FROM ORDERED TO CHAOTIC MOTION IN CELESTIAL MECHANICS

From Ordered to Chaotic Motion in Celestial Mechanics Downloaded from www.worldscientific.com FROM ORDERED TO CHAOTIC MOTION IN CELESTIAL MECHANICS

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From Ordered to Chaotic Motion in Celestial Mechanics Downloaded from www.worldscientific.com FROM ORDERED TO CHAOTIC MOTION IN CELESTIAL MECHANICS Yi-Sui Sun Li-Yong Zhou Nanjing University, China World Scientific NEW JERSEY LONDON SINGAPORE BEIJING SHANGHAI HONG KONG TAIPEI CHENNAI TOKYO

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE From Ordered to Chaotic Motion in Celestial Mechanics Downloaded from www.worldscientific.com Library of Congress Cataloging-in-Publication Data Sun, Yi-Sui, author. From ordered to chaotic motion in celestial mechanics / Yi-Sui Sun, Li-Yong Zhou, Nanjing University, China. pages cm Includes bibliographical references and index. ISBN 978-9814630542 (hardcover : alk. paper) 1. Few-body problem. 2. Celestial mechanics. I. Zhou, Li-Yong, author. II. Title. QB362.F47S87 2015 521--dc23 2015001993 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Cover image credit: ESO/M. Kornmesser. Copyright 2016 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. Typeset by Stallion Press Email: enquiries@stallionpress.com Printed in Singapore

From Ordered to Chaotic Motion in Celestial Mechanics Downloaded from www.worldscientific.com Contents Preface 1. Qualitative analysis on motion in 3-body system 1 1.1 Equations of motion and invariants............. 1 1.1.1 Classical formulation................ 1 1.1.2 Jacobi formulation.................. 2 1.1.3 Hamiltonian formulation.............. 3 1.1.4 Integral invariants.................. 4 1.2 Condition of permissible motion............... 6 1.3 Variations of configuration and position.......... 9 1.3.1 Change of inclination of orbits........... 9 1.3.2 Change of angle between orbits........... 11 1.3.3 Changes of orbital semi-major axis and eccentricity................... 14 1.4 Restricted 3-body problem and singular points of motion........................... 18 1.4.1 Equations of motion and Jacobi integral...... 18 1.4.2 Singular points of equations of motion....... 21 1.5 Stabilities of Lagrange and Euler solutions......... 24 1.5.1 Planarproblem... 25 1.5.2 Spatial problem................... 29 1.6 Elliptic restricted 3-body problem.............. 30 1.6.1 Analyses on equations of motion.......... 30 1.6.2 Permissible region of m 3.............. 33 1.7 Hill region in 3-body problem................ 36 1.7.1 Hill-type stability in 3-body problem....... 36 1.7.2 Generalized Hill-type stability in N-body problem........................ 41 xi v

vi From Ordered to Chaotic Motion From Ordered to Chaotic Motion in Celestial Mechanics Downloaded from www.worldscientific.com 1.7.3 Hill stability in hierarchical triple system..... 42 1.8 Evolution of inertia momentum in N-body problem.... 52 1.8.1 Some relations and a theorem........... 53 1.8.2 Evolution of ρ.................... 57 1.9 Motion of isolated body in 3-body problem........ 61 1.9.1 Isolated body..................... 63 1.9.2 Acceleration of isolated body............ 66 1.9.3 Position and velocity of isolated body....... 68 1.10 Sitnikov motion and its generalization........... 74 1.10.1 Model and mappings................ 74 1.10.2 Topological structure of phase space....... 79 1.10.3 Chaotic domain................... 83 1.10.4 Extended Sitnikov problem............. 84 1.11 Central configuration of 4-body problem.......... 92 1.11.1 Elementary equations and theorem......... 92 1.11.2 General 4-body problem.............. 94 1.11.3 Numerical exploration................ 98 1.12 Central configurations of N-body problem with general attraction and homographic solutions.... 103 1.12.1 Some theoretical analysis.............. 104 1.12.2 Equivalent conditions of central configuration..................... 108 1.12.3 Homographic solutions of general N-body problem........................ 110 1.12.4 Homographic solutions and central configurations.................... 114 2. Motion of small bodies in the planetary system 117 2.1 Mapping method in Hamiltonian system.......... 117 2.1.1 Hamiltonian equation of near-conservative system........................ 119 2.1.2 Averaging system and fixed points......... 121 2.1.3 Construction of mapping.............. 124 2.2 Structure of phase space near Lagrange solutions..... 128 2.2.1 Hamiltonian equations............... 128 2.2.2 Mapping model................... 132 2.2.3 Structure of phase space.............. 133 2.3 Stability of asteroid orbits in resonances.......... 136 2.3.1 Mapping model................... 137

Contents vii From Ordered to Chaotic Motion in Celestial Mechanics Downloaded from www.worldscientific.com 2.3.2 Diffusion in the mapping.............. 143 2.3.3 Stabilities of asteroid orbits in resonances..... 147 2.4 Shepherding of Uranian ring................. 151 2.4.1 Equations of motion and the Hamiltonian..... 152 2.4.2 Viscosity in the ring and the mapping....... 156 2.4.3 Numerical results.................. 157 2.5 Formation of Kuiper belt................... 163 2.5.1 Planet migration and resonant capture...... 164 2.5.2 Stochastic planet migration............. 165 2.5.3 Resonance capture in stochastic migration.... 166 2.6 Dynamics of Neptune Trojans................ 173 2.6.1 Model and method.................. 174 2.6.2 Dynamical maps................... 176 2.6.3 Frequency analysis and resonances map...... 180 2.6.4 Origin of Neptune Trojans............. 188 2.7 Apsidal and nodal resonances in multiple planetary systems....................... 192 2.7.1 Apsidal resonance.................. 192 2.7.2 Nodal secular resonance............... 199 2.8 Apsidal corotation in 3:1 mean motion resonance..... 202 2.8.1 Numerical simulations................ 204 2.8.2 Analytical model................... 206 2.8.3 Apsidal corotation.................. 209 3. Chaotic motion of orbits 217 3.1 Conservative dynamical system............... 217 3.2 Ordered and chaotic motion................. 220 3.3 Poincaré surface of section.................. 223 3.4 Ordered and chaotic motion of stars............ 228 3.5 Application of mapping method to comet motion..... 232 3.5.1 Basic model and mapping.............. 233 3.5.2 Properties of the mapping............. 237 3.5.3 Phase plane of mapping and orbital stability ofcomets... 239 3.5.4 Elliptical case.................... 243 3.6 Global applicability of symplectic integrator........ 251 3.6.1 Model......................... 252

viii From Ordered to Chaotic Motion From Ordered to Chaotic Motion in Celestial Mechanics Downloaded from www.worldscientific.com 3.6.2 Integration schemes................. 253 3.6.3 Numerical results.................. 255 3.7 Transferofcometorbit... 260 3.7.1 Mapping model................... 263 3.7.2 Orbit transfer.................... 265 3.8 Random walk in comet motion............... 274 3.9 Chaotic region of encounter-type orbit........... 278 3.9.1 Equations of motion................. 279 3.9.2 Orbital variation in close encounter........ 282 3.9.3 D i and orbital elements............... 286 3.9.4 Size of chaotic region................ 288 4. Orbit diffusion 291 4.1 Diffusion in comet motion.................. 291 4.1.1 Comets in direct orbits............... 292 4.1.2 Comets in retrograde orbits............. 296 4.2 KS entropy of area-preserving mapping........... 300 4.2.1 Basics about LCN s................. 301 4.2.2 Numerical estimation................ 302 4.2.3 Analytical estimation................ 304 4.3 Invariant tori in volume-preserving mapping........ 307 4.3.1 Existence of invariant manifolds in T 1 and T 2... 309 4.3.2 Theoretical analysis and criterion of tube existence....................... 315 4.3.3 Implications..................... 322 4.4 Perturbed extension of area-preserving mapping...... 323 4.4.1 Two-dimensional mapping T... 324 4.4.2 Three-dimensional mapping T........... 327 4.5 KS entropy of volume-preserving mapping......... 334 4.6 Attractor in three-dimension mapping........... 340 4.6.1 Mapping model................... 341 4.6.2 LCN s and fractal dimension............ 342 4.6.3 Numerical results.................. 343 4.7 Stickiness effect and hyperbolic structure (I)........ 348 4.7.1 Two-dimensional model............... 350 4.7.2 Three-dimensional model.............. 360 4.8 Stickiness effect and hyperbolic structure (II)....... 371 4.8.1 Mapping model................... 371 4.8.2 Angle between stable and unstable manifolds... 374

Contents ix 4.8.3 Stickiness effect and characteristic angle ofhyperbolicstructure... 379 4.9 Diffusion in four-dimension mapping............ 388 4.9.1 Mapping model................... 388 4.9.2 Diffusion characters in different regions...... 389 From Ordered to Chaotic Motion in Celestial Mechanics Downloaded from www.worldscientific.com Bibliography 397 Index 403

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From Ordered to Chaotic Motion in Celestial Mechanics Downloaded from www.worldscientific.com Preface This book provides a brief introduction to some basic but important problems in celestial mechanics. It is based on the main results in some of the authors research works, which are related to the qualitative method of celestial mechanics and nonlinear dynamics. Some of these works are the interdisciplinary courses, involving the celestial mechanics and nonlinear dynamics, sometimes called nonlinear celestial mechanics. Since the era of Sir Issac Newton (1643 1727) when the universal gravitation was discovered and the calculus was invented, the celestial mechanics has become a real science. In 300 years afterwards, numerous scientists devoted their great efforts to the problem of motion of celestial bodies affected (only) by their mutual gravitational attraction. The first great success in celestial mechanics, owing to Newton, is the solution of the two-body problem, where the trajectory of a planet traveling around the Sun was fully depicted and understood. When the number of celestial objects increases by only one to three, the problem becomes much more complicated, as many great scientists like Joseph-Louis Lagrange (1736 1813), Pierre-Simon Laplace (1749 1827), Henri Poincaré (1854 1912), etc. had illustrated and proven. The new version of the gravitation theory by Albert Einstein (1879 1955), which will introduce some very tiny but important correction to Newton s theory, has not been taken into account in this book. Nevertheless, this book covers a wide variety of topics. The objects of research presented in this book passes an extensive variety including the comets, asteroids, planetary rings, Trojan asteroids, etc. Many applicable methods in celestial mechanics, such as the nonlinear dynamical method, mapping method, symplectic integrator, spectral analysis etc have been shown, mainly in a practical way as in our research works. The diverse xi

xii From Ordered to Chaotic Motion From Ordered to Chaotic Motion in Celestial Mechanics Downloaded from www.worldscientific.com content in this book is organized as follows. Chapter 1 presents some qualitative analyses on the behaviors of motion in three(n)-body system. Then we devote Chapter 2 to the motion of small bodies in the planetary system. Chapter 3 concerns the chaotic motion of orbits in celestial mechanics. And finally Chapter 4 focuses on the orbit diffusion in phase space. This book is not a textbook. We will not introduce a whole course comprehensively. When a subject will be discussed, a brief overall introduction and the necessary background about the related subject will be given, and more often our readers may be directed to related references to obtain a thorough understanding of a subject. This book is neither a simple collection of the authors work. It is the reorganization of related results according to subjects. Most of the results presented in this book were the outcomes of long term pleasurable cooperations with our colleagues. The pleasant recollections invoked by this book made the writing process a happy journey to the past. We would like to thank all the co-authors of the papers on which this book is based. Particularly, Prof. Sun sincerely cherishes the memory of working with Prof. Chen Xiang-Yan. The scientific work of the authors would be impossible without the financial supports from the Natural Sciences Foundation of China (NSFC), Ministry of Science and Technology of China and Ministry of Education of China. The authors are grateful for their continuous supports. Great thanks also go to Nanjing University where the authors get education, work, and live. Finally, we would like to thank the Editor of this book, Mr. Ng Kah Fee. His efficient efforts and kind encouragement to us made this book a reality.