Orbital Divergence and Relaxation in the Gravitational N-Body Problem

Abstract

One of the fundamental aspects of statistical behaviour in many-body systems is exponential divergence of neighbouring orbits, which is often discussed in terms of Liapounov exponents. Here we study this topic for the classical gravitational N-body problem. The application we have in mind is to old stellar systems such as globular star clusters, where N∼10⁶, and so we concentrate on spherical, centrally concentrated systems with total energy E<0. Hitherto no connection has been made between the time scale for divergence (denoted here by t _e ) and the time scale on which the energies of the particles evolve because of two-body encounters (i.e., the two-body relaxation time scale, t _r ), even though both may be calculated by similar considerations. In this paper we give a simplified model showing that divergence in phase space is initially roughly exponential, on a timescale proportional to the crossing time (defined as a mean time for a star to cross from one side of the system to another). In this phase t _e <<t _r , if N is not too small (i.e., N≫30). After several e-folding times, the model shows that the divergence slows down. Thereafter the divergence (measured by the energies of the bodies) varies with time as t ^1/2, on a timescale nearly proportional to the familiar two-body relaxation timescale, i.e., t _e ∼t _r in this phase. These conclusions are illustrated by numerical results.