One can solve the Jeans equation analytically for equilibrated dark matter structures , once given two pieces of input from numerical simulations .
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 v.s .
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 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 } \approx 0 , central density slope of \alpha _ { 0 } \approx - 0.8 , and outer anisotropy of \beta _ { \infty } \approx 0.5 .