The Dynamics of Small Bodies in the Solar System

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Series C: Mathematical and Physical Sciences - Vol. 522

The Dynamics of Small Bodiesin the Solar System

A Major Key to Solar System Studies

edited by

Bonnie A. StevesDepartment of Mathematics,Glasgow Caledonian University,Glasgow, United Kingdom

and

Archie E. RoyDepartment of Physics and Astronomy,University of Glasgow,Glasgow, United Kingdom

~.

'ISpringer-Science+Business Media, B.V.

Published in cooperation with NATO Scientific Affairs Division

Proceedings of the NATO Advanced Study Institute onThe Dynamics of Small Bodies in the Solar System: A Major Key to Solar SystemStudiesMaratea, Italy29 June - 12 July 1997

Library of Congress Cataloging-in-Publication Data

The dyna.lcs of s.all bod'es In the solar syste. a .aJor key tosolar systeo studies I edIted by BonnIe A. Steves and ArchIe E. Roy.

p . c, . - - (NATO ASI sertes SerIes C, Mathe.atlcal andphysIcal scIences ; vol. 522)

· Pr oceed l ngs of the NATO Advanced Study Instttute. held In Maratea-Acquafredda. Italy. bet.een June 29 and July 12, 1997, entitled TheOyna.lcs of S.alt BodIes tn the So'ar Systeo , a Major Key to SolarSyste. Studles--Preface .

Includes Index .

1. Solar syste. --Congresses . 2 . Asterolds--Congresses . 3 . Few-body proble~--Congresses. 4 . Astrophys1cs- -Congresses .I. Steves, BonnIe A. II. Roy. A. E. (ArchIe E.). 1924-I II . The Oyna.lcs of S.all BodIes In the Solar Sys t e. , a Major Keyto 'Solar Syste. St udtes (1997 Acquafredda dl Maratea, Italy)IV. Sertes , NATO ASI serIes . Sertes C, Mathe.atlcal and physicalscIences ; no . 522 .OB500.5 .092 1999523 .5--dc21 98-31687

001

Printedon acid-free paper

All Rights Reserved© 1999 Springer Science-Business Media DordrechtOriginally published by Kluwer Academic Publishers in 1999.Softcover reprint of the hardcover 1st edition 1999

No part of the material protected by this copyright notice may be reproduced orutilized in any form or by any means, electronic or mechanical, including photo­copying, recording or by any information storage and retrieval system, without writtenpermission from the copyright owner.

ISBN 978-90-481-5133-2 ISBN 978-94-015-9221-5 (eBook)DOI 10.1007/978-94-015-9221-5

These Proceedings of the 1997 NATO Advanced StudyInstitute held in Maratea, Italybetween June 29 and July 12

are dedicated to thememory of

Professor Victor Szebehely(1921 - 1997)

renowned celestial mechanician, cherished teacher and dear friend .

CONTENTSPrefaceI. In Memory ofYictory Szebehely

K. Zare

SECTION ONE: Asteroids and Trans-Neptunian Objects

2. IntroductionE. Belbruno and LP. Williams

3. On High-Eccentricity Small-Amplitude Asteroidal LibrationsA. Simula, S. Ferraz-Mello and C. Giordano

4. A. Symplectic Mapping Approach for the Study ofStochasticity in Three Dimensional Asteroidal Resonances

F. Roig and S. Ferraz-Mcllo

5. Chaotic Diffusion in the 2/1, 3/2 and 4/3 Jovian ResonancesD. Nesvomy and S. Ferraz-Mello

6. On the Similarities and Differences Between 3/2 and2/1 Asteroidal Resonances

T.A. Mitchenko and S. Ferraz-Mello

7. A Study of Chaos in the Asteroid BeltM. Sidlichovsky and D. Nesvorny

8. Hopping in the Kuiper Belt and Significance of the 2:3 ResonanceE. Belbruno

9. The Solar System Beyond NeptuneJ.P. Williams

10. Dynamical Study of Trans-Neptuni an Objects at the 2:3 Resonance1. hadjidcmetriou and K.G. Hadjifotinou

II . Superosculating Intermediate Orbits and their Application in theProblem of Investigation of the Motion of Asteroids and Comets

VA. Shefer

12. Tests of General Relativity using Small Bodies of the Solar SystemC. Marchal

SECTION TWO: Near Earth Objects

13. IntroductionK. Muinonen and A. Milani

14. Dynamics of Planet-Crossing AsteroidsA. Milani

Xl

Xlii

7

13

19

25

31

37

51

65

71

77

93

95

viii

15. Asteroid and Comet Encounters with the Earth . Impact Hazard andCollission Probability

K. Muinonen 127

16. Optimal Low-Thrust Interception and Deflection of Earth­Crossing Asteroids

B.A. Conway 159

17. Secular Resonances: Transport Mechanism to Earth-Crossing Orbitsp. Michel 171

18. Meteoroid Streams and Meteor ShowersI.p. Williams 179

19. Planetary Close Encounters: The Engine of CometaryOrbital Evolution

G.B. Valsecchi 187

20. Galactic Perturbations in the Motion of CometsH. Pretka 197

SECTION THREE: Natural and Artificial Satellites

21. IntroductionCD. Murray 203

22. Orbits of Saturn's Satellites : Some Aspects of Commensur-abilities and Periodic Orbits

P.l. Message 207

23. How Ideal is the Mirnas-Tethys Resonance?1. Probin 227

24. The Dynamics of Planetary Rings and Small SatellitesCD. Murray 233

25. Long-Term Evolution of Coorbital Motion1. Waldvogel 257

26. Stability of Perturbed Coorbital SatellitesM.H.M. Morais and CD. Murray 277

27. Instability and Chaotic Motion for Some Fictitious Satellites ofVenus and Mars

T.Yokoyama 283

28. Applications of a High Order Secular Perturbation Theory.The Uranian Satellites and Beyond

A.A. Christou and CD. Murray 289

29. Non-Integrability of the Motion of a Point Mass around a Planetof Arbitrary Schape

M.E. Sansaturio, I Vigo-Aguiar and 1.M. Ferrandiz 295

30. Dynamics of Satellites with Multi-Day PeriodsB. Erdi

31 . Translunar Halo Orbits in the Quasibicircular ProblemM.A. Andreu and C. Sima

32. Problems Connected with the Rotational Dynamics ofCelestial Bodies

A.J. Maciejewski

33. Closed Form Expressions for Some Gravitational Potentials.Triangle, Rectangle, Pyramid and Polyhedron.

R.A. Broucke

34. Orbital Elements of a Satellite Moving in the Potential of aHemispherical Shell

B. Zafiropoulos and Ch. Stavliotis

SECTION FOUR: Few Body Systems

ix

303

309

315

321

341

35. IntroductionA.E. Roy and B.A. Steves 349

36. Some Special Solutions of the Four-Body Problem.B.A. Steves, A.E. Roy and M. Bell 351

37. Some Equal Mass Four-Body Equilibrium Configurations:Linear Stability Analysis

1. Gomatam, B.A. Steves and A.E. Roy 373

38. The Finite-Time Stability Method Applied to Planets of Binary StarsB.A. Steves, A.D. Barnett and A.E. Roy 379

39. Dynamics and Stability of Three-Body SystemsR. Mardling and S. Aarseth 385

40. New Numerical Method for Non-Conservative SystemsT.A. Taidakova and N. N. Gor'Kavyi 393

41. Tidal Friction in Triple Systems: A Means of Producing CloseStellar and Planetary Orbits

L.G. Kiseleva and P.P. Eggleton 399

42. Special Version of the Three-Body ProblemK. Gozdziewski and A.J. Maciejewski 407

43. Bifurcations in the Mass Ratio of the Planar IsoscelesThree-Body Problem

S. Chesley and K. Zare 413

x

SECTION FIVE: Studies of Dynamical Systems

44 . IntroductionG. Contopoulos and R. Dvorak 425

45 . Dynamical SpectraG. Contopoulos and N. Voglis 429

46 . Angular Dynamical Spectra and their ApplicationsC. Efthymiopoulos, N. Voglis and G. Contopoulos 455

47 . Weak Chaos and Diffusion in Hamiltonian Systems. FromNekhoroshev to Kirkwood

C. Froeschle and E. Lega 463

48 . On the Relationship between Local Lyapunov Characteristic Numbers,Largest Eigenvalues and Maximum Stretching Parameters

C. Froeschle, E. Lohinger and E. Lega 503

49 . "Stickiness" in Dynamical SystemsR. Dvorak 509

50. Distribution of Periodic Orbits in 2-D Hamiltonian SystemsE. Grousouzakou and G. Contopoulos 535

51. The Many Faces of Dynamical Instability I: The definition of"Chaos" and the (In)equivalence of Different Indicators

P. Cipriani and M. Di Sari 545

52. The Many Faces of Dynamical Instability II: The Sources"Chaos" in Dynamical Systems

M. Di Bari and P. Cipriani 55 I

53. Scale Relativity in the Solar System. A Review and ProspectsR.P. Hermann 557

54. Possible Cosmological Origin of Counterrotating GalaxiesM. Harsoula and N. Voglis 565

55. Gylden-like Systems and DEF-Mappings1. Aparicio and L. Floria 575

56. Uniform Development of a TR-Transformation to GeneralizedUniversal Variables

L. Flor{a 581

EPILOGUE: The,Shape of Things to Come

Subject and Author Indexes

Participants and Speakers

57. Future Planetary MissionsA.A. Christou 587

595

603

PREFACE

The reader will find in this volume the Proceedings of the NATO Advanced StudyInstitute held in Maratea-Acquafredda, Italy, between June 29 and July 12, 1997, entitled THEDYNAMICS OF SMALL BODIES IN THE SOLAR SYSTEM: A MAJOR KEY TOSOLAR SYSTEM STUDIES .

This Advanced Study Institute was the latest in the 'Cortina' series of NATO ASI'sbegun in the early 1970's firstly under the directorship of Professor Victor Szebehely andsubsequently under Professor Archie Roy. All, except the latest, were held at the AntonelliInstitute, Cortina d'Ampezzo, Italy. Many of those now active in the field made their firstinternational contacts at these Institutes . The Institutes bring together many of thebrightest of our young people working in dynamical astronomy, celestial mechanics andspace science, enabling them to obtain an up-to-date synoptic view of their subjectsdelivered by lecturers of high international reputation. The proceedings from these instituteshave been well-received in the international community of research workers in the disciplinesstudied.

The present institute included 15 series of lectures given by invited speakers and some45 presentations made by the other participants. The majority of these contributions areincluded in these proceedings.

The theme of the latest ASI in the series, viz. The Dynamics of Small Bodies in theSolar System : A major key to solar system studies was chosen because of the widespreadrecognition that the small natural bodies of the solar system bear an importance in solarsystem studies far outweighing their size and mass relative to the planets . The study of thenature and dynamics of asteroids, comets, meteor streams, natural satellites and ringsystems currently provides a wealth of information concerning the history and dynamicalevolution of the solar system as a whole. Such objects also provide excellent test-beds forthe study of resonance and chaos .

It has become clear that the evidence of the bombardment of planet Earth bycomets and asteroids with the subsequent extinction of a proportion of living species is notonly a topic of historical interest for the astronomer, biologist, botanist and geologist, butis also an ongoing process that must be addressed by any scientist responsible for thesurvival of terrestrial civilisation. The spectacular bombardment of Jupiter in 1995 by thefragments of Comet Shoemaker-Levy underlines the necessity of international support foroperations such as Spaceguard for finding, listing and assessing all Earth orbit crossingasteroids and comets within the Solar System.

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The dynamics of the numerous large asteroidal bodies recently found in the outerparts of the solar system are a current subject of interest, raisingquestions of whether theyhave originated there or can have been sent there from the classical asteroidal belt. Theirrelation to the hypothetical Kuiper Belt is also of considerable interest.

It is also clear that the careful design of spacecraft trajectories for artificialsatellites not only of planets, but also of asteroids and natural satellites, is of majorsignificance in enlarging our knowledge of the solar system. Artificial statellites ofasteroids and irregularly shaped natural satellites can be used to obtain their massdistributions and shapes. In addition, recent theoretical results obtained from analternative representaion of the gravitational field of non-spherical bodies are of relevanceto such studies.

All these topics were addressed not only in the courses of lectures given by theinvited speakers, but also in the seminar presentations made by participants. It is evidentthat the majority of those attending the Institute are collaboratively engaged in many ofthese topics. Many new research bonds were formed among the participants which will bedeveloped during the years to come.

The Organising Committee (Professor A. E. Roy (Director), Professor V. SzebeheIy(Associate Director), Dr P. 1. Message, Professor A. Milani and Dr B. A. Steves) is grateful tothe NATO Scientific Affairs Division, not only for its guidance, counsel and support, but alsofor providing the opportunity to invite colleagues from a number of NATO's designatedCooperative Countries. The warm international friendship demonstrated by the participantsduring the duration of the NATO ASI was a most valuable feature of the meeting. TheOrganising Committee would like to take this opportunity to thank all those whosecontributions made the ASI a success and such a very happy occasion. In particular theCommittee would like to thank the staff of the Hotel Villa del Mare, Acquafredda, Marateawhere the ASI was held. They were unstintingly helpful and friendly. The editors would alsolike to thank Mr. A. D. Barnett of the Department of Mathematics of Glasgow CaledonianUniversity for hisvaluable help in the preparation of thisvolume.

Each of the book's sections has an introduction specially written by renowned expertsin their fields. It is hoped that the book will be regarded as a worthytext and helpful referencevolume for all students and researchers involved in the study of the nature and dynamics ofasteroids, comets,meteorstreams, natural satellites and ringsystems.

It is appropriate that the book is dedicated to one of the foremost researchers inCelestial Mechanics, Professor Victor Szebehely, who although present at Maratea among hismany friends, died shortly afterhisreturn to the United States. His books, his manypapersandabove all hisenthusiasm and help given to so many in the Celestial Mechanics community overmany years will be forever remembered.

Bonnie A. StevesDept. of MathematicsGlasgow Caledonian UniversityGlasgow G4 aBAUNITEDKINGDOM

Archie E. RoyDept.of Physicsand AstronomyGlasgow University,Glasgow G128QQUNITED KINGDOM

IN MEMORY OF VICTORY SZEBEHELY

K. ZAREA ustin, Texas

The Escape Of A Dominant Star From the Cluster of CelestialMechanics

On September 13, 1997 Victor Szebehely passed away in Austin, Texas .We had planned to be co-authors of a paper in this proceeding. Instead,under the circumstances, I decided to write a short 'memoir ' in honor ofmy distinguished co-author, mentor and friend .

On August 10, 1921, a star was born in Budapest , Hungary. He receivedhis Ph.D in 1946 at the University of Budapest , writing a dissertation onthe problem of three bodies . He was ejected to the United States in 1947where he was temporarily captured by many academic institutions includingYale University (1963-1968) and the University of Texas (1968-1997) . Hisresearch activities had an interplay between ship dynamics (1947-1957)for which he was knighted by Queen Juliana of the Netherlands (1956)and space dynamics (1957-1997) for which he received numerous nationaland international awards (1978-1997) . During his lifetime he had manyencounters with his int ernational colleagues and friends, and in particularnine encounters (1972-1997) with the 'Cort ina group' . His characteristicwas not only his unusual brightness as a lead ing researcher in his field,but also his kindness , gentleness, sense of humour, and recognizable accent.His enthusiasm for celestial mechanics and in particular for the three bodyproblem was observed by many any time he talked on the subject . Hisencouragement was always a source of energy for his students and colleaguesto do research in celestial mechanics.

His last encounter with the 'Cortina group' occurrred at Maratea wherehe walked with his friends through the streets of Pompeii. My last conver­sation with him was on the phone when he told me about his cancer andhis inevitable escape. That was the saddest day in my life. A month laterthe final escape occurred on Saturday, September 13 as it was observed byhis daughter Julie. " ... As I sit here writing to you, there is another star in

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the sky... My father died quietly at home today... On the table beside myDad was a scientific journal, folded open, with his glasses on it ... "

The dominant star escaped, but he has left behind many enthusiasticresearchers inspired by him to keep his fascination with the three bodyproblem alive .