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