Orbital Mechanics, Third Edition (AIAA Education Series)

463 Pages·2002·5.72 MB·English

by  Vladimir A. Chobotov

Orbital Mechanics
Third Edition
Edited by
Vladimir A. Chobotov
EDUCATION SERIES
J. S. Przemieniecki
Series Editor-in-Chief
Air Force Institute of Technology
Wright-Patterson Air Force Base, Ohio
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naciremA Institute of scituanoreA dna ,scituanortsA Inc.
1081 rednaxelA Bell ,evirD ,notseR ainigriV 4434-19102
American Institute of Aeronautics and Astronautics, Inc., Reston, Virginia
Library of Congress Cataloging-in-Publication Data
Orbital mechanics / edited by Vladimir A. Chobotov.--3rd ed.
p. cm.--(AIAA education series)
Includes bibliographical references and index.
.1 Orbital mechanics. 2. Artificial satellites--Orbits.
.3 Navigation (Astronautics). I. Chobotov, Vladimir A. II. Series.
TLI050.O73 2002 629.4/113--dc21
ISBN 1-56347-537-5 (hardcover : alk. paper) 2002008309
Copyright © 2002 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.
Printed in the United States of America. No part of this publication may be reproduced, distributed, or
transmitted, in any form or by any means, or stored in a database or retrieval system, without the prior
written permission of the publisher.
Data and information appearing in this book are for informational purposes only. AIAA and the authors
are not responsible for any injury or damage resulting from use or reliance, nor does AIAA or the
authors warrant that use or reliance will be free from privately owned rights.
Foreword
The third edition of Orbital Mechanics edited by .V A. Chobotov complements
five other space-related texts published in the Education Series of the American
Institute of Aeronautics and Astronautics (AIAA): Re-Entry Vehicle Dynamics by
.F J. Regan, An Introduction to the Mathematics and Methods of Astrodynamics by
R. H. Batrin, Space Vehicle Design by M. D. Griffin and J. R. French, Spacecraft
Mission Design and Spacecraft Propulsion, the last two written by C. D. Brown.
The revised text on Orbital Mechanics is specifically designed as a teaching
textbook with a significant amount of reference materials and problems for the
practicing aerospace engineer, scientist, or mission planner. The revised edition
includes now more recent developments in space exploration and the chapter on
space debris was expanded to include new developments. Also a new CD-ROM
software package is included, a useful trend particularly encouraged in the AIAA
Education Series.
The authors of this text were a team of scientists and engineers from The
Aerospace Corporation, one of the leading organizations in the U.S. space pro-
gram. The text covers both the theory and application of earth orbits and inter-
planetary trajectories, orbital maneuvers, space rendezvous, orbit perturbations,
and collision hazards associated with space debris. It represents a complete au-
thoritative exposition of the present knowledge of orbital mechanics applications
to the design of space probes and vehicles.
The AIAA Education Series of textbooks and monographs, inaugurated in 1984,
embraces a broad spectrum of theory and application of different disciplines in
aeronautics and astronautics, including aerospace design practice. The series also
includes texts on defense science, engineering, and management. It serves as
teaching texts as well as reference materials for practicing engineers, scientists,
and managers. The complete list of textbooks published in the series (over seventy-
five rifles) can be found on the end pages of this volume.
J. S. Przemieniecki
Editor-in-Chief
AIAA Education Series
AIAA Education Series
Editor-in-Chief
John S. Przemieniecki
Air Force Institute of Technology (retired)
Editorial Advisory Board
Aaron R. Byerley Robert .G Loewy
.S.U Air Force Academy Georgia Institute of ygolonhceT
Daniel .J Biezad Michael Mohaghegh
California Polytechnic State University The Boeing Company
lajaK .K Gupta Dora Musielak
NASA Dryden Flight Research Center Arlington, Texas
John .K Harvey darnoC .F Newberry
Imperial College Naval Postgraduate School
David K. Holger David .K Schmidt
Iowa State University University of ,odaroloC
Colorado Springs
Rakesh .K ainapaK Peter .J ihcruT
ainigriV Polytechnic Institute Los Alamos National Laboratory
and State University
Brian murdnaL David .M naV Wie
University of Alabama, Huntsville Johns Hopkins University
Preface
An update of Orbital Mechanics, Second Edition has been made to include more
recent developments in space exploration (e.g. Galileo, Cassini, Mars Odyssey
missions). Also, the chapter on space debris was rewritten to reflect new de-
velopments in that area. Additional example problems for student exercises are
presented in selected chapters.
A new software package is included on a CD-ROM to illustrate text material
and to provide solutions to selected problems. The software package is presented
in three folders on the CD-ROM. The first folder "HW Solutions" authored by
J. Alekshun, written in Microsoft Visual C++, can be run from the CD-ROM.
This folder presents a range of viewpoints and guidelines for solving selected
problems in the text. In some cases calculators are provided for obtaining numer-
ical results of broader scope than the problem statement. These solutions accept
a more generalized span of initial conditions. They are useful in demonstrating
cross-sensitivities between variables. Throughout the work, graphical illustrations
appear where thought helpful in projecting vector relationships and spatial trajec-
tories. The second folder entitled "Orbital Calculator" by Dr. E. T. Campbell, also
written in C++, is automatically unzipped to the "C" drive. It provides an interac-
tive environment for the generation of Keplerian orbits, orbital transfer maneuvers
and animation of ellipses, hyperbolas and interplanetary orbits. The third and final
software folder "Orbital Mechanics Solutions" by C. G. Johnson is written in C
and Fortran. It must be copied to a folder on the "C" drive to run.
The new text material and the enhanced software package provide an up-to-date
database and an improved numerical processing capability to facilitate teaching
and text problem solutions. It is hoped that the Third Edition will be a useful
textbook for students and a ready reference for the practicing professional in
orbital mechanics.
V. A. Chobotov
Spring 2002
About the Authors
Vladimir A. Chobotov
Ph.D. in Mechanical Engineering, University of Southern California
Former Manager, Space Hazards Section, Astrodynamics Department
Instructor, UCLA, The Aerospace Corporation
Hans K. Karrenberg
Engineer's Degree in Aerospace Engineering,
University of Southern California
Former Director, Astrodynamics Department
Lecturer, UCLA Extension, The Aerospace Corporation
Chia-Chun "George" Chao
Ph.D. in Astrodynamics, University of California, Los Angeles
Senior Engineering Specialist, Astrodynamics Department
Lecturer, UCLA Extension, The Aerospace Corporation
Jimmy .Y Miyamoto
M.S., University of California, Los Angeles
Former Manager, Orbit Analysis Section, Astrodynamics Department
Lecturer, UCLA Extension, The Aerospace Corporation
Thomas J. Lang
M.S. in Aeronautics and Astronautics,
Massachusetts Institute of Technology
Director, Astrodynamics Department
Lecturer, UCLA Extension, The Aerospace Corporation
Jean A. Keehiehian
Ph.D. in Aeronautics and Astronautics, Stanford University
Engineering Specialist, Astrodynamics Department
The Aerospace Corporation
Software Development
Cassandra G. Johnson
B.S. ni Computer Science, California State University at Dominguez Hills
Former Member of the Technical Staff, Astrodynamics Department
The Aerospace Corporation
Joseph Alekshun, .rJ
M.S. in Aeronautics and Astronautics, Massachusetts Institute of Technology
Former Member of the Technical Staff, The Aerospace Corporation
Eric .T Campbell
Ph.D. in Aeronautics and Astronautics, Purdue University
Senior Member of the Technical Staff, Astrodynamics Department
The Aerospace Corporation
Table of
Contents
Foreword .............................................. v
Preface ............................................... vii
About the Authors ........................................ ix
Chapter 1. Basic Concepts ................................. 1
1.1 A Historical Perspective .............................. 1
1.2 Velocity and Acceleration ............................. 5
Problems ......................................... 9
Selected Solutions ................................. 10
Chapter 2. Celestial Relationships .......................... 11
2.1 Coordinate Systems ................................ 11
2.2 Time Systems .................................... 17
References ...................................... 20
Chapter 3. Keplerian Orbits .............................. 21
3.1 Newton's Universal Law of Gravitation ................... 21
3.2 General and Restricted Two-Body Problem ................ 21
3.3 Conservation of Mechanical Energy ..................... 23
3.4 Conservation of Angular Momentum ..................... 24
3.5 Orbital Parameters of a Satellite ........................ 25
3.6 Orbital Elements .................................. 28
References ...................................... 31
Problems ........................................ 31
Selected Solutions ................................. 33
Chapter 4. Position and Velocity as a Function of Time ........... 35
4.1 General Relationships ............................... 35
4.2 Solving Kepler's Equation ............................ 40
4.3 A Universal Approach ............................... 55
4.4 Expressions with f and g ............................ 59
4.5 Summary of the Universal Approach ..................... 60
4.6 The Classical Element Set ............................ 61
4.7 The Rectangular Coordinate System ..................... 62
4.8 Modified Classical to Cartesian Transformation ............. 62
4.9 Rectangular to Modified Classical Elements Transformation ..... 66
4.10 The Spherical (ADBARV) Coordinate System .............. 67
xi
xii
4.11 Rectangular to Spherical Transformation .................. 68
4.12 Spherical to Rectangular Transformation .................. 69
4.13 The Earth-Relative Spherical (LDBARV) Coordinate System .... 70
4.14 Geodetic and Geocentric Altitudes ...................... 71
4.15 Converting from Perigee/Apogee Radii to Perigee/Apogee
Altitudes ..................................... 76
4.16 Converting from Perigee/Apogee Altitudes to Perigee/Apogee
Radii ....................................... 77
References ...................................... 82
Problems ........................................ 83
Selected Solutions ................................. 85
Chapter 5. Orbital Maneuvers ............................. 87
5.1 Orbital Energy .................................... 87
5.2 Single-Impulse Maneuvers ........................... 89
5.3 Single- and Two-Impulse Transfer Comparison for Coplanar
Transfers Between Elliptic Orbits That Differ Only in
Their Apsidal Orientation ......................... 92
5.4 Hohmann Transfer ................................. 94
5.5 The Bi-elliptic Transfer .............................. 96
5.6 Restricted Three-Impulse Plane Change Maneuver for
Circular Orbits ................................. 99
5.7 General Three-Impulse Plane Change Maneuver for
Circular Orbit ................................ 103
5.8 Hohmann Transfer with Split-Plane Change ............... 104
5.9 Bi-elliptic Transfer with Split-Plane Change ............... 107
5.10 Transfer Between Coplanar Elliptic Orbits ................ 107
References ..................................... 109
Problems ....................................... 109
Selected Solutions ................................ 115
Chapter 6. Complications to Impulsive Maneuvers ............. 117
6.1 N-Impulse Maneuvers .............................. 117
6.2 Fixed-Impulse Transfers ............................ 117
6.3 Finite-Duration Bums: Gravity Losses ................... 126
6.4 Very Low Thrust Transfers .......................... 130
References ..................................... 132
Problems ....................................... 132
Selected Solutions ................................ 134
Chapter 7. Relative Motion in Orbit ........................ 135
7.1 Space Rendezvous ................................ 135
7.2 Terminal Rendezvous .............................. 155
7.3 Applications of Rendezvous Equations .................. 162
xiii
7.4 An Exact Analytical Solution for Two-Dimensional Relative
Motion ..................................... 172
7.5 Optimal Multiple-Impulse Rendezvous .................. 177
References ..................................... 181
Problems ....................................... 182
Selected Solutions ................................ 183
Chapter 8. Introduction to Orbit Perturbations ................ 185
8.1 A General Overview of Orbit Perturbations ............... 185
8.2 Earth Gravity Harmonics ............................ 186
8.3 Lunisolar Gravitational Attractions ..................... 187
8.4 Radiation Pressure Effects ........................... 188
8.5 Atmospheric Drag ................................ 189
8.6 Tidal Friction Effects and Mutual Gravitational Attraction ..... 190
References ..................................... 192
Chapter 9. Orbit Perturbations: Mathematical Foundations ....... 193
9.1 Equations of Motion ............................... 193
9.2 Methods of Solution ............................... 195
9.3 Potential Theory .................................. 202
9.4 More Definitions of Gravity Harmonics .................. 204
9.5 Perturbations Due to Oblateness (J2) .................... 207
9.6 Integration of the Equations of Variation ................. 209
References ..................................... 213
Chapter 10. Applications of Orbit Perturbations ............... 215
10.1 Earth's Oblateness (J2) Effects ........................ 215
10.2 Critical Inclination ......... ....................... 217
10.3 Sun-Synchronous Orbits ............................ 218
10.4 ./3 Effects and Frozen Orbits ......................... 220
10.5 Earth's Triaxiality Effects and East-West Stationkeeping ...... 221
10.6 Third-Body Perturbations and North/South Stationkeeping ..... 222
10.7 Solar-Radiation-Pressure Effects ....................... 223
10.8 Atmospheric Drag Effects ........................... 227
10.9 Tidal Friction Effects .............................. 230
10.10 Long-Term Inclination Variations ...................... 233
References ..................................... 237
Problems ....................................... 238
Selected Solutions ................................ 240
Chapter 11. Orbital Systems ............................. 241
11.1 Launch Window Considerations ....................... 241
11.2 Time of Event Occurrence ........................... 253
11.3 Ground-Trace Considerations ......................... 254
xiv
11.4 Highly Eccentric, Critically Inclined Q = 2 Orbits (Molniya) . . . 256
11.5 Frozen Orbits .................................... 259
References ..................................... 263
Chapter 12. Lunar and Interplanetary Trajectories ............. 265
12.1 Introduction ..................................... 265
12.2 Historical Background ............................. 266
12.3 Important Concepts ............................... 274
12.4 Lunar Trajectories ................................ 279
12.5 Analytical Approximations .......................... 280
12.6 Three-Dimensional Trajectories ....................... 287
12.7 Interplanetary Trajectories ........................... 287
12.8 Galileo Mission .................................. 294
12.9 Cassini-Huygens Mission to Saturn and Titan .............. 296
12.10 Mars Odyssey Mission ............................. 298
References ..................................... 299
Problems ....................................... 299
Selected Solutions ................................ 300
Chapter 13. Space Debris ................................ 301
13.1 Introduction ..................................... 301
13.2 Space Debris Environment: Low Earth Orbit .............. 302
13.3 Debris Measurements .............................. 303
13.4 Space Debris Environment: Geosynchronous Equatorial Orbit... 307
13.5 Spatial Density .................................. 310
13.6 Collision Hazard Assessment Methods .................. 315
13.7 Collision Hazards Associated with Orbit Operations ......... 320
13.8 Debris Cloud Modeling ............................. 322
13.9 Lifetime of Nontrackable Debris ....................... 327
13.10 Methods of Debris Control .......................... 328
13.11 Shielding ...................................... 329
13.12 Collision Avoidance ............................... 330
References ..................................... 332
Chapter 14. Optimal Low-Thrust Orbit Transfer .............. 335
14.1 Introduction ..................................... 335
14.2 The Edelbaum Low-Thrust Orbit-Transfer Problem .......... 335
14.3 The Full Six-State Formulation Using Nonsingular Equinoctial
Orbit Elements ............................... 354
14.4 Orbit Transfer with Continuous Constant Acceleration ........ 372
14.5 Orbit Transfer with Variable Specific Impulse .............. 389
Appendix: The Partials of the M Matrix .................. 399
References ..................................... 409