The Three-Body Problem

583 Pages·1990·8.485 MB·English

by  CHRISTIAN MARCHAL (Eds.)

STUDIES IN ASTRONAUTICS
Volume 1 Optimal Space Trajectories (J. P. Marec)
Volume 2 Optimal Trajectories in Atmospheric Flight (Ν. X. Vinh)
Volume 3 Optimal Spacecraft Rotational Maneuvers (J. L. Junkins and
J. D.Turner)
Volume 4 The Three-Body Problem (C. Marchal)
Picture on front cover:
The moon in the earth's shadow. Total eclipse of August 17,1989,3 hours 45 (universal time),
7 minutes before the end of totality.
STUDIES IN ASTRONAUTICS 4
THE THREE-BODY
PROBLEM
CHRISTIAN MARCHAL
Office National d'Etudes et de Recherches Aerospatiales,
Chatillon, France
ELSEVIER
Amsterdam - Oxford - New York -Tokyo 1990
ELSEVIER SCIENCE PUBLISHERS B.V.
Sara Burgerhartstraat25
P.O. Box 211,1000 AE Amsterdam,The Netherlands
Distributors for the United States and Canada:
ELSEVIER SCIENCE PUBLISHING COMPANY INC.
655 Avenue of the Americas
New York, NY 10010, U.S.A.
Library of Congress Cataloglng-ln-PublIcatIon Data
Marchal, Christian.
The three-body problem / Christian Marchal.
p. cm. — (Studies 1n astronautics ; v. 4)
Includes Index.
ISBN 0-444-87440-2
1. Three-body problem. 2. Mechanics, Celestial. I. Title.
II. Series.
QB362.T5M37 1990
521— dc20 90-39071
CIP
ISBN: 0-444-87440-2 (Vol.4)
ISBN: 0-444-41813-X (Series)
© Elsevier Science Publishers B.V, 1990
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ν
FOREWORD
It is a pleasure to present this monograph written about the
problem of three bodies. The king of sciences is known as astrono­
my and the queen as celestial mechanics ; princes and princesses
are the fields of astronomy, astrophysics, cosmogony, stellar
dynamics, observational astronomy, planetary sciences, astrodyna-
mics, lunar theories, etc... And we can ask where is the problem
of three bodies in this beautiful family tree ? Many aspects
of our field are reversible so we might go in two directions
to find the answer and investigate the descendents of the princes
and of the princesses or alternately, we might consider the ances­
try of the kings and queens. We shall find the problem of three
bodies in the group of ancestors since no universe can originate
without three body dynamics. It is of course generally accepted
that the first lO"1'1 seconds (where the value of η is between
100 and oo) is governed by particle physics, which science is
just now maturing to the level of the problem of three bodies.
Nevertheless, this growing process is natural and unavoidable
and once the basic problems of particle physics is solved, our
colleagues will apply the stability and singularity criteria
of the problem of three bodies to explain the origin of the
universe.
Newton complained of headaches when he worked on his three
body problem (the lunar theory) and found to his surprise that
the Sun had greater effect on the Moon's orbit than the Earth.
Poincare showed non-integrability, Lagrange and Euler found equi­
librium solutions, Jacobi and Hill restricted and simplified
the basic model and the list goes on and on with the most famous
names who contributed to dynamics and celestial mechanics in
the history of human culture. New discoveries are still made
today and the problem of three bodies leads dynamics into new
fields. The most significant step for man-kind, the step from
two to three bodies separates predictability from uncertainties,
Kepler from Newton, Laplace's demon from nondeterminism and in
general, artificiality from reality. The trivial problems without
vi
possible applications, which we invent and assign as problems
to our students in our dynamic classes can be easily solved but
offer little pleasure. The subject of this book is alive today
and was alive since the origin of the universe.Supercolliders
and supercomputers do not "solve" this problem but using the
model of the problem of three bodies makes discoveries in the
related area of science.
In my 45 years dedicated to the problem of three bodies I
never had a dull moment, excepting when I investigated other,
easily solvable problems as regrettable deviations. Indeed, my
professor announced the subject of my dissertation saying : "Young
man, go and solve the problem of three bodies". I soon found
out about the problems of non-integrabili ty and in my first
despararation I considered leaving my academic career. What change
my mind and made me to fall in love with the subject was my youth-
full audacity to argue with Poincare's theorem. (After some time
my professor compromised and instructed me to find a force-law
which allowed the change of this problem from non-integrability
to integrability using a special transformation).
Satisfaction in science is not associated with solving a simple
problem but comes from understanding complex situations and enjoy­
ing the small steps made to penetrate the mysteries of nature.
The subject of this book is one of the best in science since
the problem can be simply stated and instead of obtaining its
simple solution we are guaranteed by Poincare's dictum that the
complete and general solution will remain hidden. Indeed, is
there a better challenge for the human intellect than to engage
in an "unsolvable" and at the same time very important problem?
And consequently, is there a better and more satisfying book
to read and to enjoy than Dr. Marchal's ?
Dr Marchal is probably better qualified to write this book
than anyone else in our field since he combines high level of
mathematical background with intense familiarity of the littera-
ture of dynamics and celestial mechanics. He is the author of
many papers of fundamental importance in our field and is unques­
tionably one of the scientific descendants of Poincare. His rich
cultural and scientific background is clearly visible and most
enjoyable in this book.
Victor SZEBEHELY
vii
DEDICATION
A mon epouse Frangoise, la tete et le coeur de mon
foyer, et au Liban, mon pays natal abandonne par les
autres democraties a l'invasion, la violence, la barbarie.
To my wife and love Frangoise, the head and the heart
of our home, and to Lebanon my native country, forsaked
by the other democracies to invasion, violence, barbary.
viii
ACKNOWLEDGMENTS
Je suis heureux de remercier mes amis de partout qui m'ont
aide a ecrire ce livre, et en particulier mon epouse Frangoise,
le Professeur Nguyen Xuan Vinh de l'Universite du Michigan, le
Professeur Victor Szebehely de l'Universite du Texas, Mesdames
R.M. Burke et Rina van Diemen, et Messieurs A. van der Avoird
et A.J. Oxley d'Elsevier Science Publishers B.V., ainsi que
Mesdames Raban et Josse qui ont tape ces pages sans toujours
les comprendre . . .
lis ont tous ete tres patients !
I am happy to thank my friends from so many countries and
so many centuries who helped me to write this book, and especially
my wife Fransoise, Professor Nguyen Xuan Vinh of Michigan
University, Professor Victor Szebehely of Texas University,
Mrs R.M. Burke, Mrs R. van Diemen, Mr A. van der Avoird, Mr
A.J. Oxley of Elsevier Science Publishers B.V., as well as Mrs
Raban and Mrs Josse who typed these pages without always
understanding them...
They have all been very patient !
Gaarne wil ik mijn dank uitspreken tegenover al die vrienden
uit vele landen, die mij in de diverse perioden van mijn werk
hebben geholpen dit boek te schrijven.
Speciaal bedank ik mijn echtgenote Fransoise, professor Nguyen
Xuan Vinh van de universiteit van Michigan, professor Victor
Szebehely van de universiteit van Texas, mevrouw R.M. Burke en
mevrouw R. van Diemen, de heren A.W. van der Avoird en A.J. Oxley,
alsmede de dames Raban en Josse, die het vele tikwerk voor dit
boek hebben verricht zonder het overigens allemaal te begrijpen..
Zij alien hebben veel geduld betoond !
1
Chapter 1
SUMMARIES
THE THREE-BODY PROBLEM
After a short historical presentation the first chapters recall
the usual formulations of the three-body problem, the main classi­
cal results and the corresponding questions and conjectures.
The theory of perturbations, the analytical approach and the
quantitative analysis of the three-body problem have recently
reached a high degree of perfection and have been described in
several outstanding books such as references 1 to 3. The fantastic
progress of computers has also led to many improvements in the
quantitative analysis and have disclosed the extreme complexity
of the set of solutions. These studies, as presented in the cen­
tral chapters, provided both impetus and new orientations to
the qualitative analysis that is so complementary to the quanti­
tative analysis.
The final chapters describe the remarkable recent progress
of the qualitative analysis and deals with questions such as
stability, instability, escapes, singularities, regularizations,
final evolutions, periodic motions, Arnold tori, asymptotic mo­
tions, oscillatory motions, quasi-collision motions etc...
The recent criteria (or tests) of escape are very efficient and
approach very close to the true limits of escape. As a result
the domain of bounded motions is much smaller than was previously
expected. The stability of a three or η-body system seems brittle
and the small masses seem under a great risk of being expelled.
The Arnold diffusion conjecture and the near resonance theorem
lead to a revision of the notion of stability. The essential
property is no longer an illusory indefinite stability, but rather
the certainty of only small modifications over billions of years.
This seems to be the case of the solar system -except for comets-
provided that outer stars remain very far.
Quasi-collision motions, also called oscillatory motions of
the second kind, are not rare and give a large probability to
2
the formation of novae by the collision of two stars belonging
to a multiple system or, more likely, to the formation of very
close binary stars.
The classical Hill stability appearing in the circular restric­
ted three-body problem can be partly extended to the general
three-body problem.
The remaining open conjectures and the possible further
investigations are discussed in the last chapter.
An entirely new image of the three-body problem is emerging
from the latest progress.
3
LE PROBLEME DES TROIS CORPS
Resume
Apres une courte presentation historique, les premiers chapi-
tres rappellent les formulations usuelles du probleme des trois
corps, les principaux resultats classiques, les grandes questions
et les conjectures correspondantes.
La theorie des perturbations, l'approche analytique et l'ana-
lyse quantitative ont recemment atteint un haut degre de perfec­
tion et ont ete exposees dans plusieurs livres remarquables comme
ceux des references 1 a 3. Les fantastiques progres des ordina-
teurs ont eux aussi conduit a beaucoup d1 ameliorations de l'ana-
lyse quantitative et ont devoile la complexity extreme de 1' en­
semble des solutions. Tout ceci est expose dans les chapitres
centraux et a donne une nouvelle impulsion et de nouvelles orien­
tations a l'analyse qualitative qui est si complementaire de
l'analyse quantitative.
La derniere partie est orientee vers les progres recents et
remarquables de l'analyse qualitative, vers les questions d'allu-
res finales du mouvement, evasions, singularites, regularisations,
mouvements periodiques, tores d'Arnold, mouvements asymptotiques,
bornes, osci1latoires, mouvements de quasi-collisions, stabilite
et instability etc...
Les plus recents tests d'evasion sont tres efficaces, ils
approchent tres pres de la limite d'evasion veritable et le do-
maine des mouvements bornes est beaucoup plus petit qu'on ne
le croyait generalement. La stabilite d'un systeme de trois corps
ou davantage semble fragile et les masses les plus petites ont
souvent un grand risque d'etre rejetees a l'infini.
L'hypothese "de diffusion d'Arnold" et le theoreme de quasi-
resonance conduisent a une revision de la notion de stabilite.
L'essentiel n'est plus une illusoire stabilite indefinie mais
la certitude de modifications restreintes pour des durees de
plusieurs milliards d'annees. Ceci semble £tre le cas du systeme
solaire - hormis les cometes - pourvu qu'aucune etoile ne vienne
le perturber .