Long Term Evolution of Planetary Systems: Proceedings of the Alexander von Humboldt Colloquium on Celestial Mechanics, held in Ramsau, Austria, 13–19 March 1988

419 Pages·1989·53.286 MB·English

by  Andrea Milani, Anna M. Nobili (auth.), R. Dvorak, J. Henrard (eds.)

LONGTERMEVOLUTIONOFPLANErARYSYSTEMS
LONG TERM EVOLUTION
OF
PLANETARY SYSTEMS
ProceedingsoftheAlexandervonHumboldtColloquium on
CelestialMechanics, heldinRamsau,Austria, 13-19March 1988
Editedby
R.DVORAK
InstitutfUrAstronomie,UniversitiaWien,A-I180Wien,Austria
and
J.HENRARD
DepartementdeMathemathique,Facultes UniversitairesdeNamur,
B-5000Namur,Belgium
Reprintedfrom
CelestialMechanics, Vol. 43,Nos.1-4
KLUWER ACADEMIC PUBLISHERS
DORDRECHT / BOSTON / LONDON
15B1\-13:978-94-010-7525-1 e-15B1\-13:978-94-009-2285-3
DOl:10.10071978-94-009-2285-3
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TABLEOF CONTENTS
ProceedingsoftheAlexandervonHumboldtColloquiumonCelestialMechanics
LONG TERM EVOLUTIONOFPLANETARYSYSTEMS
PREFACE ix
SESSIONONPLANETARYDYNAMICS
(Chairman: J.Henrard)
A.MILANI and A.M.NOBIU / IntegrationError over VeryLong Time Spans
c. WILLIAMS I Alternative Representation of Planetary Perturbations (abstract
only) 35
J.LASKAL andJ.L.SIMON I FittingaLine toaSine 37
D.BENEST I Stable PlanetaryOrbits Around One Componentin Nearby Binary
Stars 47
P.K.SEIDELMANNand R.S.HARRINGTON I Planet X- TheCurrentStatus 55
SESSIONONRESONANCEPROBLEM
(Chairman: J.Schubart)
S.FERRAZ-MELLO I OnResonance 69
A.LEMAITREandJ.H8\TRARD / The 3/2Resonance 91
J. HENRARD and A. de VLEESCHAUWER / Sweeping Through a Second Order
Resonance 99
Ch.FROESCHLEand H.SCHOLL I SecularResonances: New Results 113
P.J.MESSAGE I NormalCo-Ordinatesand Asymptotic ExpansionsinResonance
Cases inCelestialMechanics 119
A.H.JUPP / The Critical InclinationProblem: 30YearsofProgress 127
V.SZEBEHELY I Limits ofPredictabilityofGravitationalSystems 139
SESSIONONGENERALDYNAMICALSYSTEMS
(Chairman: R.Dvorak)
G.comoPOULOS I Short and Long PeriodOrbits 147
E.KEIL I StabilityofParticleOrbits 163
C.MARCALand F.RANNOU-MONTIGNY I ASmallExampleofArnold Diffusion 177
vi TABLE OF CONTENTS
W.NEUTSCH and J.KALLRATH / Area-preservingPoincareMappings oftheUnit
Disk 185
D. L RICHARDSON and T.J. KELLY / Two-Body Motion in the Post-Newtonian
Approximation 193
P.STUMPFF / TheGeneral Kepler Equation andItsSolutions 21I
J. HAGEL / Integration Theory for the Restricted Three-Body Problem with
ApplicationtoP-Type Orbits 223
H.EICHHORN / PhysicalTimeand Astronomical "Time" 237
SESSIONONCOMETS ANDASTEROIDS
(Chairman: V.Szebehely)
H.RICKMAN andc.FROESCHLf / CometaryDynamics 243
c. FROESCHLf and H. RICKMAN / Monte Carlo Modelling of Cometary
Dynamics 265
G. HAHN and c. 1.LAGERKVIST / Orbital Evolution Studies of Planet-Crossing
Asteroids 285
B.fRm / LongPeriodicPerturbationsofTrojan Asteroids 303
J.SCHUBART / ResonantAsteroidsBetween theMainBeltandJupiter'sOrbit 309
A. CARUSI, L. KRESAK, E. PEROZZI and G. B. VALSEccm / On the Past Orbital
History ofComet P/Halley 319
R. DVORAK and J. KRIBBEL / Long Term Evolution of Comet Halleys Orbit
(abstract) 323
c.FROESCHLfandR.GONCZl / On theStochasticityofHalleyLike Comets 325
SESSIONONSATELLITES
(Chairman: S.Ferraz-Mello)
L.DURIEZ / Long-Term EvolutionoftheOrbitsofNatural Satellites 331
M. MOONS, F. DELHASE and E. DEPAEPE / Elliptical Hill's Problem (Large and
Small ImpactParameters) 349
M. KARCH and R. DVORAK / New Results on the Possible Chaotic Motion of
Enceladus 361
SESSIONON PERIODIC ORBITS
(Chairman: C.Froeschle)
J. D. HADJIDEMETRIOU / Periodic Orbits of the Planetary Type and Their
Stability 371
J. KRIBBEL and R. DVORAK / Stability of Periodic Resonance-Orbits in the
EllipticRestricted3-Body Problem 391
J.KALLRATH / Bounded Orbits intheElliptic Restricted Three-BodyProblem 399
TABLE OF CONTENTS vii
REPORTSONEVENINGDISCUSSIONS
P.K.SEIDELMANNandc.A.WILUAMS / DiscussionofCurrentStatusof
PlanetX 409
H.RICKMAN / RelationBetweenSmallBodies:Discussion 413
H.EICHHORN / PredictabilityinDynamicalSystems 417
421
CallforPapers

PREFACE
Thesuccess ofthe 1st Alexander von Humbolt Colloquium on Celestial Mechanics (March
25-31, 1984) encouraged us to meet again this year from March the 13th to the 19th 1988
in the Alpengasthof Rosegger in the Austrian Alps (Ramsau, Styria) the very same place
where we met four years ago.
The general topic of the 2nd Alexander von Humbolt Colloquium on Celestial Mechan
ics was the long termevolution of Planetary Systems. Six sessions were held on Planetary
Dynamics,ResonanceProblems,General DynamicalSystems,Comets and Asteroids,Satel
litesand PeriodicOrbits. Eachofthesesessions was introduced by an invited reviewpaper
(in bold face in the table of content) followed by offered contributions and discussions.
Three evening discussions were held, devoted respectively to Planet X, the Predictability
in Dynamical Systems and the relations between small bodies in the Solar System.
On the evening of Wednesday March 16th we were offered a wonderful piano concert by
Ariadne Erdi from Budapest. We thank her deeply. On Friday March 18th we were cor
dially greeted by the Mayor of Ramsau and then entertained by the Steiner singers from
the village ofRamsau. Wehad already enjoyed such anjoyful"Styrian Evening" four years
ago. Thissecond rendition will certainly start a tradition.
Several institutions in Austria helped us by their financial support to organize this inter
national scientific meeting. They are :
The Bundesministerium fiir Wissenschaft und Forschung,
The Osterreichischen Forschungsgemeinschaft,
The Steierrnakischen Wissenschafts- und Forschungslandesfonds,
The Handelskammer Steiermark Sektion Industrie,
The Kammer fiir Arbeiter und Angestellte fiir Steiermark,
The Osterreichischen Nationalbank,
The Formal- und Naturwissenschaftliche Facultat of the University of Vienna.
Support was also given by "Hornig Kaffee" from Graz and the Creditanstalt-Bankverein.
Wedeeply appreciate their support.
Many thanks aredue to theChairmenofthe various sessions,eachofwhom acted as Editor
for the proceedings ofhissession. Thanks to them and to thosecolleagueswho refereed the
reviews and papers presented at the sessions, we were able to assemble these proceedings
in a very short time.
Last, but by no means least, we are very glad to express our thanks to Fritz and Barbara
Walcher, our host at the Alpengasthof Peter Rosegger. With the excellents meals and
the friendly atmosphere of their Styrian hotel they contributed a lot to the success of the
meeting.
Rudolph DVORAK Jacques HENRARD
INTEGRATION ERROR OVER VERY LONG TIME SPANS
Andrea MILANI and Anna M. NOBILl
Gruppo di Meccanica Spaziale
Dipartimento di Matematica, Universita di Pisa
Via Buonarroti 2, 56100 Pisa - Italia
[email protected]
ABSTRACT. The mainlimit to the time span ofa numerical integration of the plane
tary orbits is no longerset by the availability ofcomputer resources, but rather by the
accumulation of the integration error. By the latter we mean the difference between
the computed orbit and the dynamical behaviour ofthe real physicalsystem,whatever
the causes. The analysis of these causes requires an interdisciplinary effort: there are
physicalmodeland parameterserrors, algorithm and discretisationerrors, rounding off
errors and reliability problems in the computer hardware and system software, as well
as instabilities in the dynamical system. Welist all thesources of integration error we
are aware of and discuss their relevance in determining the present limit to the time
span of a meaningful integration of the orbit of the planets. At present this limit is
of the order of 108 years for the outer planets. We discuss in more detail the trun
cation error of multistep algorithms (when applied to eccentric orbits), the coefficient
error,the method ofEncke and the associatedcoordinate change error, the procedures
used to test the numerical integration software and their limitations. Many problems
remain open, including the one ofa realisticstatistical model ofthe rounding offerror;
at present, the latter can only be described bya semiempirical model based upon the
simple N2f formula (N =number of steps, e =machine accuracy), with an unknown
numerical coefficient which is determined only a posteriori.
1. Introduction
Thisinvestigationwasundertakenas partoftheLONGSTOPresearchproject.
LONGSTOP (LONg term Gravitational Stability Test for the Outer Planets)
is an international cooperative program with the purpose of investigating the
stability of our Solar System -over timescales comparable to its present age
using a combination of analytical and numerical techniques. The first main goal
of the project was to numerically integrate the orbits of the outer planets for
100 million years. Within this project the orbits of the outer planets have been
computed for 9.3 Myr (Milani et al., 1986j Milani et al., 1987aj Carpino et al.,
1987) and for 100 Myr (Nobili et al., 1988).
Although such an integration is an expensive exercise, the limit to the times
pan over which informations on the dynamical behaviour of the planets can be
obtained by numerical integration is not determined by the availability of com-
CelestialMechanics43(1988),1-34.
©1988byKluwerAcademicPublishers.
2 ANDREAMILANIANDANNAM.NOBIU
Table 1: Very long integrations
Problem Time span Stepsize N2E:
Outer Planets lOByr 43 days 15
Inner planets 106yr 21 hours 38
Inner planets + Moon 106yr 6h30min 397
Galilean satellites 104yr 25 min 9
Artificial satellite LAGEOS 10yr 2 min 1.2 x 10-3
Earth's rotation 2.6 X 104yr 14 min 200
puter resources, e.g. CPU time. On the contrary the integration error growth
sets the limit to the length of any integration, as long as it is required that the
final computed positions are still causally related to the initial conditions (as
opposed to investigations within the framework of statistical mechanics where
longer integrations can be used, at least in principle).
This work specifically refers to the LONGSTOP IB numerical integration
of the orbits of the outer planets (from Jupiter outwards) for 100 million years;
howeverit applies toeveryvery long numericalintegrationofHamiltonequations
of motion. By very long integration we mean a numerical integration such that
(i) the growthof the integration errorsets the limit to the integration time span,
giventhe required accuracy, and (ii) thetimespan aimed atcan beachieved only
by exploiting the available computer technology to its limits. A simple rule-of
thumb can be used to decide whether a given integration is very long. Once a
stepsize h has beenchosen (compatiblewith stability constraints and truncation
error requirements, see Section 3 and Appendix C) the integration will require
N steps to get to time T = Nh, to be performed on a computer with a relative
machine accuracy E: (E: = 2-b, with b the number of bits in the mantissa of a
floating point number as represented in the processor)j the integration is very
long in the sense (i), (ii) if N2E: is larger than the required relative accuracy. In
Table 1 we list some examples of very long integrations; the table is computed
on the assumption of a 52 bit mantissa (E: = 2.2 x 10-16) and with h equal to
1/100 of the period of the fastest angular variable.
A simple explanation of the N2E: criterion is as follows: if the error at each
step is of the order of the machine accuracy E:,since most error sources have a
quadratic law of accumulation, the result ofthe integration will be unreliable if
N2E: » 1. To keep down the error at each step below the machine accuracy
level requires to take into account how each operation used in the numerical
integration algorithm is actually performed in finite-precision arithmetic. The
analysis of the error sources and of the error accumulation processes for a very
long integration cannot rely on the traditional, and very expedient distinction of
roles betweenthephysicist (modelerrors), thepuremathematician (instabilityof
the exact solution), the applied mathematician (algorithm, or truncation,errors
and instabilities) and the computer scientist (rounding off and more generally
computer limitations). In a very long integration, by definition, all thesesources
of error are relevant and interact (nonlinearly) to give a total error for which