Statistical Mechanics of Gravitational Systems with Regular Orbits: Rigid Body Model of Vector Resonant Relaxation

TL;DR

This paper models vector resonant relaxation in gravitational systems by representing orbital planes as rigid tops, deriving their statistical mechanics, and analyzing phase transitions relevant to stellar clusters.

Contribution

It introduces a rigid body model for orbital planes in gravitational systems, extending previous zero kinetic energy models to include non-zero kinetic terms and boundary effects.

Findings

Derived equilibrium states for the effective Hamiltonian.

Analyzed velocity dispersion dependence on temperature.

Expanded the validity range of gravitational phase transitions.

Abstract

I consider a self-gravitating, N-body system assuming that the N constituents follow regular orbits about the center of mass of the cluster, where a central massive object may be present. I calculate the average over a characteristic timescale of the full, N-body Hamiltonian including all kinetic and potential energy terms. The resulting effective system allows for the identification of the orbital planes with N rigid, disk-shaped tops, that can rotate about their fixed common centre and are subject to mutual gravitational torques. The time-averaging imposes boundaries on the canonical generalized momenta of the resulting canonical phase space. I investigate the statistical mechanics induced by the effective Hamiltonian on this bounded phase space and calculate the thermal equilibrium states. These are a result of the relaxation of spins' directions, identified with orbital planes' orientations, which is called vector resonant relaxation. I calculate the dependence of spins' angular velocity dispersion on temperature and calculate the velocity distribution functions. I argue that the range of validity of the gravitational phase transitions, identified in the special case of zero kinetic term by Roupas, Kocsis & Tremaine, is expanded to non-zero values of the ratio of masses between the cluster of N-bodies and the central massive object. The relevance with astrophysics is discussed focusing on stellar clusters. The same analysis performed on an unbounded phase space accounts for continuous rigid tops.