Spherical gravitating systems

Due to both the infinite range and singularity of the Newtonian gravitation al potential, the thermodynamics of systems with predominantly gravitationa l forces differs greatly from "chemical" systems dominated by short-range i nter-atomic forces. Consider a typical star: As it ages it radiates energy, contracts, and gets hotter. Therefore, its heat capacity is negative, a si tuation which cannot exist in a chemical system. Here we will consider an i dealized class of model gravitational systems consisting of concentric, thi n, mass shells. Using both mean field theory and dynamical simulation, we w ill show that when the singularity is shielded, as it must be in nature, ev en if the shells are non-rotating, the system can undergo a phase-transitio n to a more centrally condensed state. We will describe the main features o f the transition, show how it contradicts our usual intuition, and discuss possible astrophysical applications. We then discuss the case where the she lls are allowed to rotate. We will show that if all the shells have the sam e squared angular momentum, then the rotational barrier can induce a phase- transition without the need for additional shielding. Finally we use the fa ct that for a spherical system in the mean field limit there is an addition al integral of the motion besides the energy, namely the sum of the squares of the angular momentum, to introduce and study two new ensembles. We show that gravothermal catastrophe is still possible in generalizations of both the canonical ensemble (CE) and microcanonical ensemble (MCE) which includ e the additional integral and, therefore, in the absence of additional shie lding, a phase-transition will be excluded. (C) 2001 Elsevier Science Ltd. All rights reserved.