New Developments in the Dynamics of Planetary Systems: Proceedings of the Fifth Alexander von Humboldt Colloquium on Celestial Mechanics held in Badhofgastein (Austria), 19–25 March 2000

375 Pages·2001·8.96 MB·English

by  William H. Jefferys (auth.)

NEW DEVELOPMENTS IN THE DYNAMICS
OF PLANETARY SYSTEMS
New Developments in the Dynamics
of Planetary Systems
Proceedings of the Fifth Alexander von Humboldt Colloquium
on Celestial Mechanics
held in Badhofgastein (Austria), 19-25 March 2000
Edited by
RUDOLF DVORAK
Unversity of Vienna, Austria
and
JACQUES HENRARD
University of Namur (FUNDP), Belgium
Partly reprinted from Celestial Mechananics and Dynamical Astronomy
Volume 78, Nos. 1--4 (2000)
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
A c.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-90-481-5702-0 ISBN 978-94-017-2414-2 (eBook)
DOI 10.1007/978-94-017-2414-2
Printed an acid-free paper
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© 2001 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 2001
No part of the material protected by this copyright notice may be reproduced or
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Cover figure: Alexander von Humboldt, oiI painting
by Friedrich Georg Weitsch (1806) in the Staatliche Museen Zu Berlin
(Courtesy of the Staatliche Museen Zu Berlin)
NEW DEVELOPMENTS IN THE DYNAMICS OF PLANETARY SYSTEMS
Proceedings of the Fifth Alexander von Humboldt
Colloquium on Celestial Mechanics
Preface 1
W. H. JEFFERYS I Statistics for Twenty-First Century Astrometry 3-16
Z. KNEZEVIC and A. MILANI I Synthetic Proper Elements for Outer
Main Belt Asteroids 17--46
U. LOCATELLI and A. GIORGILLI I Invariant Tori in the Secular Motions of
the Three-Body Planetary Systems 47-74
J. CHAPRONT I Improvements of Planetary Theories Over 6000 Years 75-82
G. B. VALSECCHI, A. MILANI, G. F. GRONCHI and S. R. CHESLEY I
The Distribution of Energy Perturbations at Planetary Close En
counters 83-91
P. MICHEL and Ch. FROESCHLE I Dynamics of Small Earth-Approachers on
Low-Eccentricity Orbits and Implications for their Origins 93-112
Z. SANDOR, B. ERDI and C. EFTHYMIOPOULOS I The Phase Space
Structure Around L4 in the Restricted Three-Body Problem 113-123
R. DVORAK and K. TSIGANIS I Why Do Trojan ASCS (not) Escape? 125-136
J. HADJIDEMETRIOU and G. VOYATZIS /The 2/1 and 3/2 Resonant Asteroid
Motion: A Symplectic Mapping Approach 137-150
G. STAGIKA and S. ICHTIAROGLOU I Twist and Non-Twist Bifurcations in
a System of Coupled Oscillators 151-160
E. MELETLIDOU I The Mel'nikov Subharmonic Function and the Non-
Existence of Analytic Integrals in Non-Autonomous Systems 161-166
C. FROESCHLE and E. LEGA I On the Structure of Symplectic Mappings.
The Fast Lyapunov Indicator: A Very Sensitive Tool 167-195
G. CONTOPOULOS, M. HARSOULA and N. VOGLIS I Crossing of Various
Cantori 197-210
F. FREIS TETTER I Fractal Dimensions as Chaos Indicators 211-225
A. CELLETTI and C. FALCOLINI I Normal Form Invariants Around Spin-
Orbit Periodic Orbits 227-241
G. CONTOPOULOS, C. EFTHYMIOPOULOS and N. VOGLIS I The Third
Integral in a Self-Consistent Galactic Model 243-263
N. VOGLIS, M. HARSOULA and Ch. EFTHYMIOPOULOS/ Counterrotating
Galaxies and Memory of Cosmological Initial Conditions 265-278
C. MARCHAL I The Family P12 of the Three-Body Problem-The Simplest
Family of Periodic Orbits, with Twelve Symmetries Per Period 279-298
A. E. ROY and B. A. STEVES I The Caledonian Symmetrical Double
Binary Four-Body Problem 1: Surfaces of Zero-Velocity using the
Energy Integral 299-318
N. A. SOLOVAYA and E. M. PITTICH I Application of the Nonrestricted
Three-bodies Problem to the Stellar System~ UMA 319-324
T. MICHTCHENKO and S. FERRAZ-MELLO I Periodic Solutions of
the Planetary 5:2 Resonance Three-body Problem 325-328
E. PILAT-LOHINGER I Limits of Stability for Planets in Double Star
Systems Using the Fast Lyapunov Indicators 329-335
K. TSIGANIS, A. ANASTASIADIS and H. VARVOGLIS I On a Fokker-
Planck Approach to Asteroidal Transport 337-340
J. F. NAVARRO and JACQUES HENRARD I Fractality in a Galactic
Model 341-347
H. SMITH, JR I Transformation Methods fo Trigonometric Parallaxes 349-359
P. K. SEIDELMANN I The Search for Exosolar systems 361-362
H. VARVOGLIS, CH. VOZIKIS, K. WODNAR and E. DIMITRIADOU
I The Two Fixed Centers Problem Revisited 363-366
J. KALLRATH and R. DVORAK I The Phase Space Structure of the
General Sitnikov Problem 367-379
PREFACE
It is now a well-established tradition that every four years, at the end of winter, a
group of 'celestial mechanicians' from all over the world gather in the Austrian
Alps at the invitation of R. Dvorak.
This time the colloquium was held at Badhofgastein from March 19 to March
25, 2000 and was devoted to the 'New Developments in the Dynamics of Planetary
Systems'. The papers covered a large range of questions of current interest: the-
oretical questions (resonances, KAM theory, transport, ... ) and questions about
numerical tools (synthetic elements, indicators of chaos, ... ) were particularly well
represented; of course planetary theories and Near Earth Objects were also quite
popular.
Three special lectures were delivered in honor of deceased colleagues whom,
to our dismay, we will no longer meet at the 'Austrian Colloquia'. W. Jefferys
delivered the Heinrich Eichhorn lecture on 'Statistics for the Twenty-first Century
Astrometry', a topic on which Heinrich Eichhorn was a specialist. A. Roy delivered
a lecture honoring Victor Szehebely on 'Lifting the Darkness: Science in the Third
Millenium', in which in wove anecdotes and remembrances of Victor which moved
the audience very much. A. Lemaitre spoke in honor of Michele Moons on 'Mech
anism of Capture in External Resonance'. The end of her talk was devoted to a
short and moving biography of Michele illustrated by many slides.
You will find in the following pages a pot-pourri of what we listened to; you
will miss of course the charm of the little mountain village, Badhofgastein, the
pleasant and always friendly Austrian welcome, and the warm atmosphere of the
shared meals and the long evening discussions. On one evening we had a classical
concert given by the 'Vienna Mozart-Trio International' playing Schumann and
Schubert (the piano was provided by the Musikhaus Pilat from Leoben).
To be able to organize this 'Fifth Alexander von Humbolt Colloquium', we have
to thank primarily the Austrian ministry of science, the 'Osterreichische Lotterien',
the Osterreichische Forschungsgemeinschaft, the 'Osterreichische National Bank'
and especially the Bank-Austria. The latter supported our meeting also in the
form of 'Tagungsunterlagen'; coffee during the whole meeting time was offered
by Hornig-Kaffee Graz.
Many thanks are due to G. Contopoulos, B. Erdi, S. Ferraz-Mello, J. Hadjidemet
riou, W. Jefferys, P. K. Seidelmann. They helped us in the editorial task of arranging
for competent and fast refereeing so that the papers could be reviewed and, when
necessary, corrected.
R. Dvorak J. Henrard
Celestial Mechanics and Dynamical Astronomy 18: I, 2000.
© 2001 Kluwer Academic Publishers.
STATISTICS FOR TWENTY-FIRST CENTURY ASTROMETRY
(2000 Heinrich K. Eichhorn Memorial Lecture)
WILLIAM H. JEFFERYS
University of Texas at Austin. Austin, TX USA, e-mail: [email protected]
Abstract. H. K. Eichhorn had a lively interest in statistics during his entire scientific career, and
made a number of significant contributions to the statistical treatment of astrometric problems. In the
past decade, a strong movement has taken place for the reintroduction of Bayesian methods of stat
istics into astronomy, driven by new understandings of the power of these methods as well as by the
adoption of computationally-intensive simulation methods to the practical solution of Bayesian prob
lems. In this paper I will discuss how Bayesian methods may be applied to the statistical discussion
of astrometric data, with special reference to several problems that were of interest to Eichhorn.
Key words: Eichhorn, astrometry, Bayesian statistics
1. Introduction
Bayesian methods offer many advantages for astronomical research and have at
tracted much recent interest. The Astronomy and Astrophysics Abstracts web
site (http://adsabs.harvard.edu/) lists 117 articles with the keywords 'Bayes' or
'Bayesian' in the past 5 years, and the number is increasing rapidly (there were 33
articles in 1999 alone). At the June, 1999 meeting of the American Astronomical
Society, held in Chicago, there was a special session on Bayesian and Related
Likelihood Techniques. Another session at the June, 2000 meeting also featured
Bayesian methods. A good introduction to Bayesian methods in astronomy can be
found in Loredo ( 1990).
Bayesian methods have many advantages over frequentist methods, including
the following: it is simple to incorporate prior physical or statistical information
into the analysis; the results depend only on what has actually been observed and
not on observations that might have been made but were not; it is straightforward
to compare models and average over both nested and unnested models; and the
interpretation of the results is very natural, especially for physical scientists.
Bayesian inference is a systematic way of approaching statistical problems,
rather than a collection of ad hoc techniques. Very complex problems (difficult or
impossible to handle classically) are straightforwardly analyzed within a Bayesian
framework. Bayesian analysis is coherent: we will not find ourselves in a situation
where the analysis tells us that two contradictory things are simultaneously likely
to be true. With proposed astrometric missions (e.g., FAME) where the signal can
be very weak, analyses based on normal approximations may not be adequate. In
..._. Celestial Mechanics and Dynamical Astronomy 78: 3-16, 2000.
'' © 2001 Kluwer Academic Publishers.
4 WILLIAM H. JEFFERYS
such situations, Bayesian analysis that explicitly assumes the Poisson nature of the
data may be a better choice than a normal approximation.
2. Outline of Bayesian Procedure
In a nutshell, Bayesian analysis entails the following systematic steps:
(1) Choose prior distributions ('priors') that reflect your knowledge about each
parameter and model prior to looking at the data.
(2) Determine the likelihood function of the data under each model and parameter
value.
(3) Compute and normalize the full posterior distribution, conditioned on the data,
using Bayes' theorem.
(4) Derive summaries of quantities of interest from the full posterior distribution
by integrating over the posterior distribution to produce marginal distributions
or integrals of interest (e.g., means, variances).
2.1. PRIORS
The first ingredient of the Bayesian recipe is the prior distribution. Eichhorn was
acutely aware of the need to use all available information when reducing data, and
often criticized the common practice of throwing away useful information either
explicitly or by the use of suboptimal procedures. The Bayesian way of preventing
this is to use priors properly. The investigator is required to provide all relevant
prior information that he has before proceeding with the analysis. Moreover, there
is always prior information. For example, we cannot count a negative number of
photons, so in photon-counting situations that may be presumed as known. Paral
laxes are greater than zero. We now know that the most likely value of the Hubble
constant is in the ballpark of 60-80 km/s/mpc, with smaller probabilities of its
being higher or lower. Prior information can be statistical in nature, for example,
we may have statistical knowledge about the spatial or velocity distribution of stars,
or the variation in a telescope's plate scale.
e
In Bayesian analysis, our knowledge about a parameter is encoded by a prior
probability distribution on the parameter, for example, p(O I B), where B is back
ground information. Where prior information is vague or uninformative, a vague
prior generally recovers results similar to a classical analysis. However, in model
selection and model averaging situations, Bayesian analysis usually gives quite
different results, being more conservative about introducing new parameters than
is typical of frequentist approaches.
Sensitive dependence of the result on reasonable variations in prior information
should be tested, and if present indicates that no analysis, Bayesian or other, can
give reliable results. Since frequentist analyses do not use priors and therefore are
STATISTICS FOR ASTROMETRY 5
incapable of sounding such a warning, this can be considered a strength of the
Bayesian approach.
The problem of prior information of a statistical or probabilistic nature was ad
dressed in a classical framework by Eichhorn ( 1978) and by Eichhorn and Standish
(1981). They considered adjusting astrometric data given prior knowledge about
some of the parameters in the problem, for example, that the plate scale values
only varied within a certain dispersion. For the cases studied in these papers (mul
tivariate normal distributions), the result is similar to the Bayesian one, although
the interpretation is different.
In another example, Eichhorn and Smith (1996) studied the Lutz-Kelker bias.
The classical way to understand the Lutz-Kelker bias is that it is more likely that
we have observed a star slightly farther away with a negative error that brings it
closer in to the observed distance, than that we have observed a slightly nearer star
with a positive error that pushes it out to the observed distance, because the number
of stars increases with increasing distance. The Bayesian notes that it is more likely
a priori that a star of unknown distance is farther away than that it is nearer, which
dictates the use of a priori that increases with distance. The mathematical analysis
gives a similar result, but the Bayesian approach, by demanding at the outset that
we think about prior information, inevitably leads us to consider this phenomenon,
which classical astrometrists missed for a century.
2.2. THE LIKELIHOOD FUNCTION
The likelihood function C is the second ingredient in the Bayesian recipe. It de
scribes the statistical properties of the mathematical model of our problem. It tells
us how the statistics of the observations (e.g., normal or Poisson data) are related
to the parameters and to any background information. It is proportional to the
sampling distribution for observing the data Y, given the parameters, but we are
interested in its functional dependence on the parameters:
C(e; Y, B) ex p(Y I e, B).
The likelihood is known up to a constant but arbitrary factor which cancels out
in the analysis.
Like Bayesian estimation, maximum likelihood estimation (upon which Eich
horn based many of his papers) is also developed by using the likelihood function.
This is good, because the likelihood function is always a sufficient statistic for the
parameters of the problem. Furthermore, according to the important 'Likelihood
Principle' (Berger, 1985), it can be shown that under very general and natural
conditions, the likelihood function contains all the information in the data that can
be used for inference. However, the likelihood is not the whole story. Maximum
likelihood by itself does not take prior information into account, and it fails badly
in some notorious situations, like errors-in-variables problems (i.e., both x and y
have error), when the variance of the observations is estimated. Bayesian analysis