Lecture Notes in Physics Editorial Board H. Araki Research Institute for Mathematical Sciences Kyoto University, Kitashirakawa Sakyo-ku, Kyoto 606, Japan E. Br6zin Ecole Normale Sup6rieure, D6partement de Physique 24, rue Lhomond, F-75231 Paris Cedex 05, France J. Ehlers Max-Planck-Institut ftir Physik und Astrophysik, Institut fur Astrophysik Karl-Schwarzschild-Strasse ,1 D-85748 Garching, FRG U. Frisch Observatoire de Nice B. .P 229, F-06304 Nice Cedex 4, France K. Hepp Institut for Theoretische Physik, ETH H~nggerberg, CH-8093 ZUrich, Switzerland R. L. Jaffe Massachusetts Institute of Technology, Department of Physics Center for Theoretical Physics Cambridge, MA 02139, USA R. Kippenhahn Rautenbreite 2, D-37077 G6ttingen, FRG H. A. Weidenmtiller Max-Planck-Institut fur Kernphysik Saupfercheckweg ,1 D-69117 Heidelberg, FRG J. Wess Lehrstuhl for Theoretische Physik Theresienstrasse 37, D-80333 Miinchen, FRG J. Zittartz Institut fur Theoretische Physik, Universitat K~ln Ztilpicher Strasse 77, D-50937 KOln, FRG Managing Editor .W Beiglb6ck Assisted by Mrs. Sabine Landgraf c/o Springer-Verlag, Physics Editorial Department II Tiergartenstrasse ,71 D-69121 Heidelberg, FRG .V .G Gurzadyan .D Pfenniger (Eds.) cidogrE stpecnoC ni Stellar scimanyD Proceedings of an International Workshop Held at Geneva Observatory University of Geneva, Switzerland, 3-1 March 3991 galreV-regnirpS Berlin Heidelberg NewYork London Paris Tokyo Hong Kong Barcelona Budapest Editors .V G. Gurzadyan Department of Theoretical Physics Yerevan Physics Institute 375036 Yerevan, Armenia D. Pfenniger Geneva Observatory University of Geneva CH- 1290 Sauverny, Switzerland ISBN 3-540-57929-X Springer-Verlag Berlin Heidelberg New York ISBN 0-387-57929-X Springer-Verlag New York Berlin Heidelberg CIP data applied for This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustra- tions, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1994 Printed in Germany SPIN: 10080337 55/3140-543210 - Printed on acid-free paper Preface Although stellar dynamics is a well established discipline with quite a long history behind it, the topic of the meeting we chose to hold in March 1993 in Geneva could have met with some surprise a few decades ago. During recent years radical changes have occurred in many areas of scientific re- search, changes following from the realization of the underlying universality of the concepts of chaos, randomness and unpredictability. The determinism of Laplace which once dominated physical sciences has been replaced by the image of Arnold's cat. At the same time spectacular advances in the theory of dynamical systems have provided rigorous tools to study those complex phenomena. This has gone hand in hand with the increased observational accuracy of stellar systems, which offers the opportunity to put serious em- pirical constraints on the theory. The idea of bringing together specialists covering these topics is initially attractive because of the possibility of ob- taining the mutually useful information first hand. However, there was also an element of danger: that of having monologues instead of dialogues. But we were encouraged by the previous experience of meetings on stellar dynamics held in 1966, 1967 (Besan~on, Paris). Turning now the pages of those pro- ceedings, after more than a quarter of century, one cannot avoid the feeling of respect for the participants, how correctly they anticipated the forthcom- ing problems to be solved, guessed their possible answers... And if that was the time of the amazing discoveries of Hdnon z~ Heiles, Lorenz and other pioneers, today we have the same feeling when we find regular behaviour of complex systems. This was the logical and historical origin of the idea of our Workshop. The volume we are now presenting contains the talks given at the Work- shop, with the emphasis on their review character, aiming to provide a refer- ence book useful for as many specialists as possible from the different areas. It concerns first of all, the representation of tools of ergodic theory: along with the results ofl atest observations it contains sophisticated mathematical techniques usually not found in books on stellar dynamics. This is perhaps the first distinguishing feature of this volume. To maintain the emphasis on review, we therefore unfortunately had to omit the interesting discussions following almost all of the talks. However, we do present the discussion held at the last session oft he Workshop concerning 10 essential unsolved problems of stellar dynamics: in theory, computer simulations and observations. This can be considered as another non-traditional aspect of the book. Finally, let us mention that, to fulfil the conditions imposed by our aims, in accord with the Scientific Organizing Committee, all the submitted papers were refereed, almost all were revised and a few rejected, a fact again not usual for conference proceedings. Only the texts of discussion talks have been kept untouched. As a result we have a collection of papers of an even better IIIV ytilauq than those submitted .yllanigiro This experience may etacidni ot eht srotide and publishers that eht gnieerefer of niatrec conference proceedings si not only elbarised but can even be more important than rof ,slanruoj fi eht former era intended to serve longer and rof a wider audience. Let time judge this further attempt at deciphering the truth about eht world of rallets systems, rof us associated perhaps with eht face of another mysterious ,tac the Cheshire tac fo Alice ni Wonderland. We era lufetarg to lla those who helped us ni the organizing of the work- shop. In particular our thanks go ot the Director of Geneva Observatory, rosseforP Andrd Maeder, e~lt members of the Local Organizing Committee, rosseforP Louis Martinet and Dr. Daniel ,ildeirF and the ,seiraterces Ms. enerI erffehcS and Ms. Elisabeth Teichmann. We would osla ekil ot mention the help of Dr. Armen Kocharyan and eht anonymous seerefer ni the preparation of siht volume. We acknowledge the laicnanif supports of the Swiss Academy of Natural secneicS (ASSN), the Swiss National Science Foundation (FNRS), and eht ytisrevinU of Geneva. They have been evisiced ni enabling siht conference ot take .ecalp ,yllaniF ti si a ralucitrap pleasure ot record the fact that the daughter of one of ,su Karin Pfenniger, had the good taste to delay her ,htrib tsrif announced rof the beginning fo eht workshop, by four weeks. Geneva Vahe Gurzadyan October 3991 Daniel Pfenniger Contents Foreword A. Maeder ................................................... 1. Observations Structural and Dynamical Forms of Elliptical and Dwarf Galaxies S. Djorgovski ................................................. Some Clues About the Dynamics of Globular Clusters from High-Resolution Observations G. Meylan ................................................... 22 DifFusion of Stellar Orbits in the Galactic Disk B. Fuchs, C. Dettbarn, z~ R. Wielen ............................. 34 2. Tools of Ergodic Theory Ergodic Methods in Stellar Dynamics V.G. Gurzadyan .............................................. 43 On a Notion of Weak Stability and Its Relevance for Celestial Mechanics and Molecular Dynamics L. Galgani & A. Gi0rgilli ...................................... 56 Recent Developments in the Dynamics of Nonlinear Hamiltonian Systems with Many Degrees of Freedom M. Pettini ................................................... 64 Numerical Exploration of the Circular Billiard with Gravity A. Hayli & A. Vidovi~ ......................................... 85 Ergodicity and Mixing in Gravitating Systems V.A. Antonov & L.P. Ossipkov ................................. 19 3. Computer Simulations and Mappings Chaotic Itineracy and Clustered Motion in Globally Coupled Symplectic Map System T. Konishi & K. Kaneko ....................................... 95 Lyapunov Analysis of Stable Chaos in Self-Gravitating Systems N. Gouda, T. Tsuchiya, :& T. Konishi ........................... 100 Stability of the Modified Konishi-Kaneko System S. Inagaki .................................................... 105 Mixing Transformations of N Particles Conserving Almost All Classical Integrals D. Pfenniger ................................................. 111 Symplectic Integration Without Roundoff Error D. J.D. Earn ................................................. 122 Discreteness Noise Versus Force Errors in N-Body Simulations J. Makino .................................................... 131 4. Instabilities Core Motions and Global Chaotic Oscillations R.H. Miller ................................................... 137 N-Body Systems: Computer Image and Reality V.G. Gurzadyan & A.A. Kocharyan ............................. 151 The Approach to Integrability in N-Body Systems with a Central Point Mass H. Smith, Jr., H.E. Kandrup, M.E. Mahon, & C. Siopis ............ 158 On the Non-Trivial Concept of Relaxation in N-body Systems P. Cipriani & G. Pucacco ...................................... 163 Gravothermal Oscillations R. Spurzem .................................................. 170 Recent Results on the Stability of Anisotropic Stellar Systems J. Perez :& J.-J. Aly ........................................... 177 1× 5. The Few-Body Problem The Stability of the Solar System J. Laskar .................................................... 183 The One-Dimensional Three-Body Problem: Numerical Simulations J.L. Rouet, R. Dufour, & M.R. Feix ............................. 193 6. Large-N Limit Order and Chaos in "Collisionless" Numerical Simulations D. Pfenniger ................................................. 201 On the Permissible Percentage of Chaotic Orbits in Various Morphological Types of Galaxies L. Martinet .................................................. 217 Minimum Energy States of a Self-Gravitating System J.-J. Aly ..................................................... 226 Effective Collision Term Induced by Coarse-Graining T. Tsuehiya .................................................. 230 Theoretical and Numerical Investigation of the Stability of Flattened Galaxies E. Griv ~ W. Peter ........................................... 235 The Evolution of Orbits in the Stellar Disk as a Purely Discontinuous Random Process I.V. Petrovskaya .............................................. 239 7. Galactic and Extragalactic Problems Interacting Spherical Stellar Systems J.C. Muzzio .................................................. 243 How Faithful Are N-Body Simulations of Disc Galaxies? -Artificial Suppression of Stellar Dynamical Instabilities A.B. Romeo .................................................. 257 llX SPH Simulations of the Gas Flow in Normal Spiral Galaxies P.A. Patsis & N. Hiotelis ....................................... 261 Regular Orbits and Cantori in the Potential of the Barred Galaxy NGC 936 H. Wozniak .................................................. 264 The Role of Stochastic Motion in a Central Field with a Bar-Like Perturbation J.A. Ntifiez, P.M. Cincotta, :& J.C. Muzzio ....................... 267 A Hierarchical Model of Patchy-Structured Galaxies and Evolutionary Processes E.M. Nezhinskij & A. Ollongren ................................ 268 Evolution of Clusters of Galaxies Y. Funato, J. Makino, & T. Ebisuzaki ........................... 270 Smoothing of the Cosmic Background Radiation by Multiple Gravitational Scattering J. Makino .................................................... 274 Angular Momentum of Galaxies Within the Local Supercluster W. Godtowski ................................................ 278 8. Concluding Remarks 10 Key Problems V.G. Gurzadyan .............................................. 283 Comments on "10 Key Problems" R.H. Miller, A. Hayli, L. Martinet, Y. Pesin, J.C. Muzzio, S. Inagaki, D. Pfenniger, & G.S. Djorgovski ................................ 285 Indexes Author Index Subject Index List of Participants ALY Jean-Jacques DE ZEEUW P.T. Service d'Astrophysique Sterrewacht Leiden CE Saclay Huygens Laboratorium F-91191 Gif Sur Yvette, C~dex Postbus 9513 FRANCE NL-2300 RA Leiden THE NETHERLANDS ATHANASSOULA E. DJORGOVSKI S. Observatoire de Marseille Palomar Observatory ,2 Place Le Verrier Caltech F-13248 Marseille, C~dex 4 Pasadena CA 91125 FRANCE USA BALCELLS Marc EARN David J.D. Kapteyn Astronomical Institute Institute of Astronomy Postbus 008 Madingley Road NL-9700 VA Groningen Cambridge CB3 OHA THE NETHERLANDS UK BIESIADA Marek FRIEDLI Daniel Nicolaus Copernicus Astronomical Geneva Observatory Center University of Geneva Bartycka 81 ,15 ch. des Maillettes PL-00-716 Warsaw CH-1290 Sauverny POLAND SWITZERLAND CIPRIANI Pietro FUCHS Burkhard Dipartimento di Fisica Astronomisches Rechen-Institut Universita La Sapienza MSnchhofstrat~e 12-14 Piazzale Aldo Moro 2 D-W 6900 Heidelberg L00185 Roma GERMANY ITALY