Linear Temperature-Mass relation and Local virial relation: Two   hypotheses for self-gravitating systems
  We propose two hypotheses which characterize the quasi-equilibrium state that realizes after a cold collapse of self-gravitating system. The first hypothesis is the linear temperature-mass (TM) relation, which yields a characteristic non-Gaussian velocity distribution. The second is the local virial (LV) relation, which, combining the linear TM relation, determines a unique mass density profile as $\rho(r)={\rho}%_{0}r^{-4}e^{-r_0/r}$. Although this density profile is unphysical in the central region, the region is just inner a few percent around the center of a bound region in cumulative mass, which is beyond the resolution of our numerical simulations. Hence posing two hypotheses is compatible to the numerical simulations for almost the whole region of the virialized bound state. Actually, except for this inner part, this density profile fits well to the data of cold collapse simulations. Two families of spherical and isotropic models, polytropes and King models, are examined from a view point of these two hypotheses. We found that the LV relation imposes a strong constraint on these models: only polytropes with index $n \sim 5$ such as Plummer's model are compatible with the numerical results characterized by the two hypotheses among these models. King models with the concentration parameter $% c \sim 2$ violate the LV relation while they are consistent with the $R^{1/4} $ law for the surface brightness. Hence the above characteristics can serve as a guideline to build up the models for the bound state after a cold collapse, besides the conventional criteria concerning the asymptotic behavior. 
Y. Sota, O. Iguchi, M. Morikawa and A. Nakamichi