Long-Term Dynamical Behaviour of Natural and Artificial N-Body Systems

515 Pages·1988·15.31 MB·English

by  Bruce A. Conway

Long-Term Dynamical Behaviour of Natural and
Artificial N-Body Systems
NATO ASI Series
Advanced Science Institutes Series
A Series presenting the results of activities sponsored by the NA TO Science Committee,
which aims at the dissemination of advanced scientific and technological knowledge,
with a view to strengthening links between scientific communities.
The Series is published by an international board of publishers in conjunction with
the NATO Scientific Affairs Division
A Life Sciences Plenum Publishing Corporation
B Physics London and New York
C Mathematical Kluwer Academic Publishers
and Physical Sciences Dordrecht, Boston and London
D Behavioural and Social Sciences
E Applied Sciences
F Computer and Systems Sciences Springer-Verlag
G Ecological Sciences Berlin, Heidelberg, New York, London,
H Cell Biology Paris and Tokyo
Series C: Mathematical and PhYSical Sciences -Vol. 246
Long.:rerm Dynamical Behaviour
of Natural and Artificial
N-Body Systems
edited by
Arch ie E. Roy
Department of Physics and Astronomy,
University of Glasgow,
Glasgow, U.K.
Kluwer Academic Publishers
Dordrecht / Boston / London
Published in cooperation with NATO Scientific Affairs Division
Proceedings of the NATO Advanced Study Institute on
Long-Term Dynamical Behaviour of Natural and Artificial N-Body Systems
Cortina d'Ampezzo, Italy
August 2-13, 1987
Library of Congress Cataloging in Publication Data
NATO Advanced Study Institute (1987 : Cortina d'Ampezzo, Italy)
Long-term dynamical behaviour of natural and artificial n-body
systems proceedings of the NATO Advanced Study Institute held at
Cortina d'Ampezzo, Italy, August 2-13, 1987 ! edited by Archie E.
Roy.
p. cm, -- (NATO advanced science institutes. Serles C,
Mathematical and physical sciences; 246)
Includes indexes.
ISBN-13: 978-94-010-7873-3
1. Many-body problem--Congresses. 2. MechaniCS, Celestial
-Congresses. 3. Dynamics--Congresses. 4. Planets, Theory of-
-Congresses. I. Roy, A. E. (Archie E.), 1924- II. Title.
III. Serles: NATO ASI series. Series C, Mathematical and physical
sciences; no. 246.
OB362.M3N37 1987
521' .1--dc19 88-8212
CIP
ISBN-13: 978-94-010-7873-3 e-ISBN-13: 978-94-009-3053-7
001: 10.1007/978-94-009-3053-7
Published by Kluwer Academic Publishers,
P.O. Box 17, 3300 AA Dordrecht, The Netherlands.
Kluwer Academic Publishers incorporates the publishing programmes of
D. Reidel, Martinus Nijhoff, Dr W. Junk, and MTP Press.
Sold and distributed in the U.SA and Canada
by Kluwer Academic Publishers,
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In all other countries, sold and distributed
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All Rights Reserved
© 1988 by Kluwer Academic Publishers.
Softcover reprint of the hardcover 1s t edition 1988
No part of the material protected by this copyright notice may be reproduced or utilized
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TABLE OF CONTENTS
Preface
l.X
Group Photograph xii
List of Speakers and Participants xiii
PART I: PLANETARY DYNAMICS
B.A. CONWAY and T.J. ELSNER / Dynamical Evolution of 3
Planetary Systems and the Significance of
Bode sLaw
I
B.A. CONWAY and R.E.ZELENKA / Further Numerical Investigations 13
into the Significance of Bode's Law
K.A. INNANEN and S. MIKKOLA / Where are the Saturnian Trojans? 21
J. KOVALEVSKY / Orbital Evolution 27
P.J. MESSAGE / Planetary Perturbation Theory from Lie Series, 47
including Resonance and Critical Arguments
A. MILANI I Secular Perturbations of Planetary Orbits and 73
their Representation as Series
A.M. NOBILl/The Accumulation of Integration Error 109
D. E. SNOW / Numerical Regularization of the Lagrange 117
Planetary Equations
PART II: DYNAMICS OF NATURAL AND ARTIFICIAL SATELLITES
K. AKSNES I General Formulas for Three-Body Resonances 125
M. ARRIBAS and M.L. SEIN-ECHALUCE / Analytical Methods for 141
the Radial Intermediaries
E. BOIS / Analytical Theory of the Rotation of an 149
Artificial Satellite
R. BROUGKE and A. KONOPLIV / Some Models for t1-)e Hot ion of 155
the Co-orbital Satellites of Saturn
M. CARPINO / Long-period Changes in the Semimajor Axis 171
of Lageos
VI
PART II: DYNAMICS OF NATURAL AND ARTIFICIAL SATELLITES
F. DELHAISE and M. MOONS / Effects of a Non-circular Shepherd 173
upon a Planetary Ring
J.M. FERRANDIZ and M.E. SANSATURIO I General Motion of a 181
Triaxial Rigid Body in a Newtonian Force Field
D. HARPER / The Motion of the Orbit Plane of Iapetus 189
D.A. KAYA, L.A. McCARTER and J.J. F.LIU / Astrodynamics in 193
the Real-World Environment (The Whole Truth and
Nothing but the Truth).
A.E. ROY and B.A. STEVES / A Finite-Time Stability Criterion 197
for Sun-perturbed Planetary Satellites
M.L. SEIN-ECHALUCE and J.M. FRANCO / A New Radial Inter 217
mediary and its Numerical Integration
J. WALDVOGEL and F. SPIRIG / Co-orbital Satellites and 223
Hill's Lunar Problem
PART III: DYNAMICS OF ASTEROIDS
P. FARINELLA, CL. FROESCHLE and Z. KNE~EVIC / The Puzzle 237
of Asteroid Families
S. FERRAZ-MELLO / High Eccentricity Libration 245
CH. FROESCHLE and H. SCHOLL / Evolution of Asteroidal 251
Orbits Located in the Main Secular Resonances
vS,v6 and v16
J.D. HADJIDEMETRIOU I Algebraic Mappings Near a Resonance 257
with an Application to Asteroid Motion
J. POJMAN and V. SZEBEHELY I The Hierarchical Restricted 277
Problem with Applications
M. YOSHIKAWA I The Motions of Asteroids in the Secular 2M
Resonance v6 : an Analytical Model and Numerical
Calculations
PART IV: DYNAMICS OF STARS AND GALAXIES
N. CARANICOLAS and CH. VOZIKIS I Families of Periodic Orbits 297
in a Model of Interacting Galaxies
G. CONTOPOULOS / Qualitative Characteristics of Dynamical 301
Systems
F.M.F. EL-SABAA and H.H. SHERIEF / On the Galactic Motion 317
D.C. HEGGIE / The N-Body Problem in Stellar Dynamics 329
B. ZAFIROPOULOS / Rotationally and Tidally Perturbed Orbital 349
Elements of Close Binary Stars
vii
PART V: OTHER SPECIFIC TOPICS IN DYNAMICS
A. BENSENY and A. OLVERA I Eseape and Capture in the 357
Restricted Three Body Problem (RTBP)
J.G. BRYANT I A Formulation of the N-body Problem where the 363
Ve loci ties are Bounded
A. CELLETTI and L. CHIERCHIA / A Report on Some New KAM 371
Estimates
J. M. FERRANDIZ / Extended Canonical Trans formations 377
Increasing the Number of Variables
J. FONT and M. GRAU / Non Local Spiraling Characteristic 385
Curves
C. FROESCHLE and J.L. GAUTERO /Connectance and Stability 393
of Linear Differential Systems
J. HENRARD I Resonances in the Planar Elliptic Restricted 405
Problem
C. MARCHAL I The Near-Resonanee Theorem: Analysis of the 427
Vicinity of Periodic Solutions of Analytic
Differential Systems
J. MASDEMONT I Homoclinic and Heteroclinic Solutions of 449
the RTBP Joining the Triangular Equi librium Points
M. OLLE / Double Collision Orbits and Second Species 457
Solutions in the Restricted Three-Body Problem
A.D. PINOTSIS I Bifurcations and Instabilities in the 465
Restricted Three-Body Problem
F. PUEL I Three Dimensional Equations of Szebehely of the 471
Inverse Problem and Frenet Reference Frame
J.F. SEGREST, J.J.F. LIU and V.G. SZEBEHELY I Fourier 477
Transform in Astrodynamics
C. SIMO I Estimates of the Error in Normal Forms of 481
Hamiltonian Systems. Applications to Effective
Stability and Examples
A. SUSIN I Passages Near Triple Collision 505
H. YOSHIDA I Non-Integrability of Hamiltonian Systems Based 513
on Zig lin s Theorem
I
Index of Names 515
Index of Subjects 521
PREFACE
The reader will find in this volume the Proceedings of the NATO
Advanced Study Institute held in Cortina d'Ampezzo, Italy between
August 3 and August 13, 1987 under the title "Long Term Dynamical
Behaviour of Natural and Artificial N-body Systems".
The Institute was the latest in a series held in 1972, 1975, 1978,
1981, 1984 in dynamical astronomy, theoretical mechanics and celestial
mechanics under the Directorship of Professor Victor Szebehely. These
previous institutes, held in high esteem by the international community
of research workers, have resulted in a series of well-received and
valuable Proceedings.
In correspondence with Professor Szebehely and in long discussions
with him in Colorado in August 1985, I agreed to his request that I
undertake the preparation of a new ASI. I was happy to do so knowing
I could call upon his vast experience in overseeing such ASI's.
The last quarter century has been a period in which increasingly
rapid progress has been made in celestial mechanics and related subjects
not only because of the appearance of new problems urgently requiring
solution but also because of the advent of new analytical techniques
and powerful computer hardware and software. The present Institute's
subject, the long-term dynamical behaviour of natural and artificial
n-body systems, was shown to be relevant to (i) the older problems of
the age and stability of the solar system and its sub-systems of planets,
satellites and asteroids, (ii) the dynamics and evolution of planetary
rings, (iii) the long-term prediction of the behaviour of artificial
satellites, (iv) the dynamics of star clusters and galaxies.
Apart from the series of lectures given by the invited speakers,
some 38 presentations on various aspects of the subject were made.
One aspect emphasised by a number of contributions was the impor
tance in the stability of n-body dynamical systems of their hierarchical
structure. The progress made in recent years in our understanding of
the restricted and general three body problem was also the subject of
many papers.
ix
x
The successful use of the present generation of computers to
carry out massive programmes of research into the evolution of the
outer solar system over a period of 108 years was described as well
as their use in allowing the behaviour of cluster and galactic models
containing large numbers of stars to be studied. In both fields of
computer application, it was emphasised that the efficient use of
the computer depended in a highly important way on a careful prelim
inary selection of the correct model to be integrated, the choosing
of suitable analytical and numerical techniques together with the
selection of an appropriate data-processing method. A difference of
orders of magnitude in computer time, effective capacity, accuracy
of results, numbers of bodies processed, real time interval studied
could be gained by such careful preliminary studies.
The importance of chaotic behaviour in dynamical systems was
the subject of a number of presentations, particularly with respect
to the asteroid belt and the stability of planetary and satellite
orbits.
Algebraic mapping techniques in the study of n-body dynamical
systems, and the study of periodic solutions of analytic Hamiltonian
systems with many degrees of freedom were two other approaches shown
to be of importance in predicting the long-term behaviour of such
dynamical systems.
The wide variety of problems discussed, matched by the number of
new techniques, shows clearly that the fields of study embraced by
celestial mechanics, theoretical mechanics and dynamical astronomy
are in a most healthy state of activity, a view supported by the many
researchers at work in these fields.
The high standard of preparation and presentation of invited
lectures and other contributions were particularly pleasing: as
pleasing were the frequent and sustained discussions and the warm
international friendship enjoyed by the participants.
For myself I pay grateful tribute for the help and support I
received from the Organising Committee (Professor V. Szbehely,
Dr. P. J. Message, Dr. A. Milani). I also thank Dr. G. Volpi and
Ms. B. A. Steves for their unstinting efforts before and during the
meeting. As always the staff of the Antonelli Institute, where the
ASI was held, made our stay a very pleasant one and we extend our
thanks to them.
We are also very grateful to the Scientific Affairs Division
of NATO for their guidance, counsel and support.
Finally I would like to take this opportunity to thank all
those who came to Cortina and helped to make this ASI so enjoyable
and productive. Their support and cooperation was very much
appreciated.
Cortina d'Ampezzo, Italy Archie E. Roy,
and Glasgow, United Director, NATO Advanced
Kingdom. Study Institute and Editor
of the Proceedings.