Chaos and the continuum limit in the gravitational N-body problem. I. Integrable potentials

Abstract

This paper summarises a numerical investigation of the statistical properties of orbits evolved in `frozen,' time-independent N-body realisations of smooth, time-independent density distributions, allowing for 10^(2.5)<N<10^(5.5). Two principal conclusions were reached: (1) In the limit of a nearly `unsoftened' two-body kernel, the value of the largest Lyapunov exponent does NOT appear to decrease systematically with increasing N. (2) Nevertheless, there is a clear quantifiable sense in which, on the average, as N increases chaotic orbits in the frozen-N systems come to more closely approximate characteristics in the smooth potential. When viewed in configuration or velocity space, or as probed by collisionless invariants like angular momentum, frozen-N orbits typically diverge from smooth potential characteristics as a power law in time on a time scale proportional to N^(1/2)t_D, with t_D a characteristic dynamical, or crossing, time.