Optimal Softening for N -body Simulations

We propose a strategy for determining the optimal softening length in N -body simulations. This strategy ensures that the ensemble-average acceleration error is minimal at a small radius r0, and that the error never exceeds the optimal value beyond r0. Analytic formulae are derived for calculating the acceleration error for any halo profile or softened gravity so that the optimal softening length ǫopt can be determined without N -body force evaluations. For Navarro-Frenk-White halos with concentration parameter c = 5, we find, by setting r0 to one percent of the virial radius rv, that ǫopt = 0.11N −0.20 for S2 softening (Hockney & Eastwood) and ǫopt = 0.065N −0.26 for Plummer softening, where N is the number of particles within the virial radius, and ǫopt is in units of rv. We also estimate that current state-of-the-art halo simulations suffer & 6% acceleration errors at 0.01rv, which grow rapidly toward the center of the halo. Given that the mass within 0.01rv is only a few thousandths of the halo virial mass, even a few percent acceleration errors may contribute significantly to the uncertainties in the inner slope of the halo. Subject headings: dark matter — galaxies: halos — methods: analytical — methods: N -body simulations