Spherical gravitating systems

Due to both the infinite range and singularity of the Newtonian gravitational potential, the thermodynamics of systems with predominantly gravitational forces differs greatly from “chemical” systems dominated by short-range inter-atomic forces. Consider a typical star: As it ages it radiates energy, contracts, and gets hotter. Therefore, its heat capacity is negative, a situation which cannot exist in a chemical system. Here we will consider an idealized class of model gravitational systems consisting of concentric, thin, mass shells. Using both mean field theory and dynamical simulation, we will show that when the singularity is shielded, as it must be in nature, even if the shells are non-rotating, the system can undergo a phase-transition to a more centrally condensed state. We will describe the main features of the transition, show how it contradicts our usual intuition, and discuss possible astrophysical applications. We then discuss the case where the shells are allowed to rotate. We will show that if all the shells have the same squared angular momentum, then the rotational barrier can induce a phase-transition without the need for additional shielding. Finally we use the fact that for a spherical system in the mean field limit there is an additional 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 include the additional integral and, therefore, in the absence of additional shielding, a phase-transition will be excluded.