture, or disc brakes becoming corroded. As a first approximation in spacecraft design, one may assume that the failure frequency λ of the dormant spare is 1/10 of that of the active unit, in which case – Two units A in cold redundancy:

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1   PAA = PA 1 + (1 − e − 0.1λt )  10  (according to the 1/10 rule)

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(13.8)

Numerical example: Compute the system reliabilities PAA when two units A are connected (a) in hot redundancy, (b) in “ideal” cold redundancy, and (c) in a more realistic cold redundancy using the 1/10 rule. The nominal FIT rate of each unit is 1000, and the lifetime is 12 years. -9

-λt

(hot) (cold, no failures while dormant) (cold, 1/10 of the failure rate of active unit)

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(a) PAA = 0.990 (b) PAA = 0.995 (c) PAA = 0.994

= 0.900.

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We have λt = 1000 x 10 x 12 x 365.25 x 24 = 0.105, and PA(12 yrs) = e

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From Figure 13-7 it is evident that the advantage of cold redundancy over hot ditto diminishes with shorter lifetimes.

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In addition to the three redundancy cases (a), (b), (c) in the above example, there is a fourth operational scenario. Suppose we have n power amplifiers configured in active redundancy, of which k amplifiers must operate at any one time (Figure 13-8).

Figure 13-8 A total of n redundant amplifiers, out of which k amplifiers (shaded) are working.

The probability that exactly k amplifiers will operate (not necessarily the first k, but any k) is found from the binomial equation – Pk ,n =

n! k Pampl (1 − Pampl ) n −k k!(n − k )!

(13.9)

For example, if Pampl = 0.9, n = 6 and k = 4, then: P4,6 =

6! 0.9 4 (1 − 0.9)6 − 4 = 0.10 4!(6 − 4)!

This low value may come as a surprise, since intuitively one would expect 4 such reliable amplifiers to work with a high probability. Let us plot Pn,k for all values of k from 0 to 6 and see what happens.

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Figure 13-9 Probability of exactly k amplifiers out of n working.

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The graph confirms our calculation, i.e. P4,6 = 0.10. The reason for the low value is that, because the amplifiers are so reliable, the most likely situation is that all of them work, so the probability that 4 (and only 4) function is actually lower. That said, the probability that all 6 amplifiers work is only 0.5 in our example.

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The engineer in charge of designing the amplifier system is probably more interested in knowing the probability that at least 4 out of 6 amplifiers work, rather than the probability that exactly 4 out of 6 function. In this case the question may be rephrased as follows: What is the probability that exactly 4 work, or exactly 5 work, or exactly 6 work. Remember from our earlier discussion that the word “or” corresponds to the plus sign in probability theory. Therefore: P4,6 =

6! 6! 6! 0.9 4 (1 − 0.9) 6− 4 + 0.9 5 (1 − 0.9) 6−5 + 0.9 6 (1 − 0.9) 6−6 = 4!(6 − 4)! 5!(6 − 5)! 6!(6 − 6)! = 0.10 + 0.35 + 0.53 = 0.98

given that 0! = 1 and 0.90 = 1. In other words, the probability that at least 4 out of 6 amplifiers operate correctly is 0.98, which in most missions would be an acceptable value. Generally speaking, the probability that at least k out of n redundant units work is found from the following identity: n

Pk ,n = ∑ i =k

n! P i (1 − P ) n −i i!(n − i )!

(13.10)

In Figure 13-10 we have plotted Eq 13.10 for various values of unit probability P.

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1.00 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.00

Punit = 0.5

0

1

2

0.6

3

0.7

0.8

4

0.9

5

6

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Figure 13-10 Probability of at least k units out of n working.

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For example, the probability of at least 5 out of 6 units working is 0.9 if the unit’s reliability Punit = 0.9. If the unit’s reliability is very poor, say 0.5, then we can only expect at least 2 units out of 6 to work with a 0.9 probability. And so forth.

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We are now ready to examine the need for redundancy in the various subsystems of a hypothetical communications satellite shown in Figure 13-11. We will use the reliability figures from Table 13-1.

Figure 13-11 Hypothetical GEO communications satellite with deployable solar panels driven by BAPTAs, a deployable parabolic antenna, a deployable TT&C mast, two thrusters, and a set of reaction wheels.

At the beginning of this Section we made the assumption that a satellite owner might stipulate a 10-year lifetime with a probability of 0.7. For this requirement to be met, we calculated that the average reliability of each of the seven subsystems should be = 0.95, and that if one subsystem fell below that threshold, the others would have to make up for it. Let us begin with the power management subsystem (Figure 13-12).

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Battery charge regulator P5=0.983

Array switching regulator P4=0.991

Battery P6=0.991

Solar panel P3=0.991

Array switching regulator P4=0.991

Battery discharge regulator P7=0.983

Power distribution P8=0.996

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Solar panel P1=0.991

Ppower = P1P2P3P4P5P6P7P8 = 0.920

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Figure 13-12 Reliability diagram for the power management subsystem.

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Note the absence of solar panel deployment mechanisms in Figure 13-12. In Chapter 9 we stated that mechanisms are notoriously unreliable, so why are they not included? The reason is that here we are dealing with reliability during the random failure phase that lasts for years, while deployments take place only once immediately after launch. Failure of once-only deployments therefore belongs in the infant mortality phase of a satellite’s lifetime.

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Also note that we have omitted the readily predictable (i.e. not random) phenomenon of solar cell degradation due to UV radiation. This phenomenon belongs in the wear-out phase of the satellite’s lifetime. However, solar cell failures due to open or short circuits are random and are therefore included in the reliability figure for solar panels. Even though the solar panel & array switching regulator pairs are connected in parallel in the actual satellite, the satellite usually requires the power output from both. This means that both pairs are mission critical, and they are therefore represented in series in Figure 13-12. The other units are also assumed to be mission critical and are connected in series. The BAPTAs are found under Mechanisms in Figure 13-16 below. The reliability diagram for the attitude management subsystem is shown in Figure 13-13.

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Sun sensor P1=0.983

Earth sensor P2=0.974

Star tracker P3=0.974

Earth sensor P2=0.974

Star tracker P3=0.974

Gyro P4=0.996

Sensor data conversion P5=0.991

Gyro P4=0.996

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Sun sensor P1=0.983

Reaction wheel 1 P7=0.949

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Reaction wheel 4 P7=0.949

Thruster valve 2 P8=0.949

Thruster 1 P9=0.966

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Reaction wheel 3 P7=0.949

Thruster valve 1 P8=0.949

Thruster 2 P9=0.966

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Attitude controller & actuator P6=0.987

Reaction wheel 2 P7=0.949

Pattitude = [1 – (1-P1)2] [1 – (1-P2)2] [1 – (1-P3)2] [1 – (1-P4)2]P5P6 4 4! i 4− i ·[ ] [1 – (1-P8P9)2] = 0.956 P7 (1 − P7 ) i = 3 i ! ( 4 − i )!

∑

Figure 13-13 Reliability diagram for the attitude management subsystem.

The first three attitude sensors are not as reliable as one would wish (P < 0.995), so they have been duplicated. The gyro, which is small, light and inexpensive, has also been doubled up for good measure. One might argue that the sun sensor, the earth sensor and the star tracker are not all needed at the same time, i.e. if one of them were to fail, one might manage with the remaining two after some software modifications on the ground. If this were the case, there would be a justification for omitting the redundancy. The four reaction wheels are configured as in Figure 4-55, such that only 3 out of four are needed; hence the binomial formula applies with n = 4 and k = 3 → 4.

Only one redundant set of thrusters has been included for simplicity in this example, whereas a real GEO satellite would have at least six sets. The propulsion subsystem in Figure 13-14 runs on monopropellant hydrazine. The two tanks exist not so much for redundancy as to create dynamic balance. For practical reasons, each branch has its own latch valve and filter.

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Prop tank P4 =991

Latch valve P3 =0.96

Prop tank P4 =991

Filter P5 =1.0

Thruster valve 2 P6 =0.949

Thruster 2 P7 =0.966

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Thruster 1 P7 =0.966

Filter P5 =1.0

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Pyro valve P2 =0.996

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Pressurant tank P1 =0.991

Latch valve P3 =0.96

Ppropulsion = P1 P2 [1 – (1-P3P4P5)2] [1 – (1-P6P7)2] = 0.977

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Figure 13-14 Reliability diagram for the propulsion subsystem.

The two thrusters in Figure 13-14 are intended for orbit control, while those in Figure 13-13 are for attitude control. One of the most mission-critical parameters is the amount of propellant carried in the tanks. In fact, propellant depletion is the most common reason for mission termination. However, propellant depletion is not a random failure but a predictable one – unless it is caused by random failure of the thruster or its valve. Both of those events are covered in Figure 13-14. In Figure 13-15 we show the combined TT&C and OBDH subsystems. The telecommand and telemetry functions are obviously mission critical, and cross-strapping is the norm. The deployment of antennas does not figure in the diagram for the same reason as that given above for solar panels. Once deployed, antennas are, for all intents and purposes, totally reliable.

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TC| decoder P3 =0.932

TM transmitter P11 = 0.949

TM formatter P10 =0.957

TM transmitter P11 = 0.949

TM formatter P10 =0.957

Propulsion RTU P5 =0.998

Central processor P9 = 0.974

Mass memory P8 = 0.991

Attitude RTU P6 =0.998

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TM antenna P12 =1.0

TC receiver P2 = 0.941

Thermal RTU P4 =0.998

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Power RTU P7 =0.998

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POBDH = P1 [1 – (1-P2)2] [1 – (1-P3)2] P4P5P6P7P8P9 [1 – (1-P10)2] [1 – (1-P11)2] P12 = 0.945 Figure 13-15 Reliability diagram for the combined TT&C and OBDH subsystems.

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If the structure is going to fail, it will do so during launch, and the failure therefore belongs in the “infant mortality” phase. The BAPTAs are of course mechanisms, but they differ from deployment mechanisms by running continuously throughout the satellite’s operational life. In Figure 13-16 they are shown connected in series, since both are mission critical. Structure P1 =1.0

BAPTA P2 = 0.957

BAPTA P2 = 0.957

PS$M = P1 P22 = 0.916

Figure 13-16 Reliability diagram for the combined structure and mechanisms subsystem.

The thermal management subsystem is largely passive, give or take a dozen simple heaters and an associated driver. Some of these heaters may not even be mission critical. A survival probability of 0.995 is therefore realistic. The probability of the satellite surviving for 10 years is obtained by multiplying the various subsystem probabilities. Thus: Psat = PpayloadPpowerPattitudePpropulsionPOBDHPS&MPthermal

= 0.960 x 0.920 x 0.956 x 0.977 x 0.945 x 0.916 x 0.995 = 0.710

Our design therefore just barely meets our satellite owner’s functional specification that the satellite shall function in a satisfactory manner for 10 years with a probability of 0.700. 430

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There is no incentive to surpass the requirements of the functional specifications by a wider margin because, one way or another, the over-compliance will cost weight and/or money for redundancy or reliability that is not needed. Moreover, unnecessary dry weight will translate into added propellant mass to operate the satellite in orbit during the stipulated lifetime. Taken together, the extra dry and propellant mass are likely to trigger an increase in the launch cost. Few satellite owners are willing to pay extra for an overcompliant spacecraft.

13.5. Quality Assurance

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Quality assurance (QA) is a catch-all name for the many tasks of ensuring that satellite design, development, integration and test occur according to established PA rules.

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A typical day in the life of QA may involve a review of proposed parts and materials, inspection of an assembly line of printed circuit boards, checking that the assembly staff are certified in skills such as soldering, participating in a change review board to ensure that proposed design changes are notified to everybody concerned, and attending a failure review board to help assess the consequences of a test failure.

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The role of QA extends from work carried out at home base to that being performed at the facilities of subcontractors and suppliers. QA also includes incoming inspection of delivered equipment, and surveillance of activities associated with handling, packaging, shipping and storage of satellite equipment at all levels.

13.6. Configuration Management

It goes without saying that the tens or hundreds of functional units onboard a satellite must function flawlessly together, but this is not a trivial task for two reasons: 1) Each new satellite design is unique and contains a large amount of cutting edge technology. In the course of a satellite’s design and development phase, it is only natural that the design of some units may have to be modified, either because the contemplated design appears not to meet the specifications, or because the specifications themselves have been changed. With each modification there is a possibility that the electrical or mechanical interfaces change in a manner that affects the design of other functional units. 2) Dozens of manufacturers are involved in the design and development of individual satellite units. Usually these companies are geographically dispersed nationally and internationally, so that day-to-day cooperation between the various engineers is impracticable.

To ensure that everything fits, it is essential to maintain a centralized register of all design and manufacturing documentation, and to record changes over time. The register must be readily accessible by engineers throughout the industrial consortium. Maintaining an upto-date documentation database is known as configuration management, and the responsibility usually falls under PA. 431

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13.7. Safety and Risk

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Many activities during satellite processing are hazardous for the people doing the work, for the facilities in which they operate, and for the satellite itself. Given the values involved, an elaborate set of safety rules has been developed for each hazardous operation. It is the task of PA to ensure that these rules are being followed.

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For example, during hydrazine fuelling of the satellite at the launch site, the operators wear special “space suits”, called SCAPE suits (SCAPE stands for Self Contained Atmospheric Protection Ensemble) that protect them from toxic fumes in case of a spill or a leak. The suits are equipped with radio links for operator communication. All fuelling activities take place in reinforced and ventilated bunker-style buildings. Non-essential personnel are barred from entering during hazardous operations.

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Another kind of risk is that of satellite failure, with the attendant loss of the mission. In the course of spacecraft development, design engineers are made to contribute sections to a document called Failure Mode Effects and Criticality Analysis (FMECA). The authors endeavour to identify every plausible failure mode that might affect a component or a unit, and to hypothesize what the adverse consequences might be. Each failure mode is classified according to the perceived level of criticality for the mission. For example, if a TV satellite in GEO experiences abnormal battery capacity degradation, the consequence might be that eclipse operation is no longer possible, and that the television broadcast service has to be interrupted for max 72 minutes around local midnight during the 6weekly spring and autumn eclipse seasons. The aim of producing the FMECA at an early stage is to identify risk factors and mitigate them as far as possible. In many cases the FMECA may conclude that the cost of eliminating a particular risk is too great, given the low probability of its occurrence. In other cases, risks are dealt with by making the OBDH detect, identify and rectify anomalies on its own. This process is known as failure detection, isolation and recovery (FDIR) and is implemented in both hardware and software. Managing the FMECA is a PA responsibility.

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13.8. Solved Problems See Appendix C for solutions.

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13.8.1 The final stage of a satellite transponder consists of five power amplifiers connected in parallel in hot redundancy. Each amplifier has a reliability Pampl = 0.9. What is the probability that exactly 3 out of the 5 amplifiers will work? 13.8.2 What is the probability that at least 3 out of the above 5 amplifiers will work?

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13.8.4 Write the formula for the subsystem’s reliability.

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13.8.3 The attitude control subsystem of a geostationary satellite consists of three attitude sensors (earth, sun, star), two redundant A/D converters, a momentum wheel, a nutation damper, and redundant pairs of thrusters for attitude control in pitch, roll and yaw. Draw the reliability block diagram.

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13.8.5 Calculate the subsystem’s reliability, assuming that each unit has a reliability of 0.9.

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14. Development and Test

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14.1. Introduction

Subsystems

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Components Units

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It takes upwards of three years to design, develop, build and launch a medium-sized satellite. During the first 2 years, the work described in Chapters 2 through 13 is carried out. By then, all the satellite’s flight units should have been delivered by their manufacturers. These are the satellite’s building blocks. Figure 14-1 illustrates the hierarchical levels of satellite hardware and, where applicable, software.

Figure 14-1 Satellite hardware hierarchy.

It is useful to recapitulate the definition of components, units and subsystems. A component, also known as a piece part, is an electrical, electronic or mechanical device that performs a single function and is made of elements which cannot be separated without destroying its functionality. Examples are semiconductors, transformers, crystals and filters. A unit is a complex assembly of components mounted in boxes or on substrates which may be moved around easily. Examples are solar panels, batteries, encoders, decoders, microprocessors, and attitude sensors. (In some American literature, a unit is referred to as a component.) A subsystem is made up of units interconnected by electrical harnesses or structural members. They perform major functions and are not readily removable from the satellite. Chapters 3 – 10 of the present book cover individual subsystems. Within each chapter we have dealt with the major units that make up the subsystems. In Chapters 12 and 13 we have also touched on the subject of components. In this chapter we shall focus on the development and AIT of the completed satellite, while keeping in mind that much of the discussion is also applicable to the lower levels. The complete range of activities at lower levels is a vast subject and falls outside the scope of the present book. 435

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14.2. Satellite Development

Brassboard

Engineering Model

Qualification Model

Flight Model

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Early in a satellite programme, the satellite owner and his prime contractor agree on a model philosophy for the satellite down to the unit level. The model philosophy spells out how many prototypes of each item must be built, as well as their level of authenticity in regard to the items that are to be flown eventually. The requirements vary depending on a particular item’s previous flight heritage, or its similarity with previously flown designs. A brand new design may call for several prototypes to be built, while no prototyping is needed for a well-proven design. Figure 14-2 summarizes the classes of prototypes in common usage.

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Figure 14-2 Prototype models.

The breadboard model is a first attempt to turn a paper design into a functioning piece of hardware. The expression “breadboard” stems from the days when electrical components were nailed or glued to a piece of wood and were interconnected with ordinary wires. Nowadays a breadboard model consists of a printed circuit board (PCB) with tracks being cut or wire-connected as required to make the circuit function in a rudimentary fashion. A brassboard model is a cleaner second iteration of the breadboard model with improved functionality. It is sometimes referred to as an elegant breadboard. The engineering model is fully representative of the final product, including the housing where applicable, but uses commercial off-the-shelf (COTS) components. The qualification model differs from the previous models by using space-qualified components rather than commercial ditto. It is representative of the flight model in every respect, but its severe test programme will render it unsuitable for flight.

The flight model, also known as the acceptance model, is the ultimate flight article. It is also subject to testing, but the test levels are more lenient so as not to impair the reliability. Nowadays it is common practice to combine the qualification model and the flight model into a single protoflight model. This is done to save cost as well as time, but is sometimes the subject of controversy since it represents a compromise at the expense of design safety margins. The breadboard and brassboard models exist only at unit level, while the engineering, qualification, protoflight and flight models may be found also at subsystem and satellite level. The latter three models are built and tested under full product assurance scrutiny.

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14.3. Qualification and Acceptance

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The three most important test concepts in spacecraft engineering are qualification testing, acceptance testing and protoflight testing.

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Qualification testing is intended to verify that the design meets the performance requirements in all foreseen flight regimes, and that a healthy design margin exists to accommodate a degree of regime uncertainties. This test is performed only once for any new design. To probe the design margins, the test levels are quite severe – typically 1½ or 2 times the expected flight levels in terms of electrical, mechanical, thermal or acoustic loading. The article subjected to qualification testing is the qualification model described in Section 14.2 above. As already mentioned, the severity of the test renders the article unsuitable for flight.

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Acceptance testing aims to ascertain that the workmanship in manufacturing has been impeccable. Since workmanship problems may arise any time, each flight article is subjected to acceptance testing before being cleared for flight. The test levels are usually in the range 1.10 – 1.25 times the anticipated flight loading, which is thought to be sufficient to flush out most workmanship defects but not enough to impair reliability. The test duration is shorter than in the case of qualification. Protoflight testing is a compromise between qualification and acceptance to save cost. The compromise usually consists in exposing the article to qualification test levels at acceptance durations.

14.4. Assembly, Integration and Test

Once the integration of the satellite is complete, a lengthy test campaign begins, first on the prime contractors premises, and later at the launch site. If all goes well, the entire assembly, integration and test (AIT) phase requires typically a year to complete. When design or manufacturing flaws are discovered in the course of testing, a programme delay lasting from a few days to several years may ensue, sometimes with disastrous consequences for the overall mission. This is a simplified description of development and AIT at the satellite level. Similar development and AIT programmes exist at subsystem, unit and component level. Sometimes the programmes overlap, especially when problems are encountered along the way.

14.5. AIT Facilities

Satellites are integrated in large halls equipped with the necessary mechanical ground support equipment (MGSE), consisting mostly of cranes, slings, turntables and containers. The air in the halls is maintained within tight temperature, humidity and cleanliness limits. Electrical ground support equipment (EGSE) is accommodated in adjacent rooms. The EGSE centerpiece is a bank of computers and peripherals linked to 437

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the satellite via cables. The computers are used to validate the progress of integration, and to perform extensive functional tests of the finished spacecraft.

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Testing proper consists in exposing the satellite to the various parameters in outer space that are likely to affect its ability to function, such as vacuum, weightlessness, temperature extremes, and static as well as dynamic launch loads. Not only are these simulations difficult to perform realistically, but they are also very costly, and the test programme is therefore by necessity a compromise between the achievable and the affordable. 14.5.1. Static Loads

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The loads to be simulated are those encountered during the ascent of the launch vehicle (Figure 14-3), as well as those that the satellite might experience during ground transportation. 6

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Acceleration (g)

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1200

1400

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Figure 14-3 Typical longitudinal acceleration profile of a three-stage launch vehicle.

While Figure 14-3 shows the longitudinal (axial) acceleration, there are also transverse (lateral) accelerations to contend with. There are two approaches to simulating static loads on the ground. One method is to place the satellite on a horizontal centrifuge, the other is to use hydraulic jacks that act on the satellite in strategic places. The centrifuge has the advantage of exposing all parts of the satellite to acceleration, although unrealistic acceleration gradients arise due to the different radii from the centrifuge hub to the various parts of the satellite (Figure 14-4). This is so because the centrifugal force Fc = mωr2, while the equivalent acceleration force Fa = ma, such that a = ωr2.

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Figure 14-4 Acceleration gradients on a centrifuge.

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Other drawbacks with the centrifuge are the high cost and the difficulty in measuring structural deflections in the satellite.

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Hydraulic jacks mounted on a jig (“whiffle-tree”) overcome the disadvantages of the centrifuge, and this approach is therefore the most common. However, with the number of jacks being limited for practical reasons, it is not possible to achieve a fully realistic distribution of loads across the satellite. A degree of interpolation is possible using the mathematical model of the satellite’s structure. 14.5.2. Dynamic Loads

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Recall from Section 12.2.2 that dynamic loads are broken down into sinusoidal, random, acoustic and shock loads. The first two kinds of load are simulated by placing the satellite on a shaker, which is a computer-driven electromechanical vibration table with three degrees of freedom (along the x, y and z axes). The shaker may be driven in sinusoidal or random mode, and the frequency may be swept across a preset spectral range over a given time (Figure 14-5). The amplitude and frequency response of the satellite is measured using accelerometers placed in strategic locations. Accelerometer

Satellite

Vibration table

Motor

Figure 14-5 Satellite shaker.

Additional accelerometers sense out-of-limit conditions which are fed back to the shaker drive mechanism to reduce loading. This process is called notching the shaker’s vibration level (Figure 14-6) at the satellite’s natural frequencies to prevent damage due to resonance (refer to Figure 9-9).

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Amplitude

Notch

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Notch Notch

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Figure 14-6 Vibration level notching.

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Frequency

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Q: The shaker is presumably simulating the rocket’s vibration input to the satellite. If the satellite is going to shake to pieces as a result, isn’t notching a bit like sticking one’s head in the sand?

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A: Not if notching is performed judiciously. In reality, both the satellite and the rocket are semi-flexible structures and will absorb some of the resonant vibration energy. The shaker, on the other hand, is totally rigid, so to simulate the structural flexing, it is necessary to reduce the vibration amplitude in proportion to the predicted energy absorption.

To perform acoustic testing, the satellite is placed in a large hall equipped with horns (“trumpets”) of various sizes to cover the different parts of the noise spectrum. The horns are fed with nitrogen gas under electromechanical valve control. A microphone on the hall floor provides control feedback to the horn drivers, as do accelerometers mounted on the satellite. Random and acoustic vibration cover approximately the same spectral range (Figure 12-3 and Figure 12-4) and are therefore somewhat redundant. Random vibration on a shaker is preferred for relatively small, compact satellites whose main noise input arrives through the interface adapter with the launch vehicle and therefore affects the backbone of the satellite structure. Acoustic noise, on the other hand, enters mainly through the rocket’s heatshield. Acoustic testing is therefore more appropriate for larger satellites with flimsy appendages such as folded solar panels and antennas. Shock testing is carried out by feeding the shaker with a representative impulse. Alternatively, the satellite may be suspended from a ceiling crane a few centimetres above the support surface, and then be dropped. 14.5.3. Temperature and Vacuum

The thermal design of a satellite is verified by placing the spacecraft in a solar simulation chamber, while workmanship problems are flushed out in a thermal vacuum chamber. Temperature simulation is conducted in near-vacuum to imitate conditions in space and avoid thermal convection in the atmosphere. The satellite is therefore placed in a

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14 / Development & Test hermetically sealed vacuum chamber, inside which the air is evacuated to between 10-4 and 10-6 torr (1 torr = 133.3 Pascal = 133.3 N/m2).

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A solar simulation chamber is a vacuum chamber whose short wall is covered with highintensity xenon lamps to simulate the sun’s radiation intensity and spectrum. The satellite is mounted on a sting – a form of motor-driven gimbal that allows the satellite to experience various solar incidence angles (Figure 14-7). The cold temperature of space is achieved by flowing nitrogen through pipes inside the chamber walls.

Satellite

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Sting

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Figure 14-7 Solar simulation chamber.

Solar simulation allows the thermal mathematical model to be accurately calibrated against a well-defined set of illumination cases. A thermal vacuum chamber resembles the solar simulation chamber, except that there is no sting, and the xenon lamps have been replaced by electric heating elements mounted on a shroud around the satellite. In some applications, these heating elements consist of infrared lamps that may be individually switched on and off to create rudimentary thermal gradients in the satellite. The steady-state temperature is regulated within a range covering the extremes that the satellite is expected to encounter in orbit, plus a small temperature margin on either side of hot and cold. Figure 14-8 shows a typical thermal cycling profile during a thermal vacuum test.

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Temperature Chamber 10° above evacuation max mission & temperature electrical discharge check 20°C

Cold soak

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Time (days)

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Figure 14-8 Thermal vacuum temperature profile. (Adapted from Gordon & Morgan [7] with permission from John Wiley & Sons, Inc.)

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As illustrated in Figure 14-8, a thermal vacuum test may take up to two weeks, with the short cycles lasting 12 hours and the longer “soaks” requiring 72 hours. During the first two short cycles, the aim is to observe the satellite’s response to thermal gradients while ensuring that it reaches the equilibrium temperature at both ends. The hot and cold soaks probe the satellite’s ability to function in a sustained thermal environment, while the final short cycle flushes out any residual problems caused by the soaks. The satellite remains electrically activated throughout the test. While the chamber is being pumped down, attention is given to any tendency of high-voltage satellite equipment to discharge abruptly via an arc to adjacent structures. Arcing is most likely to occur at atmospheric pressures between ambient and vacuum, and is of concern during the first few days after launch when air trapped inside the satellite is outgassing. 14.5.4. Weightlessness

Simulating weightlessness is highly desirable but virtually impossible on the ground. The functions that would benefit the most from a zero-g simulation are deployments of solar panels and antennas, heat pipe operation, and propellant management, not to mention payload tasks related to microgravity. In practice, only deployments are tested under any semblance of weightlessness. Figure 14-9 shows a solar panel being deployed while suspended from a ceiling-mounted “curtain rail” and supported by frictionless air cushions. The aim is to create the equivalent of weightlessness in the horizontal direction to ensure that the spring-loaded hinges and the lock-up mechanisms work properly.

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Figure 14-9 Solar panel deployment under simulated weightlessness.

14.5.5. Antenna Pattern Tests

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Deployable antennas are maintained "weightless" in the AIT area by attaching large helium balloons to them.

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Antenna patterns are measured by transmitting a selection of signals through the antenna being tested, and using a calibrated receiving antenna for power measurement. The primary aim of the test is to establish the shape of the main antenna lobe and the side lobes. For antenna pattern measurements to yield accurate results, they must be performed free of reflexions and radio interference. Conducting the tests outdoors eliminates the problem of reflexion but invites interference. Conversely, performing them indoors poses the problem of reflexions from the surrounding walls. The solution to both problems lies in building an anechoic chamber (anechoic = echo-free) whose floor, ceiling and walls are covered with material that absorbs radio waves. The material consists of a foam-like substance shaped into pyramids, as illustrated in Figure 14-10.

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Figure 14-10 Anechoic chamber.

14.6. AIT at the Launch Site

Although the satellite to be launched has undergone extensive testing at home base, AIT activities continue at the launch site. From the moment of satellite arrival until lift-off, a launch campaign takes 4 – 7 weeks depending on the chosen launch vehicle. An abbreviated functional test is conducted upon arrival to ascertain that the satellite has survived the mechanical stresses due to handling and shipping by air. The solar panels are normally transported separately and must be re-installed. Hazardous operations follow, which consist of installing pyrotechnic devices and filling the spacecraft with propellants. Thereafter, the satellite is brought to the launch pad and is installed on the launch vehicle, where it is encapsulated inside the heatshield. The EGSE remains connected to the satellite via an umbilical cable, and is used for keeping the batteries charged and for continuous monitoring of the satellite’s health up until the moment of lift-off.

14.7. In-Orbit Testing

After the rough ride into space, a careful step-by-step approach is adopted to “commission” the satellite, i.e. to switch it on and assess its health. The extent to which the satellite is tested – e.g. whether to test redundant units – is up to debate, considering the risk involved in any activation, especially if nothing can be done should a unit prove to have failed. 444

14 / Development & Test In some projects, in-orbit testing (IOT) refers to the satellite's payload, while the platform checkout activities in orbit are often referred to as commissioning.

14.8. Overall AIT Sequence

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In many satellite projects, the prime contractor is rewarded financially if the satellite performs according to specifications after launch. Alternatively, he may be penalized if non-compliant performance is discovered. In either case, specialized in-orbit testing is often conducted to establish performance parameters such as EIRP, image quality or solar array output.

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A typical AIT programme for a telecommunications satellite destined for GEO is shown in Figure 14-11.

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Payload module Install receivers Install transmitters Install IF section Install antennas Ship to Prime Contractor

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Integrated satellite Electrical functional test EMC test Solar panel illumination test Propulsion leak check Install thermal control Install solar panels Acoustic noise test Separation shock test Sine vibration test Antenna deployment test Solar panel deployment test Spin test and weighing Ground station compatibility test Ship satellite to launch site

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Platform module Install propulsion s/s Install OBDH s/s Install TT&C s/s Install power s/s Install attitude s/s Mate with payload module

Launch campaign Visual inspection Electrical functional test Propulsion leak test Install solar panels Install apogee kick motor Install pyrotechnics Propellant filling Transportation to launch pad Install satellite on rocket Heatshield encapsulation Recharge batteries Countdown Lift-off In-orbit testing Commissioning Performance-related tests

Figure 14-11 AIT outline schedule. The time line is typically weeks.

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14.9. Solved Problems See Appendix C for solutions.

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14.8.1 Name the four construction features onboard a geostationary telecommunications satellite that are most likely to limit its lifetime. Explain in some detail their uses, as well as the causes and consequences of their limitations. Also discuss reasons why it hasn’t been possible to overcome these limitations.

14.8.2 Give at least three examples of how the platform equipment onboard geostationary satellites differs from that of low-orbiting satellites, and explain why.

First stage malfunction. The cause of the failure was traced to combustion instability in one of the four first stage engines 5.75 seconds after ignition. High frequency vibrations (POGO effect) had a destructive effect on Viking engine injectors. Autodestruct of launch vehicle and payload occurred after 108 seconds. The third stage fuel turbopump failed 9 minutes 20 seconds after lift-off. Failure of the gearing or lubrication system was given as the most probable cause. The failure resulted in a premature cut-off of the third stage engine so that insufficient velocity was imparted to orbit the payloads, causing them to fall into the Atlantic,

Justification

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May 23, 1980

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Design (D) or workmanship (W)?

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14.8.3 Ariane has been one of the most successful and reliable launch vehicles on the market. The causes of Ariane 1, 2, 3 and 4 launch failures are summarized in the table below. Indicate in the two right-hand columns which failures can be attributed to design (D) or workmanship (W), and justify your verdicts.

Sep 10, 1982

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May 31, 1986

Third stage motor failed to ignite because of a sealing defect of the engine's hydrogen feeding valve. Vehicle destroyed by the activation of the on-board destruct system due to the vehicles deviation from its programmed trajectory. Mission terminated by the range safety officer 4 minutes 36 seconds after launch. The Board of Inquiry set up to investigate the failure believes that third-stage ignition occurred but did not continue normally. Fourteen recommendations were submitted for modifications to the third stage and a review of the third-stage engine acceptance process. Malfunction in the first stage 1 minute 40 seconds after launch. The vehicle exploded about 10 km from the launch pad. The Board of Enquiry determined the cause of the failure to be an almost total obstruction of the water supply line to one of the four first stage engines by a piece of cloth and because of a faulty connection which led to the fire inside the No.3 liquid propellant strap-on booster. Premature shutdown of the third-stage engine. The report of the Board of Enquiry set up to investigate the failure, established the cause of the shutdown was probably the result of insufficient cooling of the immersed liquid oxygen bearing in the turbo-pump, combined with other aggravating factors resulting in an overload to the bearing.

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Sep 12, 1985

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Feb 22, 1990

Jan 24, 1994

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After a nominal lift-off and normal performance of the first and second stages, the third stage motor ignited, but when the gas generator ignition valve opened, the pressure in the gas generator was low, resulting in an engine performance which stabilized at only 70 % of its nominal steady state value. Consequently the thrust generated was insufficient to complete the mission after a total flight time of the third stage of 740 seconds. The findings of the Board of Inquiry were that a blockage in the flow of oxygen to the gas generator in the third stage engine was the most probable cause of the failure.

Co p by yri Lu ght leå © Un 201 ive 4, rsi ty o

Dec 1, 1994

olo gy .

14 / Development & Test

449

olo gy .

hn

ec

fT

Co p by yri Lu ght leå © Un 201 ive 4, rsi ty o

Appendix A / Spherical Trigonometry

Appendix A: Spherical Trigonometry

Longitude

Equator

Subsatellite track

Co p by yri Lu ght leå © Un 201 ive 4, rsi ty o

a

a

b

B

b

c

Latitude

fT

a

C

ec

hn

olo gy .

The sides of a spherical triangle are segments of great circles. The plane of a great circle includes the centre of the sphere. Hence, the circles represented by earth longitudes are great circles, as are the equatorial circle and the subsatellite track of satellite orbits. However, with the exception of the Equator, the latitude circles are not great circles.

b

co-B

c

A

co-A

Sine & cosine rules

Sine rule:

Cosine rules:

co-c

Napier’s rules

sin a sin b sin c = = sin A sin B sin C

cos c = cos a cos b + sin a sin b cos C

cos C = − cos A cos B + sin A sin B cos c

Napier’s rules with C = 90°:

The sine of an angle equals the product of the tangent of the adjacent angles. The sine of an angle equals the product of the cosine of the opposite angles. Examples of Napier’s rules:

cos A = tan b cot c → tan b = tan c cos A. cos c = cos a cos b (also obvious from the first cosine rule).

451

olo gy .

hn

ec

fT

Co p by yri Lu ght leå © Un 201 ive 4, rsi ty o

Appendix B / Matrix Algebra

Appendix B: Matrix Algebra

B.1

olo gy .

This Appendix covers basic matrix algebra with emphasis on vector transformation through successive rotations of orthogonal coordinate systems. Rotation methods using Euler angles, eigenvectors and quaternions are illustrated. The quaternion method is the norm in modern satellite attitude determination and control.

Matrix Inversion

hn

Matrices are useful for solving simultaneous linear equations. For example, suppose we have three unknown variables x, y, z and three equations with which to determine them:

ec

4x + 5 y − z = 8 − 2 x + 3 y + 3z = 5 x − 2 y + 2 z = −1

fT

(B.1)

Co p by yri Lu ght leå © Un 201 ive 4, rsi ty o

One way of solving for x, y, z is to begin by eliminating x, and then y, such that we are left with a unique value for z. In our example, multiply the second equation by 2 and then add it to the first equation: 4x + 5 y − z = 8

− 4 x + 6 y + 6 z = 10 x − 2 y + 2 z = −1

4x + 5 y − z = 8

⇒

0 x + 11y + 5 z = 18 x − 2 y + 2 z = −1

Similarly, multiply the last equation in Eq B.1 by –4 and add it to the first equation: 4x + 5 y − z = 8

0 x + 11y + 5 z = 18 ⇒ − 4x + 8 y − 8z = 4

4x + 5 y − z = 8

0 x + 11y + 5 z = 18 0 x + 13 y − 9 z = 12

(B.2)

Next, multiply the last equation in Eq B.2 by –11/13 and add it to the second equation:

4x + 5 y − z = 8

0 x + 11 y + 5 z = 18 11 11 0 x − 11y + 9 z = − 12 13 13

4x + 5 y − z = 8

⇒

0 x + 11 y + 5 z = 18 0 x + 0 y + 12.61z = 7.85

(B.3)

The third equation in Eq B.3 now tells us that z = 7.85/12.61 = 0.62. Inserting this value in the second equation of Eq B.3 gives us y = 1.35. Lastly, inserting these values for z and y in the first equation yields x = 0.46. In matrix form, Eq B.1 may be written as follows:

453

Appendix B / Matrix Algebra

5 − 1 x   8   4      3  y  =  5  − 2 3  1 − 2 2  z   − 1     

olo gy .

(B.4)

or, in shorthand notation, [A]V = [B], where V represents the vector (x, y, z). A quicker way of finding x, y, z is to compute the inverse matrix [A]-1, such that: −1

⇒

(B.5)

hn

V = [A]-1[B]

5 − 1  8   x  4       3 5  y = − 2 3  z   1 − 2 2   − 1      

fT

ec

Unfortunately, calculating [A]-1 by hand can be a cumbersome process even for a simple 3 x 3 matrix, and becomes exponentially more difficult as the order of the matrix increases. We must therefore resort to computer programmes such as Matlab or Excel to obtain [A]1 , and thereby x, y, z. (Some advanced pocket calculators also have this capability.) In our example, the Excel function MINVERSE yields:

 0.146 − 0.098 0.220    [A] =  0.085 0.110 − 0.122   0.012 0.159 0.286  

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-1

 0.146 − 0.098 0.220  8   0.463   x         Hence: V = [A] [B] =  0.085 0.110 − 0.122  5  =  1.353  =  y   0.012 0.159 0.286  − 1  0.622   z   -1

Matrix inversion is greatly simplified in the case of orthonormal matrices, since [A]-1 = [A]T, i.e. the inverse matrix is identical to the transposed matrix. Recall that transposing a matrix amounts to inverting its rows and columns:

 a11  [A] =  a 21 a  31

a12

a 22 a 32

a13   a 23  ; a33 

 a11  [A] =  a12 a  13 T

a 21

a 22 a 23

a31   a32  a33 

A matrix [A] is orthonormal if multiplying it with its transpose yields the unity matrix [1]. A unity matrix has all the elements = 0 except for the diagonal ones which are = 1. Therefore, in the case of an orthonormal 3x3 matrix [A] we have:

 1 0 0   [A]⋅[A]T = [I] =  0 1 0  0 0 1  

454

Appendix B / Matrix Algebra

0 1  For example, the matrix  0 cos α  0 − sin α  0   sin α  cos α 

0 1   0 cos α  0 sin α 

  1 0 0    − cos α sin α + cos α sin α  =  0 1 0   0 0 1 sin 2 α + cos 2 α    0

hn

0 1  2 cos α + sin 2 α 0  0 − sin α cos α + sin α cos α 

  − sin α  = cos α  0

olo gy .

0 1   0 cos α  0 − sin α 

0   sin α  is orthonormal, because cos α 

B.2

ec

All the matrices in Sections B.2 and B.3 below are orthonormal.

Coordinate Transformations

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fT

Often it is useful to translate a vector from one coordinate system to another. For example, it is easier to predict the eclipse duration of a satellite if its radius vector is expressed in a heliocentric coordinate system rather than in a geocentric ditto. The translation is performed via a series of matrix multiplications, whereby each multiplication corresponds to a rotation of one coordinate system to align it with another. Take a vector V as defined in a coordinate system whose orthogonal axes are defined by the unit vectors i, j, k. We have: V = V1 i + V2 j +V3 k. The vector length V = |V| = (V12 + V22 + V32)½. Now assume that we wish to express V in a different coordinate system x, y, z, where x = i. Here we are dealing with a simple rotation of the i, j, k system around the i-axis by some angle α. z

z

k

k

j

α

α

y

α

α

x=i

j

y

x=i

Figure B-1 Positive and negative rotation of coordinate system.

Looking at the figure to the left, we are rotating the i, j, k system in a positive direction (according to the right-hand rule) in order to align it with the x, y, z system. We notice that: 455

Appendix B / Matrix Algebra

⇒

i j = k

1 0 0 0 cosα -sinα 0 sinα cosα

x y z

(B.6)

olo gy .

i = 1x + 0y + 0z j = 0x + cosα y – sinα z k = 0x + sinα y + cosα z

The figure to the right in Figure B-1 shows a negative rotation. The corresponding rotation matrix is obtained in the same manner:

⇒

i j = k

1 0 0 0 cosα sinα 0 -sinα cosα

x y z

(B.7)

hn

i = 1x + 0y + 0z j = 0x + cosα y + sinα z k = 0x - sinα y + cosα z

fT

ec

The difference between the rotation matrices in Eq B.6 and B.7 is the sign between the two sinus functions. It is therefore important to keep in mind whether the rotation is positive or negative according to the right-hand convention. (It is sufficient to memorize the matrix in Eq B.6, since the matrix in B.7 is arrived at naturally by replacing α with – α.)

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The elements in an orthonormal matrix constitute direction cosines, i.e. they are the cosine of the angles between each set of axes i, j, k and x, y, z. Thus Eq B.7 may be written out as follows (with o as the vector dot product operator):   i   i  x i  y i  z  x   cos 0        j  =  j  x j  y j  z  y  =  cos 90  k   k  x k  y k  z  z   cos 90       

cos 90  cos 90   x   1 0      cos α cos(90 − α )  y  =  0 cos α      z   0 − sin α cos(90 + α ) cos α 

0  x    sin α  y  cos α  z 

So far we have assumed that the coordinate transformation occurs by simple rotation around the i = x axis by an angle α. If instead a positive rotation takes place around the j = y axis by some angle β, we find through similar reasoning that:

i = cosβ x + 0y + sinβ z j= 0x + 1y + 0z k = -sinβx + 0y + cosβ z

⇒

i j = k

cosβ 0 sinβ 0 1 0 -sinβ 0 cosβ

x y z

(B.8)

while a positive rotation around the k = z axis by an angle γ axis takes the form:

i = cosγ x - sinγ y + 0 z j = sinγ x + cosγ y + 0 z k= 0x + 0y + 1 z

⇒

i j = k

cosγ -sinγ 0 sinγ cosγ 0 0 0 1

x y z

(B.9)

The angles α, β, γ, etc. are known as Euler angles. An example of Euler angles is roll, pitch and yaw, as discussed in Chapter 5. These angles may serve to illustrate why [A][B] ≠ [B][A], i.e. why the order of rotations is important (Figure B-2).

456

Appendix B / Matrix Algebra

Roll 90°

Roll 90°

Pitch 45°

hn

Level flight

Pitch 45°

olo gy .

Level flight

Figure B-2 Different rotation sequence leads to a different result.

ec

In shorthand notation, we may write [Rx(α)] to denote the positive rotation matrix in Eq B.6, [Ry(β)] for the matrix in Eq B.8, and [Rz(γ)] for the matrix in Eq B.9. For negative rotation we will write [Rx(-α)], etc.

Co p by yri Lu ght leå © Un 201 ive 4, rsi ty o

fT

Returning to our vector V = V1 i + V2 j +V3 k, we may now be faced with the problem of translating it into a different coordinate system u,v,w involving several elemental rotations of the kind illustrated in Eq B.6, B.8 and B.9 above. The transformation might look like this:

Vuvw = [Ry(α)][Rx(β)][Rz(-γ)][Ry(δ)] Vijk

(B.10)

Using Matlab, Excel or similar programmes, this transformation is relatively straightforward, provided the input data is correctly defined in terms of sequence and signs. Note that it makes no difference whether we multiply the matrices in Eq B.10 as follows: Vuvw = [Ry(α)][Rx(β)][Rz(-γ)][Ry(δ)] Vijk

or as follows:

Vuvw = [Ry(α)][Rx(β)][Rz(-γ)][Ry(δ)] Vijk

although the first method is preferable, as each multiplication results in only 3 elements instead of 9. However, reversing the order of internal pairs of matrices is not admissible, since in general [A][B] ≠ [B][A] according to Figure B-2 above.

Example: Transfer the radius vector r from the geocentric orbital coordinate system to the geocentric-equatorial rotating system (cf. Section 2.4.1).

457

Appendix B / Matrix Algebra

rφ =

 r1    r2   r3  φ

= [Rz(φ)] [Rz(-Ω)] [Rx(-i)] [Rz(-ω)] [Rz(-ν)]• rs

olo gy .

We have already performed this exercise in Section 2.4.2, as we sought to plot a subsatellite track on the earth’s surface. Using mostly negative rotations, we arrived at Eq 2.20. Expressed differently, and using the convention in Eq B.10:

(B.11)

Using an even more compressed notation, we may write Eq B.11 as:

rφ = [R]φ/s• rs

hn

(B.12)

ec

where [R]φ/s is the rotation matrix that brings a vector in the s-system to the φ-system. Inserting rs = (1, 0, 0)T in Eq B.11, we obtain rφ as:

fT

 cφ cΩ cω cν − cφ cΩ sω sν − cφ sΩ ci sω cν − cφ sΩ ci cω sν + sφ sΩ cω cν − sφ sΩ sω sν + sφ cΩ ci sω cν + sφ cΩ ci cω sν    rφ =  − sφ cΩ cω cν + sφ cΩ sω sν + sφ sΩ ci sω cν + sφ sΩ ci cω sν + cφ sΩ cω cν − cφ sΩ sω sν + cφ cΩ ci sω cν + cφ cΩ ci cω sν    si sω cν + si cω sv  

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with s_ denoting sin_ and c_ being shorthand for cos_. The first vector component is the direction cosine for the r-vector with respect to the xφ-axis; the second component is the direction cosine with respect to the yφ-axis; and the third component is the direction cosine with respect to the zφ-axis.

B.3

Eigenvalues and Eigenvectors

As we have seen above, transforming the r-vector from the s-frame to the φ-frame using Euler angles is cumbersome indeed. Arriving at the overall rotation matrix [R]φ/s is even more arduous, since in the above example it involves 5 coordinate transformations with each matrix having 9 elements. Considering that each transformation entails 33 = 27 multiplications, we have had to undertake no less than 4 x 27 = 108 trigonometric multiplications to find [R]φ/s. Is there not an easier way? A shortcut is available to us by employing a so-called eigenvector. An eigenvector may be viewed as a shaft E, around which the rotation of the s-frame to the φ-frame by some angle Φ is performed in a single step. The question is how to find the eigenvector E and the rotation angle Φ.

458

Appendix B / Matrix Algebra zφ

E

ys

xφ

xs

ec

Figure B-3 Eigenvector E.

hn

yφ

olo gy .

zs

or

[R]E = λ[1]E

Co p by yri Lu ght leå © Un 201 ive 4, rsi ty o

[R]E = λE

fT

It is clearly in the nature of the eigenvector that the direction cosines between E and the (x,y,z)s axes are the same as between E and the (x,y,z)φ axes, since E serves as a fixed rotation shaft. Consequently: (B.13)

1 0 0   with [1] being the identity matrix =  0 1 0  . (Note that a vector or a matrix multiplied 0 0 1  

by the identity matrix yields the same vector or matrix.) The scalar quantity λ is known as the eigenvalue and determines the length of E. Eq B.13 may be rewritten as follows: [R]E - λ[1]E = {[R] - λ[1]}E = 0

or, written out in full:  r11   r21  r  31

r12

r22 r32

r13   λ 0 0   x       r23  −  0 λ 0    y  r33   0 0 λ   z 

 r11 − λ 

=  r21  r  31

r12 r22 − λ r32

r13   x     r23    y  = 0 r33 − λ   z 

(B.14)

Let us apply Eq B.14 to find the eigenvector and the eigenvalue for a simple 2 x 2 matrix:  3 2  , such that [R] =  6 7  

3 − λ   6

2   x   = 0 7 − λ   y 

(B.15)

This equality implies that the determinant of [R] = 0, i.e. (3-λ)(7-λ)-12 = 0, or λ2 – 10λ + 9 = 0, which gives us the eigenvalues λ = 9 and λ = 1. Choosing λ = 9, and returning to Eq B.13, we have:  3 2  x   x      = 9  which gives us y = 3x. 6 7  y  y

459

Appendix B / Matrix Algebra

 3 2  x  x      = 9  , i.e.  6 7   3x   3x 

 3 2 1 1      = 9   6 7   3  3

olo gy .

Hence:

hn

Comparing this result with Eq B.13, we find that the eigenvector corresponding to the 1 eigenvalue λ = 9 is E =   . Similarly, the eigenvector corresponding to λ = 1 is E =  3 1   . (A quick rule for checking the result: The sum of the diagonal values = the sum of  −1 the eigenvalues, in this example = 10.)

ec

Example: Make a complex coordinate transformation first by making a 90° turn around the x-axis, then a 90° turn around the y-axis, and lastly a 90° turn around the z-axis. z

x

y

z

x

x

Co p by yri Lu ght leå © Un 201 ive 4, rsi ty o

x

fT

z

y

z y

y

Figure B-4 Successive 90° coordinate transformations.

Applying Eq B.7 through B.9, and noting the negative directions of rotation in the above figures, we have:

0 0   cos 90° 0 − sin 90°   cos 90° sin 90° 0  1       1 0  0 cos 90° sin 90°    0    − sin 90° cos 90° 0  =  0 − sin 90° cos 90°   sin 90° 0 cos 90°   0 0 1      

 1 0 0   0 0 − 1  0 1 0   0 0 − 1         0 0 1   0 1 0    −1 0 0 = 0 1 0   0 − 1 0  1 0 0   0 0 1  1 0 0         

0 0 − λ  1− λ To find the eigenvalue, we set the determinant of  0  1 0 

−1   0 =0 0 − λ 

which yields λ = 1 as the only real root. Next, we look for the eigenvector. From the diagrams above, we may conclude that the three rotations have achieved no more than what one rotation around the y-axis would have accomplished in the first place. Figure B-4 leads us to surmise that the eigenvector E = (0, 1, 0)T. Let us try it out for size.

460

Appendix B / Matrix Algebra

or

olo gy .

[R]E = λE

 0 0 − 1  0  0       0 1 0   1 = 1 1  1 0 0   0 0      

We stated earlier that the eigenvector is like a fixed shaft, around which the overall rotation takes place. Having found the eigenvector E = (0, 1, 0)T in our example, we have yet to establish the rotation angle Φ, which is readily calculated from the equation 1  Φ = cos −1  (trace[ R ] − 1) 2 

hn

(B.16)

where trace[R] is the sum of the diagonal elements of [R]. In our example, trace[R] = 1, and consequently Φ = 90°, a result which is confirmed in Figure B-4.

ec

It can be shown that: [ R ] = (1 − cos Φ)EE T − sin ΦE × + cos Φ[ I ]

fT − e3 0 e1

e2  1 0 0    − e1 ; [1] =  0 1 0  0 0 1 0   

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 e1   0    T × where E =  e2 ; E = (e1 , e2 , e3 ); E =  e 3 e  − e  3  2

(B.17)

In our example:

0  0 0 0 0 1 0 0 0 0  0 0 1           T T × E =  1  =  1 0 0 ; E = (0,1, 0) =  0 0 0 ; EE =  0 1 0 ; E =  0 0 0  0  0 0 0 0 0 0 0 0 0  −1 0 0          

 0 0 − 1   From Eq B.17: [ R] = EET − E× =  0 1 0  1 0 0    which is the same matrix [R] that we obtained following the “long” route via three separate transformations. In summary, therefore, knowing the eigenvector E and the rotation angle Φ around that vector allows us to compute the rotation matrix [R] in a single step.

B.4

Quaternions

The eigenvector method of coordinate transformation, while computationally more efficient than the multiplication of matrices, still leaves us with a trigonometric function and some annoying singularities as we move through the various Euler angles. A quaternion is a mathematical device which frees us from singularities and some trigonometry, thereby speeding up computation.

461

Appendix B / Matrix Algebra

olo gy .

The definition of the quaternion q relies on the eigenvector E = (e1, e2, e3)T whose eigenvalue λ = 1. The quaternion also needs the rotation angle Φ of the rotation matrix [R]. It is deceptively similar to a vector, in that it has a vector part q = (q1, q2, q3)T, as well as a scalar component q4:

q = (q, q4)= (q1, q2, q3, q4)

(B.18)

Φ  Φ Φ Φ =  e1 sin , e2 sin , e3 sin  (B.19) 2 2 2 2  Φ q4 = cos (B.20) 2 As before, the eigenvector E = (e1, e2, e3)T represents the axis of rotation, while Φ is the rotation angle. However, quaternions differ from vectors by their rules of multiplication.

hn

where q = E sin

1 1 + trace[ R ] 2

 R23 − R32     R31 − R13  R −R  21   12

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1 q= 4q 4

(B.21)

fT

q4 =

ec

We may also derive the components from the Euler rotation matrix [R], if known:

(B.22)

Conversely, the rotation matrix [R] is obtained from: 2

[ R ] = (q 4 − q T q)[ I ] + 2qq T − 2q 4 q ×

with

qTq = q12 +q22 + q32.

In analogy with Eq B.17, we have:

and:

(B.23)

 q1  T qq =  q 2 q  3

0 0  q1  0 0  0 0 0  0

q2 0 0

 0  q =  q3 − q  2

− q3

×

0 q1

2 q3   q1   0  =  q1 q 2  0   q1 q3

q1 q 2 q2

2

q 2 q3

q2   − q1  0 

q1 q 3   q 2 q3  2  q3 

Eq B.23 may now be written out in full:

 q1 2 − q 2 2 − q 3 2 + q 4 2  [R] =  2(q1q 2 − q 3 q 4 )   2(q1 q3 + q 2 q 4 )

462

2(q1 q 2 + q3 q 4 ) 2

2

2

− q1 + q 2 − q3 + q 4 2(q 2 q 3 − q1 q 4 )

   (B.24) 2(q 2 q3 + q1 q 4 ) 2 2 2 2 − q1 − q 2 + q 3 + q 4  2(q1 q3 − q 2 q 4 )

2

Appendix B / Matrix Algebra

 0 0 − 1   Borrowing from our example in Section B.3, where [R] =  0 1 0  , we obtain from 1 0 0    1 1 T Eq B.21 and B.22: q = 0, 2 ,0, 2 . Eq B.24 gives us back our [R].

B.5

)

olo gy .

(

Quaternion Multiplication

− q' 2 q '1 q' 4 − q'3

q '1  q1    q ' 2  q 2  q '3  q 3    q ' 4  q 4 

(B.25)

ec

q '3 q' 4 − q '1 − q' 2

fT

 q"1   q ' 4     q"2   − q '3  q"  =  q '  3  2  q"   − q ' 1  4 

hn

If one rotation is represented by [R] and another by [R’] as per Eq B.24, then it can be shown that the resulting rotation [R”] = [R][R’], whereby

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The quaternion components q1, q2, q3, q4 are found from Eq B.21 and B.22 as follows for the three standard rotation matrices:

Rotation axis

Around xaxis Around yaxis Around zaxis

Direction cosine matrix, right-hand rotation 0 0  1    0 cos α − sin α   0 sin α cos α     cos α 0 sin α    1 0   0  − sin α 0 cos α     cos α − sin α 0     sin α cos α 0   0 0 1  

Quaternion

q1

− sin α

2 + 2 cos α

0

0

q2

q3

0

0

− sin α

2 + 2 cos α

0

0 − sin α 2 + 2 cos α

q4 1 2 1 2 1 2

1 + cos α

1 + cos α

1 + cos α

Table B-1 Conversion from Euler matrices to quaternions.

Whatever the transformation method, successive rotations are accomplished by multiplying the corresponding rotation matrices [R]. A 3x3 direction cosine matrix contains 9 elements (a11, a12, ... a33), whereas only 4 elements are needed to describe an equivalent rotation using quaternions (namely q1, q2, q3, q4). It takes 27 computations to multiply two direction cosine matrices, and only 16 computations when quaternions are used, so the amount of computing power is reduced accordingly. Note that Table B-1 applies to right-hand rotations. If a left-hand rotation is involved, it is necessary to replace the argument α with –α. B.6 Summary of Rotation Matrix Derivations

463

Appendix B / Matrix Algebra

Using Euler angles (Eq B.12): Using eigenvectors (Eq. B.17):

[R] = [R1(α)][R2(β)][R3(γ)][R...] [ R ] = (1 − cos Φ )EE T − sin ΦE × + cos Φ[1]

Using quaternions (Eq. B.23):

[ R ] = (q 4 − q T q)[1] + 2qq T − 2q 4 q ×

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fT

ec

hn

olo gy .

2

464

Appendix C / Solved Problems

2.6.1

Eq 2.12: τ = 2πa a / µ

, i.e. a 3 =

µτ 2 , and a = 6945 km. 4π 2

Therefore: h = a – R = 574 km. µ Eq 2.14: V = = 7576 m/s. a

2 1 According to Eq 2.13: V = µ( − ) . We need to find r and a. r a

hn

2.6.2

olo gy .

Appendix C: Solved Problems

Eq 2.4: a = ½(ra + rp ) = ½(Ap + Pe) + R = 24,421 km. rp c Pe + R = 1− = 1− = 0.727 a a a a (1 − e 2 ) Eq 2.3: r = = 11,514 km 1 + e cos ν and therefore V = 7.27 km/s.

fT

ec

Eq 2.5: e =

1 mµ . mV 2 − 2 r Therefore Ekin = 26.4 x 109 Nm and Epot = -34,6 x 109 Nm. (Remember to express V in m/s rather than km/s!). The total energy E = -8.2 x 109 Nm. Let us verify this result against Eq 2.17: mµ = -8.2 x 109 Nm. E=− 2a

According to Eq 2.16: E = E kin + E pot =

2.6.4

The condition is met when dω/dt = dΩ/dt. dΩ − 10 cos i Eq 2.18: ; = dt ( a / R) 7 / 2 (1 − e 2 ) 2

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2.6.3

dω 5(5 cos 2 i − 1) = . dt ( a / R ) 7 / 2 (1 − e 2 ) 2 The denominators are the same in both equations, so the condition is met simply by manipulating the inclination i. We have: 5 cos2i + 2 cosi – 1 = 0, which leaves us with i = 73.15° or i = 133.60° regardless of a and e. However, e must be ≠ 0, since a perigee is needed for there to be a perigee drift. Note that the solution i = 133.60° calls for a launch in a westerly direction, i.e. against the rotation of the earth, and is therefore unlikely to be adopted in practice. Eq 2.19:

2.6.5 If the orbit must never see eclipse, it has to be sun-synchronous with the nodes at the dawn-dusk points. Seen from the north:

465

olo gy .

Appendix C / Solved Problems

We know from Figure 2-85 that the inclination of an SSO is always greater than 90°. Seen from the side of the ecliptic plane, the worst-case configuration that defines the minimum radial distance a is as follows:

a

R

Equator i

ec

ε

hn

N

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fT

Ecliptic

We have: R/a = cos(i - 90° + ε) = cos(i – 66.56°), with ε = 23.44° and i > 90°. From Eq 2.18: a/R = [-cos i /0.09856]2/7, and therefore: [-cos i /0.09856]2/7 = 1/cos(i66.56° ) By solving this equation iteratively, we find that i = 101.3°, and therefore a = 7753 km. This is to say that the minimum orbital height h = a – R = 1382 km. 2.6.6 Inclinations > 90° require a launch with a westward component. Vandenberg only allows southerly launches (Figure 11-8). We must therefore choose the SW quadrant for our launch. 1  360°   tan λ  Eq 2.38: HH GMT = 180° + Ω − Λ − sin −1   − ( DD + 101) 15  365.25   tan i  HHGMT = 14h20m; HHlocal = 06h20m.

 2π  The number of orbital planes P =    3α'   2π   Eq 2.55: The number of satellites per plane N =   3α'  The geocentric coverage angle α’ is obtained from Eq 2.57:

2.6.7

Eq 2.54:

 R  α' = cos −1  cos δ  − δ = 15.7° = 0.274 radians R+h 

466

Appendix C / Solved Problems With P = 7.6 ≈ 8; N = 13.2 ≈ 13, the total number of satellites S is 8 x 13 = 104.

2.6.9

olo gy .

2.6.8 With the satellite launched in 2003, the inclination drift rate will be approaching the maximum value of 0.953 deg/year (June 2006), so it is fair to assume an average drift rate of 0.9 deg/year. The drift duration is therefore 3 + 3 deg divided by 0.9, or 6.7 years. Eq 2.49 gives the radius of the inclination vector trace ρ = 7.985 deg. The minimum inclination will be encountered when the inclination vector passes the Ω = 180 deg line, such that imin = ρ – θ = 7.985 – 7.400 = 0.585 deg. Let us first verify the orbital period. From Eq 2.4: a = ½(ra + rp ) = 26621 km.

Eq 2.9:

M = E − e sin E ;

fT

ec

hn

From Eq 2.12: τ = 2πa a / µ = 43,204 s = 12.0 hours. Therefore, each satellite will pass through the apogee every 12 hours. rp r c Eq. 2.5: e = = 1 − = a − 1 = 0.742. a a a e + cos ν cos E = ; Eq 2.8: 1 + e cos ν E1 = 153.9 deg for ν1 = 169.8 deg, and E2 = 206.11 deg for ν2 = 190.2 deg.

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M1 = 135.2 deg for ν1 = 169.8 deg, and M2 = 224.8 deg for ν2 = 190.2 deg. M ∆M Eq 2.7: t=τ ; ∆t = τ = 3 hours. Therefore 4 satellites are needed in 2π 2π any 12-hour period, knowing that the pattern repeats itself every 12 hours thereafter.

Refer to pages 3 – 6 and 3 – 7. P = Fc Fs cos φ s . Eq 3.6: Pmax At BOL, φs = 0° and Fs = 1. We also know that νe ≈ 90° in the Eq 2.3: 2 a e (1 − ee ) re = , so that Fc = (ae/re)2 = 1/(1-ee2)2 = 1. 1 + ee cos ν e Therefore, PBOL = Pmax = 1000 W. 7¼ years later we have φs = 23.44° and Fs = 0.7 + 0.3 e-365.25x7.25/1000 = 0.721. 2 a (1 − ee ) We also know that νe ≈ 180° in the equation re = e , 1 + ee cos ν e so that Fc = (a/re)2 = (1-ee)2/(1-ee2)2 = 1/(1+ee)2 = 0.967. Therefore, P7/Pmax = 0.967 x 0.721 x cos(23.44°) = 0.640, i.e. P7 = 640 W.

3.6.1

3.6.2 The capacity Cused needed by the satellite is the product of the drawn current I and the maximum time tecl in eclipse, i.e. Cused = I tecl. Here, I = Psat/Ubus = 20 A. a  R The maximum time in eclipse is obtained from Eq 2.29: (t ecl )max = 2a sin −1   = µ a 2126 s = 0.59 h. Therefore Cused = 20 x 0.59 = 11.8 Ah.

467

Appendix C / Solved Problems

olo gy .

However, we are not allowed to drain the battery completely, so there is a limit to the allowable depth of discharge (DOD). According to Figure 3-20, the DOD depends on the ratio tecl/(τ – tecl). The orbital period is found from Eq 2.12: τ = 1.61 h, and tecl/tsun = 0.58. According to Figure 3-20, DOD = 25% = 0.25, which gives us Crated = Cused/DOD = 47.2 Ah.

3.6.3 We must dimension the solar panels for the worst case situation, which occurs at summer solstice when the solar power flux density is at its weakest and the obliquity  causes the greatest sun vector offset from the solar panel normal (namely 23.45°). The total required solar cell surface is (Eq 3.8):

hn

η = 0.25 according to Figure 3-8 Fc = (ae/Re)2 = 0.967 as per Problem 3.6.1 Fs = 0.71 according to Eq 3.5 (φs)max = ε = 23.45° at summer solstice.

ec

with

P 4000 = 18.77 m2 = ηSFc Fs cos φ s 0.25 ⋅ 1353 ⋅ 0.967 ⋅ 0.71 ⋅ cos 23.45°

fT

A=

The number of 4 x 6 cm solar cells required is therefore 18.77/(0.04 x 0.06) = 7823 cells.

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3.6.4 Both satellites will wish to keep their solar panels pointed in a N/S direction for maximum solar illumination. There are two stages, namely approach and docking: (a) Approach: During final approach, local shadowing will occur if the chaser gets in the way between the sun and the comsat. One solution is to choose the time of approach during the local night-time when the sun is behind the comsat, but this may prove problematic for the chaser’s docking cameras which are probably dazzled by the sun and are unable to make out the contours of the target’s docking mechanism. Alternatively, the target could turn its solar panels 90° during the approach, such that they are fully shaded and therefore insensitive to local shadowing. The target might be able to rely on its batteries for the duration of the approach. If not, the target will have to interrupt its mission until docking. (b) Docking: The only certain way to avoid local shadowing of the target’s solar panels at all times and during all seasons is to turn the chaser 90° just before docking, wind-mill fashion, relative to the target. The consequence is that the chaser’s solar panels are poorly illuminated during much of the day, and its batteries will have to be sized accordingly.

468

olo gy .

Appendix C / Solved Problems

Chaser Target

During approach

hn

After docking

fT

ec

4.6.1 By turning the wheel clockwise, you create an angular momentum vector H pointing away from you, according to the right-hand rule. When you try to bend the axis downward, you also create a torque vector T which points to the left. The momentum vector tries to align itself with the torque vector, causing the platform to turn counterclockwise as seen by yourself looking down.

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4.6.2 Any physics handbook will tell us that Iz = ½mr2, and that Ix = m/12 ·(3r2 + h2). From Eq 4.14: λ = Iz/Ix > 1 as a condition for spin stability. Consequently h/r < √3. 4.6.3

The vector s is defined by the direction cosines, i.e.

s = (cos φu , cos φv , cos φw)T = (0.799, 0.500, 0.342)T.

To ascertain that this is correct, we calculate the length of the unit vector: s = cos2φu + cos2φv + cos2φw = 0.7992 + 0.5002 + 0.3422 = 1.000, as expected.

4.6.4 Eq 4.52: H = Isat Ω + Iwheel ω. Conservation of angular momentum requires H to be constant during the incident, i.e. 0 = Isat dΩ/dt + Iwheel dω/dt, or, in our case, 0 = Isat ∆Ω + Iwheel ∆ω. Let subscript 0 denote the values before the incident and 1 afterwards. We are looking for Ω1. ∆Ω = Ω0 – Ω1 ; ∆ω = ω0 – ω1;

with Ω0 = 360°/24 h = 0.00417°/s, given that a GEO satellite makes one complete turn around the pitch axis in 24 hours; ω0 = 5000 rpm = 5000 ·360°/60 = 30,000°/s; ω1 = 0°/s; ∆ω = 30,000°/s. We have: ∆Ω = -Iwheel / Isat ∆ω, and therefore: Ω1 = Iwheel / Isat ∆ω + Ω0 = 30 + 0.00471 ≈ 300°/s, or 5 rpm.

4.6.5

Refer to Appendix B.  0 1 0   Eq B.24: [R ] =  0 0 1  ; Eq B.20: Φ = 2 cos-1(½) = 120°. 1 0 0   469

Appendix C / Solved Problems

Eq B.19: e1 sin 60° = q1 = ½;

e1 = e2 = e3 =

1 3

;

T

 1 1 1  Eigenvector E = (e1, e2, e3) =  , ,  .  3 3 3 Apparently the eigenvector is a rotation shaft that forms equal angles with each of the three coordinate axes x, y, z. The rotation angle Φ = 120° around E is 1/3 of a complete turn. This suggests that the initial coordinate system has been rotated such that all three coordinate axes have changed places, i.e. x → y, y → z , z → x .

olo gy .

T

hn

z1

Φ

fT

E

ec

z2

y2

Co p by yri Lu ght leå © Un 201 ive 4, rsi ty o

y1

x1

x2

The eigenvalue λ is obtained from Eq B.14 by setting the following determinant to zero:

0−λ 0 1

1

0

3

Hence λ = 1.

0−λ 0 = −λ + 1 = 0 ; 0 0−λ

 0 1 0   Verify by proving that  0 0 1  E = λE.  1 0 0  

5.5.1

2

2

2

2

2

2

r = rx + ry + rz = 7211 km; V = V x + V y + V z = 7.28 km/s;

Eq 5.18: e = (1/µ)[(V2 – µ/r)r – (r⋅V)V] = (-0.0228, 0, -0.0343)T; 2

2

Eq 5.22: e = e x + e y + e z

2

= 0.0412;

Eq 5.20: ε = V2/2 – µ/r ; µ = 6926 km; Eq 5.21: a = − 2ε Ap = a(1 + e) - R = 840 km; Pe = a(1 - e) – R = 270 km.

470

Appendix C / Solved Problems

5.5.2

a = R + h = 6971 km; e = 0 (circular orbit). Eq 2.14: V = 7.56 km/s.

2

olo gy .

(a) The first manoeuvre consists in rotating the nodes without changing the inclination. The manoeuvre must therefore be initiated over one of the poles. 2

hn

Eq 5.27: ∆V = V1 + V2 − 2V1V2 cos θ = 264 m/s, with V1 = V2 = V, and θ = ∆Ω = 2°. ∆V −   g Isp   Eq 5.42: ∆m1 = m0 1 − e = 43 kg.     The second manoeuvre involves changing the inclination without rotating the (b) nodes; hence this manoeuvre must be initiated over the equator. Sun-synchronism requires dΩ/dt = 360°/365.25 = 0,9856 deg/day.

2

ec

Eq 2.18 yields i = 97,76°. 2

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fT

∆V = V1 + V2 − 2V1V2 cos θ = 1023 m/s, Eq 5.27: with V1 = V2 = V, and θ = ∆i = 97,76° - 90.00° = 7.76°. ∆V −   g Eq 5.42: ∆m2 = (m0 − ∆m1 )1 − e Isp  = 134 kg.     The total consumed propellant mass ∆mtot = 43 + 134 = 177 kg.

5.5.3

Eq 5.28:

θ = cos-1[cos i1 cos i2 + sin i1 sin i2 cos ∆Ω] = 8° after inserting i1 = 90°, i2 = 97.76° and ∆Ω = 2°.

Eq 5.27:

∆V = V1 + V2 − 2V1V2 cos θ = 1055 m/s,

2

2

∆V −   g Isp   ∆mtot = m0 1 − e Eq 5.42: = 151 kg.     The propellant mass saving = 177 – 151 = 26 kg.

Eq 5.43: dm/dt = F/(g Isp) = 3.4 g/s. ∆t = ∆mtot dt/dm = 151/0.0034 = 44,412 s = 740 minutes. Note that the duration ∆t is the actual thruster burn time. If the thruster is fired during 1 minute once per orbit (τ = 96.5 minutes), then the total manoeuvre duration is t = 740 x 96.5 = 71,429 minutes = 49.6 days. This suggests that the manoeuvre in question is quite onerous for such a small thruster.

5.5.4

5.5.5 Let the satellite be allowed to drift to the edge of the window before a correction is initiated. The window half-size ∆φ = 0.1°. Eq 2.47: d2Λ/dt2 = 0.002 deg/day2 in a westerly direction (Figure 2-78). Eq 5.44: ∆t = 28.3 days. Eq 5.45: ∆V = 0.16 m/s Eq 5.42: ∆m = 217 g. 5.5.6 E/W: The worst case (i.e. the strongest E/W pull) applies to a GEO satellite

= 0.020 deg/day2. Eq 5.45 located at longitude 120°E (Figure 5-25), in which case Λ gives a ∆V = 0.16 m/s per manoeuvre, and Eq 5.44 says that the manoeuvre must be 471

Appendix C / Solved Problems

repeated every 28 days to remain within the ±0.1° window. In the course of one year, the manoeuvre should therefore be repeated N = 365/28 = 13 times, such that the total velocity increment over one year becomes ∆VtotE/W = N·∆V = 2.08 m/s.

olo gy .

N/S: In the worst case, the inclination drifts 0.952 deg/year (Eq 2.52). Assuming that the drift is to be centered symmetrically around i = 0°, the inclination should be managed as in the polar diagram below: Ω = 180°

∆i = 0.952° 90°

fT

ec

270°

hn

i = 1°

0°

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Therefore, i1 = i2 = ½ · 0.952° = 0.476°, and ∆Ω ~ 180° in Eq 5.46, such that cos θ = 0.99986. With V1 = V2 = VGEO = 3075 m/s, we obtain from Eq 5.48: ∆VtotN/S = VGEO 2 − 2 cos θ = 51.45 m/s

6.6.1

Eq 5.42: Eq 6.1:

∆V −   ∆m = m0 1 − e g Isp  = 4 kg.     g I sp ∆m F dt I sp = ; F= = 20 N. g dm ∆t

6.6.2 In order to change the inclination alone, the manoeuvre must occur above the equator where, in the present case, the apogee and the perigee are located.

V1

∆i

V2

∆V

Eq 2.4:

472

a = ½(Ap + Pe) + R = 24,369 km.

Appendix C / Solved Problems

Eq 2.13:

 2 1 V Ap = µ −  = 1.60 km/s.  Ap + R a 

Eq 5.27:

∆V Ap = V Ap + V Ap − 2V ApV Ap cos ∆i = 1,042 km/s.

Eq 5.42:

∆V −   g Isp   ∆mAp = m0 1 − e = 1,193 kg.    

2

olo gy .

2

ec

hn

6.6.3 By performing the equivalent calculations for the perigee as in Problem 6.6.2, we find that VPe = 10.24 km/s, ∆VPe = 6.67 km/s, and ∆mPe = 3,585 kg. The intuitive reason for the much higher propellant consumption is that it takes more energy to divert a fastmoving projectile from its natural trajectory than a slow-moving projectile (compare with a bullet propelled by a rifle and thrown by hand). Therefore, given a choice, an orbital manoeuvre is usually performed at the point of lowest orbital speed. 2

(a)

Eq 6.1: I sp =

F∆t Ve = ; g∆m g

∆m =

F∆t Ve

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(b)

fT

6.6.4

mVe ; Eq 6.1: Ve = g Isp = 14,715 m/s Eq 6.4: U = 2q Hence U = 148.5 V

   m0  m0  = Ve ln Eq 6.3: ∆V = Ve ln  F∆t  m0 − ∆m   m0 − Ve 

   = 0.36 m/s   

6.6.5 By triggering a ∆V along the velocity vector V, the astronaut creates a new elliptic orbit in which he is at the perigee initially. Because anew is larger than aISS, we find from Eq 2.12 that the new orbital period τnew is longer, and therefore the mean motion nnew (= average angular velocity ω) is slower than that of the ISS (since ω = 2π/τ). Consequently the astronaut will trail behind the ISS the next time the station arrives at the perigee, and the situation will only get worse over time. V+∆V

Initially

Later

Later still

After one ISS orbit

Seen from the ISS:

473

Appendix C / Solved Problems

Apogee V

Apogee

Apogee

Perigee

olo gy .

ISS orbit Perigee

Perigee

Earth

hn

7.4.1 The full-scale decimal value of 8 bits is 28 = 256. Therefore, each bit offers a data resolution of 1/256 = 0,0039, or 0.39% of the full analogue value.

ρ  1  C   = ln   N  req B  2 BER  2 Pt Gt Gr  λ  2  Gr  1  λ  C     =   = EIRP  kTB  4πd   N  ach  T  kB  4πd 

Eq 7.12:

fT

7.4.3

ec

7.4.2 The oscillator frequency = 216 Hz. The clock will restart from zero after 248/216 = 232 seconds, or 232/(365.25 · 24 · 3600) = 136.01 years, i.e. on 5 February 2136.

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Eq. 7.15:

Our task is to match (C/N)ach with (C/N)req. In our example, (C/N)ach = 4·(C/N)req to comply with the requirement of a 6 dB margin (since 10 log 4 = 6). Both equations contain the bandwidth B in the denominator, so we may ignore it in the following. Thus:  C   N0

  1   = ρ ln  = 51.2 dB  2 BER   req

 C  G   = EIRP r  T  N 0  ach

Eq 7.17: Eq 7.20:

2

G 1 λ     = EIRP r  T  k  4πd 

1  FSL k

f · λ = c; λ = 0.15 m. d2 = 2Rh + h2; d = 2,829 km.

EIRP = [C/N]ach - [Gr/T] – [1/k] – [FSL] + [Margin] = = 51.2 + 10.0 - 228.6 + 167.5 + 6.0 = 6.1 dBW. 2

 λ  FSL =   = Free Space Loss.  4πd 

Working with EIRP and G/T, instead of Pr, Gr, Gt and T separately, speeds up link budget calculations, leaving the matter of transmit power and relative antenna sizes to a detailed trade-off. Though the dB is a dimensionless quantity, it is helpful to record the physical origin of the parameter by writing dBW for power, dBK for temperature, etc. The arithmetic stays the same.

7.4.4

474

Eq 2.39: ρ = 8.7°; Eq 7.22: θ3dB = 70°(λ/Dt) = 2ρ = 17.4°; Eq 7.16: Gt = 102.54, or 20.1 dB.

Dt = 0.6 m

Appendix C / Solved Problems

7.4.6

Pointing loss: Alignment Polarization loss: Faraday

Lpoint1 Lpol

Atomspheric loss O2

Latm1

Atomspheric loss H2O

Latm2

Galactic & tropospheric noise temperature

Tgalact

Hot body noise (mainly sun) Receive antenna pointing loss (1 deg) Total:

Thotbody Lpoint2 Ls, Ts

Figure 7-42 Eq 7.27 Eq 7.34

Loss (dB) 0 0

Noise temp (K)

0.2

0.04 8

0 8.2

230 3 13 407

653

Co p by yri Lu ght leå © Un 201 ive 4, rsi ty o

fT

Cosmic noise temperature

Lrain Train Tcosm

Rain loss/temp

Fig or Eq Eq 7.34 Eq 7.32 Figure 7-36 Figure 7-36 Figure 7-44

hn

Param

ec

Source

olo gy .

7.4.5 Figure 7-33: The coding gain is the difference in Eb/N0 between the uncoded and the coded signals, i.e. 10 dB – 2.4 dB = 7.6 dB.

Attenuation and noise due to rain are the most severe factors for the link quality. The hot body noise level is potentially high, but it is unlikely that the sun would ever enter the very narrow antenna beam.

9.4.1 According to readily available physics handbooks, E = 1010 Pa = 1010 N/m2 for aluminium; the thermal expansion coefficient α = 23.2 · 10-6; the limit yield tensile stress σty = 450 · 106 N/m2; ultimate tensile stress σtu = 520 · 106 N/m2. Eq 9.5: Eq 9.4: Eq 9.2: Eq 9.3:

Eq 9.1:

∆L = 0.7 · 10-3 m ε = 0.7 · 10-3 m/m σ = 7 · 106 N/m2 > Fc with I = πr4/4 = πD4/64 = 4.9 · 10-6 m4 and Le = 0.5 m.

The beam will neither yield (σ