  We demonstrate that the quasi-equilibrium state in self-gravitating $N$-body
system after cold collapse are uniquely characterized by the local virial
relation using numerical simulations. Conversely assuming the constant local
virial ratio and Jeans equation for spherically steady state system, we
investigate the full solution space of the problem under the constant
anisotropy parameter and obtain some relevant solutions. Especially, the local
virial relation always provides a solution which has a power law density
profile in both the asymptotic regions $r\to 0$ and $\infty$. This type of
solutions observed commonly in many numerical simulations. Only the anisotropic
velocity dispersion controls this asymptotic behavior of density profile.

Astrophysics