Chaos in the N-Body Problem of Stellar Dynamics

  • Douglas C. Heggie
Part of the NATO ASI Series book series (NSSB, volume 272)


Stellar dynamics uses several different models of stellar systems, but in this paper we consider the most fundamental, which is governed by the N-body equations:
$$\ddot r_i = - G\sum\limits_{j = 1,j \ne i}^N {m_j \frac{{r_i - r_j }}{{\left| {r_i - r_j } \right|^3 }}} $$
Stellar dynamics shares these equations with celestial mechanics (a term which is used here to denote the study of the orbital dynamics of bodies in the solar system), but there are important differences of emphasis. In stellar dynamics all masses are comparable, whereas in celestial mechanics one mass tends to dominate (either the sun or a primary). This has an effect on the methods used and the types of motion which result. Approximate analytical methods are of immense value in celestial mechanics, but not in stellar dynamics, where numerical methods predominate. In celestial mechanics motions tend to be very nearly regular for long intervals of time, whereas in stellar systems motions are highly irregular.


Globular Cluster Celestial Mechanic Star Cluster Stellar System Close Encounter 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Aarseth, S.J., 1974, Dynamical Evolution of Simulated Star Clusters I. Isolated Models, Astron. Astrophys., 35:237.ADSGoogle Scholar
  2. Aarseth, S.J., 1985, Direct Methods for JV-Body Simulations, in: Multiple Time Scales, J.U. Brackbill & B.I. Cohen, eds.. Academic Press, New York.Google Scholar
  3. Arnold, V.I., 1978,Mathematical Methods of Classical Mechanics, Springer-Verlag, New York.zbMATHGoogle Scholar
  4. Binney, J. & Tremaine, S., 1987, Galactic Dynamics, Princeton University Press, Princeton.zbMATHGoogle Scholar
  5. Breeden, J.L., Packard, N.H. & Cohn, H., 1990, Chaos in Astrophysical Systems: Core Oscillations in Globular Clusters, preprint, CCSR-90–2 (Center for Complex Systems Research, Dept. of Physics, Beckman Institute, University of Illinois at Urbana-Champaign).Google Scholar
  6. Carnevali, P. & Santangelo, P., 1980, Automated Graphical Plots for the Study of the Gravitational N-body Problem,Mem.S.A.It., 51:529.ADSGoogle Scholar
  7. Channel, P.J. & Scovel, C., 1988, Symplectic Integration of Hamiltonian Systems, preprint, LA-UR-88–1828 (Los Alamos).Google Scholar
  8. Dejonghe, H. & Hut, P., 1986, Round-Off Sensitivity in the N-Body Problem, in:The Use of Supercomputers in Stellar Dynamics, P. Hut & S. McMillan, eds.. Springer-Verlag, Berlin.Google Scholar
  9. Goodman, J., Heggie, D.C. & Hut, P., 1990, On the Exponential Instability of N-Body Systems, preprint.Google Scholar
  10. Guckenheimer, J. & Holmes, P., 1983, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York.zbMATHGoogle Scholar
  11. Gurzadyan, V.G. & Savvidy, G.K., 1986, Collective Relaxation of Stellar Systems,Astron. Astrophys., 160:203.ADSzbMATHGoogle Scholar
  12. Heggie, D.C., 1988, The N-Body Problem in Stellar Dynamics, in: Long-Term Dynamical Behaviour of Natural and Artificial N-Body Systems, A.E. Roy, ed., Kluwer, Dordrecht.Google Scholar
  13. Heggie, D.C. & Ramamani, N., 1989, Evolution of Star Clusters After Core Collapse, M.N.R.A.S., 237:757.ADSGoogle Scholar
  14. Krylov, N.S., 1979,Works on the Foundations of Statistical Physics, Princeton University Press, Princeton.Google Scholar
  15. Lecar, M., 1968, A Comparison of Eleven Numerical Integrations of the Same Gravitational 25-Body Problem, Bull Astron., 3:91.Google Scholar
  16. Lichtenberg, A.J. & Lieberman, M.A., 1983, Regular and Stochastic Motion, Springer- Veriag, New York.zbMATHGoogle Scholar
  17. Miller, R.H., 1964, Irreversibility in Small Stellar Dynamical Systems, Ap.J., 140:250.ADSCrossRefGoogle Scholar
  18. Press, W.H., Flannery, B.P., Teukolsky, S.A. & Vetterling, W.T., 1986, Numerical Recipes, Cambridge University Press, Cambridge.Google Scholar
  19. Smith, H., Jr., 1977, The Validity of Statistical Results from N-Body Calculations, Astron. Astrophys., 61:305.ADSGoogle Scholar
  20. Spitzer, L., Jr., 1987, Dynamical Evolution of Globular Clusters, Princeton University Press, Princeton.Google Scholar
  21. Standish, E.M., 1968, Numerical Studies of the Gravitational Problem of N Bodies, Ph.D. Thesis, Yale University.Google Scholar
  22. Szebeheley, V.G. & Peters, C.F., 1967, Complete Solution of a General Problem of Three Bodies, A.J., 72:876.ADSCrossRefGoogle Scholar
  23. Valtonen, M.J., 1974, Statistics of Three-Body Experiments, in: The Stability of the Solar System and of Small Stellar Systems, Y. Kozai, ed., Reidel, Dordrecht.Google Scholar
  24. Wielen, R., 1968, On the Escape Rate of Stars from Clusters, Bull. Astron., 3:127.Google Scholar

Copyright information

© Plenum Press, New York 1991

Authors and Affiliations

  • Douglas C. Heggie
    • 1
  1. 1.Department of MathematicsUniversity of EdinburghEdinburghUK

Personalised recommendations