On the sensitivity of the N-body problem toward small changes in initial conditions. 4

Abstract

Earlier papers in this series summarized numerical simulations which indicate that, largely independent of the form of the initial state or the initial perturbation, the exponential instability of the gravitational N-body problem toward small changes in initial conditions is an extremely robust phenomenon which proceeds on a timescale t_*_ ~ t_cr_, with t_cr_ a typical crossing time. This paper continues these investigations. It is shown that the total perturbations in energy and angular momentum grow on essentially the same timescale as the total perturbations in position and velocity until the configuration space perturbation of a typical particle becomes comparable to the "size" of the system. At this point the growth rate decelerates. The subsequent growth of the perturbation is tracked by examining the loss of correlation between pairs of perturbed and unperturbed simulations. It is shown that the unperturbed and perturbed positions, velocities, energies, and angular momenta all decorrelate rapidly, although the timescale for the energy and angular momentum is somewhat longer than that for position and velocity. Additional simulations were also analyzed to reinforce two earlier conclusions. (1) A systematic investigation of the effects of varying the initial virial ratio Q_0_ indicates that, as far as this instability is concerned, virial equilibrium is not special. (2) An investigation of the dependence on total particle number N varying from 30 to 4000 indicates that, when expressed in units of t_cr_, the timescale for the instability is essentially independent of N, for N>200 or so.