ON THE SENSITIVITY OF THE N-BODY PROBLEM TO SMALL CHANGES IN INITIAL CONDITIONS

This paper summarizes the results of a considerable number of numerical simulations which investigate the instability of the gravitational N-body problem towards small, "random" changes in the initial positions of the particles. The specific aims are (1) to follow the growths of the perturbations of individual particles in the simulation, analyzing their statistical properties, (2) to investigate how these statistical properties vary as the number of particles N in the simulation is changed from 16 to 340, and (3) to determine how these properties vary if one changes the "softening parameter." Principal conclusions obtained for the simulations considered include the following: (1) The mean e-folding time t* for the perturbation in the location of some particle is comparable to the dynamical, or crossing, time t(cr) for all N. (2) There is, however, a weak N-dependence, t*/t(cr) decreasing as N increases but tending towards an asymptotic value for large N. (3) The distribution of e-folding times for a given N has a distinctive and comparatively reproducible shape, the reproducibility improving as N grows. (4) Increasing the "softening parameter" leads to a systematic increase in t*, but this time scale remains comparable to t(cr) even when the "softening parameter" becomes so large as to be comparable to the interparticle spacing.