On the stability of motion of N-body systems: the effect of the variation of particle number, softening and rotation

Abstract

Using the Ricci and scalar curvatures of the configuration manifold of gravitational N-body systems, we study the exponential instability in their trajectories. It is found that the exponentiation time-scale for isotropic Plummer spheres varies very little with particle number if the softening is small. Large softening on the other hand has a marked effect and, if large enough, can cause the curvatures to become positive. This last result confirms the previous observations for self gravitating sheets and suggests that the qualitative behaviour of large-N and continuum systems may be different, and that their equivalence is only obtained in the limit of infinite $N$ and finite softening. It is also found that the presence of a large fraction of the kinetic energy in rotational motion increases the exponentiation time-scales significantly --- an effect that should be expected given the regular nature of nearly circular motion. In the light of the results of this and of previous studies, it is suggested that the exponential instability may arise from low order resonances between the period of the variation of the gravitational field due to distant encounters and the orbital period of a test particle. For periods long compared to the exponentiation time but short compared to the diffusion time-scales of the action variables, the standard picture of collisionless dynamics may be valid in an averaged sense. Instead it is suggested that, at least for systems with well defined final states, the relaxation time should scale as $\sim N^{1/2}$.