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

International audience; We consider a system of hard spheres with gravitational interactions ina stationary state described in terms of the microcanonical ensemble. Weintroduce a set of similar auxiliary systems with increasing sizes and numbersof particles. The masses and radii of the hard spheres of the auxiliary systemsare rescaled in such a way that the usual extensive properties are maintaineddespite the long-range nature of the gravitational interactions, while the massdensity and packing fractions are kept fixed. We show, within that scalinglimit, that a local thermalization spontaneously emerges as a consequence ofboth extensive properties and the relative smallness of the fluctuations. Theresulting mass density profile for the infinite system can be determined withina hydrostatic approach, where the gradient of the local hard-sphere pressure isbalanced by the average gravitational field. The derivation sheds light on themechanisms which ensure that the local equilibrium in the infinite system isentirely controlled by hard-core interactions, while gravitational interactionscan be treated at the mean-field level. This allows us to determine the conditionsunder which the hydrostatic approach is also valid for the actual finite systemof interest. We provide simple tests of such conditions for a few astrophysicalexample