Chaos in Complex Astrophysical Systems

Ana Nunes

Departamento de Fisica, Faculdade de Ciencias da Universidade de Lisboa,
Campo Grande, Edificio C8, 1749-016 Lisboa, Portugal.

Nuno Pereira

Area Departamental de Matematica, ESTIG/IPB,
Rua Afonso III, no. 1, 7800-050 Beja, Portugal.

Abstract: The dynamics of some astrophysical systems, such as star clusters (N = 102-106) or planetary systems (N = 10), can be modeled by a set of differential equations known as the N-body gravitational problem. The impossibility of solving analytically the general problem when N>2 [1] requires the use of numerical integration when studying these systems.

The sensitivity of the trajectories of the stars to small changes (perturbations) on the initial conditions was first reported by Miller [2]. The improved computational power now available allowed a sistematic study of the typical time scale of this instability in terms of the characteristic crossing time of the system [3].

In this work we present results obtained from numerical integration of the N-body gravitational problem and the associated variational equations of motion, using the NNEWTON package [4]. We use the ``Lyapunov Characteristic Indicator'' [5] to estimate the time scale of the instability associated with the exponential growth of perturbations (variations) on the initial conditions.

Our preliminary results are in good agreement with those of [6] and show a simple relation between the time scale of the instability, the number of particles and the crossing time of the system.

References :

Poincare, H. (1892): Les Methodes Nouvelles de la Mecanique Celeste. Gauthier-Villars, Paris, Vol. (I).
Miller, R. H. (1964): Irreversibility in Small Stellar Dynamics Systems. Astrophysical Journal, 140, 250.
Kandrup, H. E. (1994): Chaos, Regularity, and Noise in Self-Gravitating Systems. An invited plenary talk at: The Seventh Marcel Grossmann Meeting.
[4] Pereira, N. S. A. (2001): Master's Thesis. Faculty of Sciences, University of Lisbon, Portugal.
Heggie, D. C. (1991): Chaos in the N-Body Problem of Stellar Dynamics. Predictability, Stability, and Chaos in N-Body Dynamical Systems, Edited by A. E. Roy, Plenum Press, New York.
Goodman, J., Heggie, D. C., Hut, P. (1993): On the Exponential Instability of N-Body Systems. The Astrophysical Journal, 415:715-733.