Predictability, Stability, and Chaos in N-Body Dynamical Systems

580 Pages·1991·16.34 MB·English

by  J. Casasayas

Pred ictabi lity, Stabi lity,
and Chaos in N-Body
Dynamical Systems
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Predictability, Stability,
and Chaos in N-Body
Dynamical Systems
Edited by
Archie E. Roy
University of Glasgow
Glasgow, United Kingdom
Plenum Press
New York and london
Published in cooperation with NATO Scientific Affairs Division
Proceedings of a NATO Advanced Study Institute on
Predictability, Stability, and Chaos in N·Body Dynamical Systems,
held August 6-17, 1990,
in Cortina d' Ampezzo, Italy
LIbrary of Congress CatalogIng-In-PublIcatIon Data
NATO Advanced Study Institute on Predictability. Stabllity. and Chaos
in N-Body Dyna~ical Systems (1990 : Cortina d'Aapezzo. Italy)
Predictability. stability. and chaos in N-body dynamical systems I
edited by Archie E. Roy.
p. cm, -- (NATO ASI series. Series B. Physics; vol. 272)
"Proceedings of a NATO Advanced Study Institute on Predictability.
Stability. and Chaos in N-Body Dyna.ical Systems. held August 6-17.
1990. in Cortina d'Aapezzo. Italy"--T.p. verse.
"Published in cooperation with NATO Scientific Affairs Division."
Includes bibliographical references and indexes.
ISBN-13: 978-1-4684-5999-9 e-ISBN-13: 978-1-4684-5997-5
001: 10.1007/978-1-4684-5997-5
1. Many-body problea--Congresses. 2. Astronoay--Congresses.
1. Roy. A. E. (Archie E.). 1924- n. North Atlantic Treaty
Organization. Scientific Affairs Division. III. Title.
IV. Series: NATO ASI series. Series B. Physics; v. 272-.
CB362.M3N38 1990
521--dc20 91-28278
CIP
ISBN-13: 978-1-4684-5999-9
© 1991 Plenum Press, New York
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PREFACE
The reader will find in this volume the Proceedings of the NATO
Advanced Study Institute held in Cortina d'Ampezzo, Italy between August
6 and August 17, 1990 under the title "Predictability, Stability, and
Chaos in N-Body Dynamical Systems".
The Institute was the latest in a series held at three-yearly inter
vals from 1972 to 1987 in dynamical astronomy, theoretical mechanics and
celestial mechanics. These previous institutes, held in high esteem by
the international community of research workers, have resulted in a series
of well-received Proceedings. The 1990 Institute attracted 74 participants
from 16 countries, six outside the NATO group. Fifteen series of lectures
were given by invited speakers; additionally some 40 valuable presentations
were made by the younger participants, most of which are included in these
Proceedings.
The last twenty years in particular has been a time of increasingly
rapid progress in tackling long-standing and also newly-arising problems
in dynamics of N-body systems, point-mass and non-point-mass, a rate of
progress achieved because of correspondingly rapid developments of new
computer hardware and software together with the advent of new analytical
techniques. It was a time of exciting progress culminating in the ability
to carry out research programmes into the evolution of the outer Solar
System over periods of more than 108 years and to study star cluster and
galactic models in unprecedented detail. Notwithstanding these successes,
however, in recent years it has become increasingly acknowledged that the
influence of chaos in dynamical systems is of supreme importance in under
standing their behaviour.
The present Institute therefore had as its main goal the examination
of the relationships between predictability, stability and chaos in N-body
dynamical systems. Among the systems studied were the Solar System's
planets and satellites, comets, asteroids, and meteors, and the dynamics
of star clusters and galaxies.
In real dynamical systems, imprecision of starting conditions and
the inevitable accumulation of round-off error conspire with the non
linearity of the differential equations to produce limits on the predict
ability of the future behaviour of such systems. The 'predictability
horizon' depends in a hypersensitive manner on the proximity of the system
to chaotic regions.
In the few-body problem, as in sub-systems within the Solar System,
it was recognised that resonance also played a major part in limiting
predictability in, for example, the evolution of the orbits of asteroids
disturbed by Jupiter and Saturn, where such asteroids evolved to become
Mars- or Earth-crossers, becoming essentially chaotic systems.
vii
In models of star clusters, it was evident that in spite of the
powerful computer power now available, the predictability horizon for
individual stars was surprisingly limited though a number of properties
of the cluster's evolution could be explored for longer periods of time
that were surprisingly independent of the precision aimed at for the
individual stars.
The Institute concluded that for periods of time less than clearly
identified predictability horizons, modern analytical methods, together
with computer hardware and software, could produce results accurate
enough for all practical purposes. For many of the most interesting and
important problems, however, such as the long-term evolution of the Solar
System and of star clusters, our modern techniques have demonstrated their
inadequacy in the presence of chaos, necessitating a search for new
approaches. Dynamics, in short, has recognised its arrival at, hopefully,
the beginning of a new era in its history, one perhaps leading to new and
fruitful analytical techniques. Ironically, the modern concept of chaos
was found by Poincare in his study of the three-body problem almost 100
years ago; his brilliant farsightedness and insight were highlighted by
the number of times participants had occasion to refer to his work in
this Institute. He himself predicted that a long time would pass before
this kind of concept would be relevant for actual computations to be used
in dynamical astronomy. Quoting Andrea Milani, in this volume: "Whether
we like it or not, this time is now, and therefore the basic tools needed
to build a theory of chaotic dynamics •.. are now becoming essential tools
of our trade."
Finally, let me record my thanks to my colleagues.
The high quality of preparation and presentation of invited lectures
and other contributions was particularly pleasing. As pleasing were the
frequent and sustained discussions and the warm international friendship
enjoyed by the participants.
I am also glad to have this opportunity to pay grateful tribute for
the help and support I received from the Organising Committee (Professor
V. Szebehely, Dr. P.J. Message, Professor A. Milani). I also want to thank
Dr. G. Volpi and Dr. B.A. Steves for their unstinting efforts before and
during the meeting. The staff of the Antonelli Institute, where the ASI
was held, were always more than helpful: they made our stay a very
pleasant one and we extend our thanks to them.
The Committee are also very grateful to the Scientific Affairs
Division of NATO for their guidance, counsel and support.
Lastly, 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, cooperation and friendship were very much appreciated.
Cortina D'Ampezzo, Italy Archie E. Roy
and Glasgow, United Kingdom Director, NATO Advanced Study
Institute and Editor of the
Proceedings
viii
CONTENTS
PART I: ASPECTS OF CHAOS
Chaos in a Restricted, Charged Four-Body Problem . 3
J. Casasayas and A. Nunes
Chaos in the Three-Body Problem 11
A. Milani
A New Route to Chaos: Generation of Spiral Characteristics. . • •• 35
G. Contopoulos
Chaos in the N-Body Problem of Stellar Dynamics • . • • . • . . • • • 47
D.C. Heggie
Chaos, Stability, and Predictability in Newtonian Dynamics .•..•. 63
V. Szebehely
Predictability, Stability, and Chaos in Dynamical Systems . . • . •• 73
C. Marchal
Analytical Framework in Poincare Variables for the Motion
of the Solar System • • • • . • • • • • • • . • . • 93
J. Laskar
Origin of Chaos and Orbital Behaviour in Slowly Rotating
Triaxial Models . • • . • . • . • • • • • • • . • . • • • • • • 115
S. Udry and L. Martinet
PART II: DYNAMICS OF ASTEROIDS, COMETS, AND METEORS
Modelling: An Aim and a Tool for the Study of the Chaotic
Behaviour of Asteroidal and Cometary Orbits . • . • . • • • • • 125
C. Froeschle
Mapping Models for Hamiltonian Systems with Application
to Resonant Asteroid Motion . • • • • • • • . • . • • • • • • • 157
J.D. Hadjidemetriou
A Model for the Study of Very-High-Eccentricity Asteroidal Motion:
The 3:1 Resonance .••.•.•..•••..•..••.••• 177
S. Ferraz-Mello and J.C. Klafke
The Location of Secular Resonances • . . . • • • . . • . . • • • • • • 185
A. Morbidelli
ix
Temporary Capture into Resonance • • • • • • • • • • • • • • • • • • • 193
J. Henrard
Applications of the Restricted Many-Body Problem to
Binary Asteroids • • • • • • • • • • • • • ••••••••• 197
V. Szebehely and J.R. Pojman
The Wavelet Transform as Clustering Tool for the Determination
of Asteroid Families • • • • • • • • •• ••••••••••• 205
Ph. Bendjoya, E. Slezak, and CI. Froeschle
Delivery of Meteorites from the v6 Secular Resonance
Region Near 2 AU • • • • • • • • • • • • • • • • • • • • • • 215
Ch. Froeschle and H. Scholl
The Dynamics of Meteoroid Streams • • • • • • • • • • • • • • • • ••• 225
I.P. Williams and Z. Wu
Perturbation Theory, Resonance, Librations, Chaos,
and Halley's Comet • • • • • • • • • • • • • • • • • • • • • • • 239
P.J. Message
Rotational Behaviour of Comet Nuclei • • • • • • • • • • • • • • • • • 249
P. Oberti, E. Bois, and C. Froeschle
PART III: DYNAMICS OF NATURAL AND ARTIFICIAL SATELLITES
The Moon's Physical Librations - Part 1: Direct Gravitational
Perturbations • • • 257
I. Wytrzyszczak and E. Bois
The Moon's Physical Librations - Part II: Non-Rigid Moon and
Direct Non-Gravitational Perturbations • • • • • • • • • • • • • 265
E. Bois and I. Wytrzyszczak
Significant High Number Commensurabilities in the Main Lunar
Problem: A Postscript to a Discovery of the Ancient
Chaldeans • • • • • • • • • • • •• • • • • • • 273
A.E. Roy, B.A. Steves, G.B. Valsecchi, and E. Perozzi
Moon's Influence on the Transfer from the Earth to a Halo
Orbit ~ound Ll • • • • • • • • • • • • • • • • • • • •• 283
G. Gomez, A. Jorba, J. Masdemont, and C. Sima
First Order Theory of Perturbed Circular Motion: An Application
to Artificial Satellites • • • • • • • • • • •• 291
E. Bois and I. Wytrzyszczak
Poincare-Similar Variables Including J2-Secular Effects • • • • • •• 297
L. Flor!a and J.M. Ferr~ndiz
Measuring the Lack of Integrability of the J2 Problem
for Earth's Satellites • • • • • • ••••••••••••• 305
C. Sima
The Effects of the J3-Harmonic (Pear Shape) on the Orbits of
a Satellite • • • • • • • • • • • • • • • • • • • • 311
R.A. Broucke
x
Stability of Satellites in Spin-Orbit Resonances and
Capture Probabilities . • • • • • • • . . . . • 337
A. Celletti
Statistical Analysis of the Effects of Close Encounters of
Particles in Planetary Rings • • • • • • . • • • • • • • . 345
F.P. Gama and J.-M. Petit
The Three-Dipole Problem • • • • • • • • • . • • • • • • • • • • • • • 355
C.L. Goudas and E.G. Petsagourakis
The N-Dipole Problem and the Rings of Saturn . • • • • • • . • • • • • 371
C.L. Goudas
Long-Time Predictions of Satellite Orbits by Numerical Integration • • 387
J.M. Ferrandiz, M.E. Sansaturio, and J. Vigo
Chaos in Coorbital Motion • • 395
F. Spirig and J. Waldvogel
PART IV: THE THREE-BODY PROBLEM
Remarkable Termination Orbits of the Restricted Problem 413
V• V. Markellos
Periodic Orbits in the Isosceles Three-Body Problem • • • • • • • • • 425
C.C. Monleon and J.M. Alfaro
Quasi-Periodic Orbits as a Substitute of Libration Points
in the Solar System • . • • • • • • • • 433
A.
G. Gomez, Jorba, J. Masdemont, and C. Simo
Stability Zones Around the Triangular Lagrangian Points . • • • . •• 439
R. Dvorak and E. Lohinger
Chaotic Trajectories in the Restricted Problem of Three Bodies • • • • 447
R.H. Smith and V. Szebehely
New Formulations of the Sitnikov Problem • • • • • • • • • • • • • • • 457
K. Wodnar
Periodic Solutions for the Elliptic Planar Restricted Three-Body
Problem: A Variational Approach . . . • • • . • • • • • • • • • 467
M.L. Bertotti
Hill-Type Stability and Hierarchical Stability of the
General Three-Body Problem . • • . • . . . . • • . • • • • • • • 475
Y.-C. Ge
Equilibrium Connections on the Triple Collision Manifold • • • • . • • 481
A. Susfn and C. Sima
Orbits Asymptotic to the Outermost KAM in the Restricted
Three-Body Problem • • . • • • • • • . • • • . • • 493
M. Sekiguchi and K. Tanikawa
xi