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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 Publishedby AmericanInstitute of Aeronauticsand Astronautics,Inc. 1801 AlexanderBell Drive, Reston, Virginia20191-4344 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 U.S. Air Force Academy Georgia Institute of Technology Daniel J. Biezad Michael Mohaghegh California Polytechnic State University The Boeing Company Kajal K. Gupta Dora Musielak NASA Dryden Flight Research Center Arlington, Texas John K. Harvey Conrad F. Newberry Imperial College Naval Postgraduate School David K. Holger David K. Schmidt Iowa State University University of Colorado, Colorado Springs Rakesh K. Kapania Peter J. Turchi Virginia Polytechnic Institute Los Alamos National Laboratory and State University Brian Landrum David M. Van 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. in Computer Science, California State University at Dominguez Hills Former Member of the Technical Staff, Astrodynamics Department The Aerospace Corporation Joseph Alekshun, Jr. 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
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