ON THE SENSITIVITY OF THE N-BODY PROBLEM TO SMALL CHANGES IN INITIAL CONDITIONS .3.

The first two papers in this series showed that, largely independent of initial conditions, a nontrivial mass function, or the total particle number N, the exponential instability of the gravitational N-body problem toward small changes in initial conditions proceeds on a time scale somewhat shorter than a typical crossing time t(cr). This paper investigates the effects of increasing the softening parameter epsilon in the two-body forces, with the aim of ascertaining the relative importance for this instability of close encounters and the bulk mean field. This is probed by considering both (a) the mean e-folding times t* for individual particle perturbations and (b) the e-folding times t(LAMBDA) for the total 3N-dimensional configuration space perturbation, which continues to grow exponentially until the typical perturbation \deltar\ of an individual particles becomes comparable to the size of the system. At least for values of epsilon smaller than the typical interparticle spacing 1, both time scales are well fitted by growth laws t(epsilon/l) = t(0) + A(epsilon/l)p, where p* almost-equal-to 1.4, p(LAMBDA) almost-equal-to 1.1, and, in units of t(cr), A* and A(LAMBDA) are of order unity. For larger values of epsilon, the growth of the time scales t(epsilon/l) gives indications of slowing down. In particular, there is no sense in which the instability turns off abruptly for epsilon > l. Close encounters and the bulk mean field are both important and apparently reinforce each other: the relatively smooth, systematic tidal force associated with the mean field tends to amplify divergences associated with ''kicks'' induced by shorter range effects, which remain important even for large N. This indicates that an analysis of the stability of stability of individual orbits in a mean field potential does not completely probe the true instability of the N-body problem, even for a system in a collisionless macroscopic equilibrium. One must also allow for interactions between nearby perturbed orbits separated by distances less than or similar l.