Thermodynamics of Self-Gravitating Systems with Softened Potentials

The microcanonical statistical mechanics of a set of self-gravitating particles is analyzed in mean-field approach. In order to deal with an upper bounded entropy functional, a softened gravitational potential is used. The softening is achieved by truncating to N terms an expansion of the Newtonian potential in spherical Bessel functions. The order N is related to the softening at short distances. This regularization has the remarkable property that it allows for an exact solution of the mean field equation. It is found that for N not too large the absolute maximum of the entropy coincides to high accuracy with the solution of the Lane-Emden equation, which determines the mean field mass distribution for the Newtonian potential for energies larger than $E_c\approx -0.335 G M^2/R$. Below this energy a collapsing phase transition, with negative specific heat, takes place. The dependence of this result on the regularizing parameter N is discussed.