Two-body relaxation in softened potentials

Semi-analytical calculations for the two-body relaxation in softened potentials are presented and compared with N-body simulations. With respect to a Keplerian potential the increase of the relaxation time in the modified potentials is generally less than one order of magnitude, typically between 2 and 5, if the softening length is of the order of the mean interparticle distance. A comparison between two frequently applied softening schemes, one based on a Plummer mass distribution, the other defined by a spline interpolation on a compact support, shows that the spline-based procedure gives for the same central potential depth systematically smaller increases, but the differences between both are only of the order of 20-40%. If the softening length of the spline based scheme is twice the Plummer softening length, the relaxation rates become almost identical. A simple model that assumes that the influence of softening can be described by neglecting all deflections, if the impact parameter is less than the softening length, and otherwise uses Keplerian orbits, is within a factor of 2 in agreement with the detailed calculations. Measuring the relaxation times by N-body simulations gives a fair agreement with the semianalytical results and, also with the simple model. In order to reproduce the measured relaxation time ratios the upper limit pmax of the impact parameter used to calculate the Coulomb logarithm must be chosen of the order of the system’s size instead of themean interparticle distance which corroborates the results of Farouki & Salpeter (1982, 1994).