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.

We consider a simple 1D model of hard rods with gravitational interactions. First, we consider the situation where the sytem is enclosed in a box with finite size and we exactly compute the equilibrium thermodynamical quantities. Thanks to the confining nature of gravity in 1D which prevents evaporation, the box can be released and we can study an open system with its center of mass fixed. Then, we exactly compute the corresponding equilibrium density profile within the microcanonical ensemble. All those analytical results are discussed in connection with the general issue of ensemble inequivalences for systems with long-range interactions. They also provide specific tests for the reliability of the hydrostatic approach combined with a mean-field treatment of gravitational interactions. In particular, the hydrostatic approach is shown to fail for energies close to the collapse energy where a core-halo structure emerges.

We introduce a simple model of hard spheres with gravitational interactions, for which we study a suitable scaling limit. Usual extensive properties are maintained notwithstanding the long range of gravitational interaction. We show that a local thermalization spontaneously emerges within a microcanonical description of the stationary state. In the considered scaling limit, the resulting density profile can be determined in a hydrostatic approach.