Orbital Divergence and Relaxation in the Gravitational N-Body Problem

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~10^6, 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 two-body relaxation time scale (t_r), even though both may be calculated by consideration of two-body encounters. 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 is much less than t_r, if N is not too small (i.e. N is much more than about 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.

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