QUALITATIVE AND QUANTITATIVE BEHAVIOUR OF PLANETARY SYSTEMS EDITOR-IN-CHIEF 1. HENRARD, Dept. ofM athematics, Facultes Universitaires de Namur, Namur, Belgium ASSISTANT EDITOR M. MOONS, Dept. of Mathematics, Facultes Universitaires de Namur, Namur, Belgium ASSOCIATE EDITORS N. Borderies, Jet Propulsion Laboratory, Pasadena, P. 1. Message, University ofL iverpool, U.K. U.S.A. A. Milani, Universita di Pisa, Italy R. Broucke, University of Texas, Austin, U.S.A. A. E. Roy, University of Glasgow, U.K. V. A. Brumberg, Institute ofA pplied Astronomy, D. G. Saari, Northwestern University, Evanston, Leningrad, Russia U.S.A. G. Contopoulos, University of Athens, Greece V. A. Sarychev, Institute ofA pplied Mathematics, A. Deprit, National Bureau of Standards, Moscow, Russia Gaithersburg, U.S.A. L. Sehnal, Astronomical Institute, Ondrejov, S. Ferraz-Mello, University ofS iio Paulo, Brasil Czechoslovakia Y. Kozai, National Astronomy Observatory, P. Vandervoort, University of Chicago, U.S.A. Mitaka, Tokyo, Japan CELESTIAL MECHANICS INSTITUTE Kaare Aksnes, Institute of Theoretical Astronomy, William M. Kaula, University of California, Los Oslo Angeles R. H. Battin, The Charles Stark Draper Laboratory, Hiroshi Kinoshita, National Astronomy Obser- Cambridge vatory, Tokyo R. Broucke, University of Texas, Austin Jean Kovalevsky, CERGA, Grasse 1. M. A. Danby, North Carolina State University, Jay Lieske, Pasadena Raleigh Richard H. Miller, University of Chicago Morris S. Davis, University ofN orth Carolina, P. K. Seidelmann, U.S. Naval Observatory, Chapel Hill Washington (President) L. E. Doggett, U.S. Naval Observatory, Washington Irwin I. Shapiro, Harvard-Smithsonian Center for R. L. Duncombe, University of Texas, Austin Astrophysics, Cambridge J. B. Eades, E & SA Engineering & Science E. M. Standish, Jet Propulsion Laboratory, Associates Inc., Rockville (Vice President) Pasadena (Secretary) E. Gaposchin, MIT Lincoln Laboratory, Westford V. Szebehely, University of Texas, Austin B. Garfinkel, Yale University Observatory, New B. Tapley, University of Texas, Austin Haven C. A. Williams, University of South Florida, Tampa John L. Junkins, Texas A&M University, College (Treasurer) Station John Hadjidemetriou, Department of Theoretical Mechanics, Thessaloniki ISSN 0923-2958 All Rights Reserved © 1993 Kluwer Academic Publishers No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner. QUALITATIVE AND QUANTITATIVE BEHAV IOUR OF PLANETARY SYSTEMS Proceedings of the Third Alexander von Humboldt Colloquium on Celestial Mechanics Edited by R. DVORAK InstituI fur Astronomie, Universitătsstemwarte, VieI/ilo, Austria ,nd J.HENRARD Departemem de Ma/hemarique. FNDP, Namur, Be/giwlI Reprinted from Celestial Mechanics and Oynamical Astronomy Volume 56, Nos. 1-2, 1993 •• SPRINGER SCIENCE+BUSlNESS MEDIA, B.V. A C.I.P. Catalogue record for this book is available from the Library of Congress. ISBN 978-94-010-4898-9 ISBN 978-94-011-2030-2 (eBook) DOI 10.1007/978-94-011-2030-2 Printed on acid-free paper AII Rights Reserved © 1993 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1993 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner. TABLE OF CONTENTS PREFACE ix SESSION ON PLANETARY THEORIES (Chairman: H. Eichhorn) P.K. SEIDELMANN / Review of Planetary and Satellite Theories 1 C. MARCHAL / The Mystery of Pluto's Mass - The Ring Hypothesis 13 H. YOSHIDA / Recent progress in the theory and application of symplectic Integrators 27 D. BENEST / Stable Planetary Orbits Around One Component in Nearby Binary Stars 45 A. KUBALA, D. BLACK and V. SZEBEHELY / Stability of Outer Planetary Orbits Around Binary Stars: a Comparison of Hill's and Laplace Stability Criteria 51 T. INOUE / An Excess Motion of the Ascending Node of Mercury in the Observations Used by Le Verrier 69 SESSION ON THE SITNIKOV PROBLEM (Chairman: Yi-Sui Sun) R. DVORAK / Numerical Results to the Sitnikov-Problem 71 J. HAGEL and Th. TRENKLER / A Computer Aided Analysis of the Sitnikov Problem 81 K. WODNAR / The Original Sitnikov Article - New Insights 99 SESSION ON ASTEROIDS (Chairman: J. Henrard) A. LEMAITRE / Proper Elements: What are They? 103 T. MICHTCHENKO and S. FERRAZ-MELLO / The High Eccentricity Libration of the Hildas II: Synthetic-Theory Approach 121 G. HAHN, c.1. LAGERKVIST and B.A. LINDBLAD / A Stability Study of Asteroid Families Near the 3:1 and 5:2 Resonance with Jupiter 131 M. SIDLICHOVSKY / Chaotic Behaviour of Trajectories for the Asteroidal Resonances 143 v vi TABLE OF CONTENTS J. SCHUBART / Low-Eccentricity Motion of Asteroids Near the 211 Jovian Resonance 153 Ch. FROESCHLE and H. SCHOLL: Numerical Experiments in the 311 and 1I6 Overlapping Resonance Region 163 SESSION ON RESONANCE PROBLEMS (Chairman: S. Ferraz-Mello) A. MORBIDELLI / An Introduction to Hamiltonian Dynamical Systems and Practical Perturbation Methods: New Insight by Successive Elimination of Perturbation Harmonics 177 J. LASKAR / Frequency Analysis of a Dynamical System 191 Ph. ROBUTEL / An Application of KAM Theory to the Planetary Three Body Problem 197 J. HADJIDEMETRIOU / Resonant Motion in the Restricted Three Body Problem 201 B. ERDI and J. KOY Acs / A Fourth-Order Solution of the Ideal Resonance Problem 221 SESSION ON GENERAL DYNAMICAL SYSTEMS (Chairman: P.K. Seidelmann) Ph. BENDJOYA and E. SLEZAK / Wavelet Analysis and Applications to Some Dynamical Systems 231 A.D. GILBERT, Cl. FROESCHLE and U. FRISCH / Wavelet Analysis of the Standard Map: Structure and Scaling 263 M. MOONS and A. MORBIDELLI / Asteroids: 211 Resonance and High Eccentricity 273 PJ. MESSAGE / On the Second Order Long-Period Motion of Hyperion 277 F. DELHAISE and J. HENRARD / The Problem of Critical Inclination Combined with a Resonance in Mean Motion in Artificial Satellite Theory 285 SESSION ON CHAOS AND STABILITY (Chairman: P.J. Message) P. FARINELLA, Ch. FROESCHLE and R. GONCZI / Meteorites from the Asteroid 6 Hebe 287 TABLE OF CONTENTS vii Cl. FROESCHLE, Ch. FROESCHLE and E. LOHINGER / Generalized Lyapunov-Characteristic Indicators and Corresponding Kolmogorov like Entropy of the Standard Mapping 307 E. LOHINGER, Cl. FROESCHLE and R. DVORAK / Generalized Lyapunov Exponents Indicators in Hamiltonian Dynamics: An Application to a Double Star System. 315 A. MILANI / Asteroid 522 Helga is Chaotic and Stable 323 SESSION ON MISCELLANEOUS PROBLEMS (Chairman: V. Szehebely) G. CONTOPOULOS / Classical Periodic Orbits and Quantum Mechanical Eigenvalues and Eigenfunctions 325 H. EICHHORN / Generalized Least-Squares Adjustments: A Timely but Much Ignored Tool 337 J. KALLRATH, J.P. SCHLODER and H.G. BOCK / Least Squares Parameter Estimation in Chaotic Differential Equations 353 G. VALSECCHI, E. PEROZZI, A.E. ROY and B.A. STEVES / The arrangement in Mean Elements Space of the Periodic Orbits Close to that ofthe Moon 373 H. SCHOLL, F. ROQUES and B. SICARDY / Resonance Trapping of Circumstellar Dust Particles by an Alleged Planet 381 D. LAZZARO, B. SICARDY, F. ROQUES and H. SCHOLL / Orbital Resonances and Poynting-Robertson Drag Confining Dust Particles in ,B-Pic Disk 395 Participants in the Third Alexander von Humboldt Colloquium on Celestial Mechanics PREFACE The success of the two first Alexander von Humboldt Colloquia on Celestial Mechanics (March 1984 and March 1988) encouraged us to meet again this year from March 29 to April 4, 1992 in the " Alpengasthof Peter Rosegger"" in the Styrian Alps (Ramsau, Austria). This time the colloquium was devoted to the "Qualitative and Quantitative behaviour of Planetary Systems". The papers covered a large range of questions of current interest from the behaviour of dust particles to the stability of the Solar System as a whole, without forgetting the motion of Asteroids and their classification into families. KAM theory, chaotic motions, resonances, Lyapunov characteristic exponents, perturbation theory, numerical integration were - of course - on the menu every day, served with sauces and accents from various part of the world: from China, from Brazil, from the United States and from all over Europe. To be able to organize this - now well established - meeting , we have to thank primarily the munificence of the Austrian Ministry of Science, furthermore the Steiermarkischen Wissenschafts- und Landesfonds and the Osterreichische Nationalbank for financial support. Also, the Osterreichische Forschungsgemein schaft and the University of Vienna made it possible to invite participants to this meeting from abroad. Support in form of "Tagungsunterlagen" was given by the Creditanstalt-Bankverein in Vienna and coffee during the whole meeting time was donated by Homig-Kaffee Graz. Many thanks are due to the Chairmen (unfortunately we did not have any chairwoman this time) of the sessions. They acted as Editors of the papers presented during "their" session, arranging for competent and fast refereeing so that the papers could be reviewed and, when necessary, corrected in a very short time. Our model hosts Barbara and Fritz Walcher contributed a lot to the success of the meeting, as they did their best providing pleasant and friendly surroundings. We all are looking forward to 1996 when we plan to meet again at the Alpengasthof Rosegger. R. Dvorak J. Henrard Celestial Mechanics and Dynamical Astronomy 56: ix, 1993. REVIEW OF PLANETARY AND SATELLITE THEORIES P. K. SEIDELMANN US Naval Observatory Abstract. Planetary and satellite theories have been historically and are presently intimately related to the available computing capabilities, the accuracy of observational data, and the requirements of the astronomical community. Thus, the development of computers made it possible to replace planetary and lunar general theories with numerical integrations, or special perturbation methods. In turn, the availability of inexpensive small computers and high-speed computers with inexpensive memory stimulated the requirement to change from numerical integration back to general theories, or representative ephemerides, where the ephemerides could be calculated for a given date rather than using a table look-up process. In parallel with this progression, the observational accuracy has improved such that general theories cannot presently achieve the accuracy of the observations, and, in turn, it appears that in some cases the models and methods of numerical integration also need to be improved for the accuracies of the observations. Planetary and lunar theories were originally developed to be able to predict phenomena, and provide what are now considered low accuracy ephemerides of the bodies. This proceeded to the requirement for high accuracy ephemerides, and the progression of accuracy improvement has led to the discoveries of the variable rotation of the Earth, several planets, and a satellite. By means of mapping techniques, it is now possible to integrate a model of the motion of the entire solar system back for the history of the solar system. The challenges for the future are: Can general planetary and lunar theories with an acceptable number of terms achieve the accuracies of observations? How can numerical integrations more accurately represent the true motions of the solar system? Can regularly available observations be improved in accuracy? What are the meanings and interpretations of stability and chaos with respect to the motions of the bodies of our solar system? There has been a parallel progress and development of problems in dealing with the motions of artificial satellites. The large number of bodies of various sizes in the limited space around the Earth, subject to the additional forces of drag, radiation pressure, and Earth zonal and tesseral forces, require more accurate theories, improved observational accuracies, and improved prediction capabilities, so that potential collisions may be avoided. This must be accomplished by efficient use of computer capabilities. Key words: Planetary theories - satellite theories - artificial satellites 1. Introduction There is a real solar system as part of the real universe that is moving and changing due to the real existing forces. We attempt to compute, observe, and understand the real thing by means of models and approximations. Unfortunately, our theories and computations are incomplete, and our observations are subject to systematic and random errors. There is a need to develop new theories for the planets, the moon, and artifi cial satellites. Technology has advanced such that we have significantly increased computer capabilities, both speed and memory, much better software, and alge braic manipulators. The challenge is to use these improvements to satisfy the requirements for new theories. Priorto the 1950's planetary and lunar theories were developed from two body models by expansions in tenns of small quantities and by orders of the power of the perturbing masses. Theories by Newcomb (1898), Hill (1898), Leverrier Celestial Mechanics and Dynamical Astronomv 56: 1-12,1993. © 1993 Kluwer Academic Publishers.