Steepest-descent technique and stellar equilibrium statistical mechanics. I - Newtonian clusters in a box

Abstract

A microcanonical statistical mechanics formulation has been developed for nonrotating systems of stars of equal mass. The system, placed in a confining volume and with a cutoff of the interparticle gravitational interaction at short distances, can have thermodynamic equilibrium states. Sequences of equilibrium states are presumed to simulate slowly evolving, near-equilibrium configurations of real star clusters. An exact functional expression for the entropy of such systems is derived which has also a relativistic counterpart. The entropy is evaluated in an approximation which is mean field plus fluctuations. Evaluations beyond this approximation can readily be carried out. We obtain the necessary and sufficient conditions for spherically symmetric clusters to be thermodynamically stable about a mean field solution, with respect to arbitrary fluctuations in the microcanonical ensemble. The stability conditions amount to the following quantities having definite signs: (i) a functional form, quadratic in ''mean field'' fluctuations, (ii) the derivative of the gravito-chemical potential with respect to the number of particles, at fixed temperature, being positive definite, and (iii) the heat capacity C/sub ..nu../, at fixed number of particles, being positive definite. In a sequence of equilibrium configurations in which the ratio of densities between the center and the boundary of themore » cluster is progressively increased, conditions (i) and (ii) break down simultaneously when this density contrast is equal to 1.58. Condition (i) remains unsatisfied for higher density contrasts. The limit 1.58 on the density contrast is much more stringent than that given by condition (iii) which breaks down only for a value of 32.1. Our results are in sharp contrast to those of Antonov's criterion according to which instabilities appear when the density contrast is higher than 709. Time scales of evolutions of various unstable configurations are not considered in this work. (AIP)« less