On the exponential instability of N-body systems

We reconsider the old problem of the growth of numerical errors in N-body integrations. We analyze the effects of successive encounters and show that these tend to magnify errors on a time scale which is comparable with the crossing time. This conclusion is based on an approximate treatment of encounters which can be analyzed in three ways: by construction of a master equation, by approximate analytic methods, and by simulation. However, a deeper discussion of the manner in which errors propagate from one star to another implies that the true rate of growth varies as ln ln N/t_cr. Next we study the growth of errors in N-body simulations, in which all gravitational interactions are correctly included. We confirm our earlier results and recent work of Kandrup & Smith that the rate of growth of errors per crossing time increases as N increases up to about 30, but that for larger systems the rate of growth is approximately independent of N. In this limit, errors grow with an e-folding time which is nearly one-tenth of a crossing time. This demonstrates that the N-dependence observed in the pioneering investigation by Miller, who considered systems with N ≤ 32, cannot be extended to larger N. We also investigate the rate of growth of errors in N-body systems with softened potentials. For example, when the softening radius is held at a fixed fraction of the size of the system, the rate of growth of errors varies approximately as N^-1/3 when N is large enough. In the final section of the paper we summarize arguments to show that two-body interactions, and not collective effects, are the underlying physical mechanism causing the growth of errors in spherical stellar systems in dynamic equilibrium. We explain the distinction between this instability and two-body relaxation, and discuss its implications for N-body simulations. For example, it can be shown that the accurate simulation of a system up to the time of core collapse would require computations with O(N) decimal places. After core collapse the rate of growth of errors is still faster, because of three-body interactions involving binaries.