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I consider a self-gravitating, N-body system assuming that the N constituentsfollow regular orbits about the center of mass of the cluster, where a centralmassive object may be present. I calculate the average over a characteristictimescale of the full, N-body Hamiltonian including all kinetic and potentialenergy terms. The resulting effective system allows for the identification ofthe orbital planes with N rigid, disk-shaped tops, that can rotate about theirfixed common centre and are subject to mutual gravitational torques. Thetime-averaging imposes boundaries on the canonical generalized momenta of theresulting canonical phase space. I investigate the statistical mechanicsinduced by the effective Hamiltonian on this bounded phase space and calculatethe thermal equilibrium states. These are a result of the relaxation of spins'directions, identified with orbital planes' orientations, which is calledvector resonant relaxation. I calculate the dependence of spins' angularvelocity dispersion on temperature and calculate the velocity distributionfunctions. I argue that the range of validity of the gravitational phasetransitions, identified in the special case of zero kinetic term by Roupas,Kocsis & Tremaine, is expanded to non-zero values of the ratio of massesbetween the cluster of N-bodies and the central massive object. The relevancewith astrophysics is discussed focusing on stellar clusters. The same analysisperformed on an unbounded phase space accounts for continuous rigid tops.

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