The velocity anisotropy - density slope relation

One can solve the Jeans equation analytically for equilibrated dark matter structures once two pieces of input from numerical simulations have been obtained. These inputs are ( 1) a connection between phase-space density and radius, and ( 2) a connection between velocity anisotropy and density slope, the alpha - beta relation. The first (phase-space density versus radius) has already been analysed through several different simulations, however the second (alpha - beta relation) has not been quantified yet. We perform a large set of numerical experiments in order to quantify the slope and zero-point of the alpha - beta relation. We find a strong indication that the relation is indeed an attractor. When combined with the assumption of phase-space being a power-law in radius, this allows us to conclude that equilibrated dark matter structures indeed have zero central velocity anisotropy beta(0) approximate to 0, a central density slope of a(0) approximate to - 0.8, and outer anisotropy of beta(infinity) approximate to 0.5.