Validity conditions of the hydrostatic approach for self-gravitating systems: a microcanonical analysis

Abstract

We consider a system of hard spheres with gravitational interactions in a stationary state described in terms of the microcanonical ensemble. We introduce a set of similar auxiliary systems with increasing sizes and numbers of particles. The masses and radii of the hard spheres of the auxiliary systems are rescaled in such a way that the usual extensive properties are maintained despite the long-range nature of the gravitational interactions, while the mass density and packing fractions are kept fixed. We show, within that scaling limit, that a local thermalization spontaneously emerges as a consequence of both extensive properties and the relative smallness of the fluctuations. The resulting mass density profile for the infinite system can be determined within a hydrostatic approach, where the gradient of the local hard-sphere pressure is balanced by the average gravitational field. The derivation sheds light on the mechanisms which ensure that the local equilibrium in the infinite system is entirely controlled by hard-core interactions, while gravitational interactions can be treated at the mean-field level. This allows us to determine the conditions under which the hydrostatic approach is also valid for the actual finite system of interest. We provide simple tests of such conditions for a few astrophysical examples.