ADVENTURES IN CELESTIAL MECHANICS Second Edition VICTOR G. SZEBEHELY HANS MARK Department of Aerospace Engineering and Engineering Mechanics The University of Texas at Austin LEY- WIVCH WILEY-VCH Verlag GmbH & Co. KGaA This Page Intentionally Left Blank ADVENTURES IN CELESTIAL MECHANICS This Page Intentionally Left Blank ADVENTURES IN CELESTIAL MECHANICS Second Edition VICTOR G. SZEBEHELY HANS MARK Department of Aerospace Engineering and Engineering Mechanics The University of Texas at Austin LEY- WIVCH WILEY-VCH Verlag GmbH & Co. KGaA All books published by Wiley-VCH are carefully produced. Nevertheless, authors, editors, and publisher do not warrant the information contained in these books, including this book, to be free oferrors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate. Library of Congress Card No.: Applied for British Library Cataloging-in-Publication Data: A catalogue record for this book is available from the British Library Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at <http://dnb.ddb.de>. 0 1998 by John Wiley & Sons, Inc. 0 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheini All rights reserved (including those of translation into other languages). No part ofthis book may be reproduced in any form ~ nor transmitted or translated into machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law. Printed in the Federal Republic of German); Printed on acid-free paper Printing Strauss GmbH, Morlenbach Bookbinding Litges & Dopf Buchbinderei GmbH, Heppenheim Cover Photo NASA ISBN-13: 978- 0-47 1 - 133 17-9 ISBN-10: 0-471-133 17-5 A TRIBUTE TO VICTOR G. SZEBEHELY Victor G. Szebehely 1921--1997 It is with a heavy heart that I write these words. On September 13, 1997, my friend, colleague, and mentor, Professor Victor G. Szebehely, died at his home in Austin. I have to confess that it has been very hard for me to carry on with this book without him. He was the guiding spirit of the work. Victor and I had much in common. We were both refugees from Europe-he from Hungary and I from Austria-fleeing the twin scourges of Nazism and Communism. We both became Americans and we both worked on technical projects related to the national security. We both came to love The University of Texas. Finally, we both developed a strong interest in space exploration, and Victor made important contributions to the success of our journeys to the Moon. Where Victor was unique was in his deep understanding of celestial mechanics and his ability to apply this knowledge to the solution of prac- tical problems. Victor did not hesitate to tackle the toughest scientific problem in his field which is the subject of the final chapter in this work: The problem of three bodies. He had the intellectual courage to take on the hardest challenges and the intellectual horsepower to make critical contributions of lasting value. I would be remiss if I did not mention Victor’s personal qualities. In addition to being a man of intellect, Victor was also a man of good will who was honored and respected by all who knew him. Perhaps most im- portant for his friends was his impish sense of humor. We both had our V vi ATRIBUTE TO VICTOR SZEBEHELY offices on the fourth floor of Woolrich Hall, the aerospace building on our campus. I was on one end of the floor and he was on the other. One morning I was complaining to him about something that had gone wrong with our research finding in the Congress. Suddenly, he proposed that we resurrect the Austro-Hungarian Empire on the fourth floor and raise the Imperial banner “with appropriate salutes” every morning. “Maybe,,’ he said, “that will solve your problem!” I laughed and promptly forgot what was upsetting me. Victor Szebehely was a great man whose influence was widespread. I was one of the people who came into his orbit and I am proud to have been his student. I mourn him and I miss him. Rest in peace, my friend, and go with God. September I99 7 HANSMARK CONTENTS Preface ix 1 ON THE SHOULDERS OF GIANTS: AN HISTORICAL REVIEW 1 2 CIRCULAR ORBITS 13 3 THE GENERAL PROBLEM OF TWO BODIES 33 4 ELLIPTIC ORBITS 47 5 ROCKETS 65 6 ENERGY RELATIONSHIPS: HYPERBOLIC AND PARABOLIC ORBITS 85 7 KEPLER’S EQUATION AND LAMBERT’S THEOREM 99 a ORBITAL MANEUVERING OF SPACECRAFT 139 9 ELEMENTS OF SPACECRAFT DYNAMICS 161 10 PLANETARY EXPLORATION Ia7 vii 11 GENERAL PERTURBATION THEORY AND A SPECIFIC APPLICATION TO THE MOTION OF THE PLANET MERCURY 221 12 THE MOTION OF EARTH-ORBITING SATELLITES 247 13 THE PROBLEM OF THREE BODIES AND THE STABILITY OF THE SOLAR SYSTEM 203 APPENDIX I GLOSSARY 283 APPENDIX II PHYSICAL CONSTANTS 291 APPENDIX 111 ANNOTATED LIST OF MAJOR REFERENCE BOOKS 295 INDEX 305 PREFACE This volume is the second edition of Adventures in Celestial Mechanics published by Victor G. Szebehely in 1989 at the University of Texas Press. The subject of this edition is the same as the previous one which was to quote the earlier introduction “to study the motion of natural and artificial bodies in space.” The work is still intended as a textbook for a first course in orbital mechanics and spacecraft dynamics and we have at- tempted to produce a second edition that maintains the spirit of the first. This was also stated succinctly in the introduction of the first edition: “fundamental ideas will be emphasized and will not be cluttered up with details that are available in the immense literature of this field.” Having described the similarities between this book and the previous one, we should say a word about the changes. The principal difference be- tween the two editions is that we have added some material that strength- ens the treatment of the “artificial bodies in space.” A chapter on rocket propulsion has been added that describes what must be done to get things into space. We have included a chapter on elementary spacecraft dynam- ics so that we discuss not only trajectory maneuvers but also how space- craft are stabilized and oriented. Finally, we have included a chapter on the exploration of the solar system in which the “natural” and “artificial” bodies are treated together. This area is one of the genuine triumphs of modern science and engineering, and it constitutes the most important modern application of celestial mechanics. Therefore we felt that it was necessary to address it even in an elementary course. ix X PREFACE In addition to these major changes, there are minor ones as well. In several instances (e.g., Lambert’s theorem and gravity-assist trajectories) we have included details that were not present in the first volume. We have also somewhat expanded the discussion of the three-body problem to include chaotic motion in nonlinear systems. For the most part, therefore, the second edition is similar to the first. Each of the chapters contains some numerical examples so that students will become familiar with how various calculations are performed. Prob- lems are also included at the end of each chapter. Finally, appropriate ref- erences are mentioned at the end of each chapter and also in the appen- dix. Many people helped us to write this book. We are grateful to these col- leagues in particular: Professor Roger Broucke for his help in developing the derivation of Lambert’s theorem, Professors Wallace Fowler and Bob E. Schutz for their help in writing and revising Chapter 9 (Elements of Spacecraft Dynamics), and Professor Raynor L. Duncombe for carefully reading and commenting on the manuscript. We owe a very special debt of gratitude to Ms. Maureen A. Salkin who did a superb job typing the entire manuscript. In addition, Ms. Salkin made important editorial sug- gestions that significantly improved the quality of the work. Finally, we would like to thank all of the students who were in our classes during the years that we have taught this course at The University of Texas at Austin. These young people provided us with continuing stimulation and inspira- tion which made it a great pleasure for both of us to work on this project. VICTORG. SZEBEHELY HANSMARK Austin, Texas August 1997 CHAPTER 1 ON THE SHOULDERS OF GIANTS: AN HISTORICAL REVIEW People have looked at the stars since the dawn of history. The obvious “permanence” of the heavens and the regularity of the motions executed by the Sun, the Moon, and the planets soon led people to look for expla- nations. Each of the major civilizations produced a “cosmology” that was based on more or less crude observations and was melded with the myths of the civilization. These “theories” of the cosmos were important in that they were early attempts to understand how the universe works. While many of these had philosophical and perhaps literary value, they lacked what is essential in a modern scientific theory: predictive value. None of these theories were able to make really accurate predictions of phenome- na such as eclipses or were able to explain why the observed regularities in the planetary motions exist. During the fourth and third centuries before the birth of Christ, there was a great flowering of civilization in Greece. Philosophical schools were established by a number of people, and one of the major topics of interest was cosmology. Many theories were set forth, including at least one that put the Sun at the center of the solar system.