Statistical Mechanics of Gravitational Systems with Regular Orbits: Rigid Body Model of Vector Resonant Relaxation. (arXiv:1910.05735v1 [astro-ph.GA])
<a href="http://arxiv.org/find/astro-ph/1/au:+Roupas_Z/0/1/0/all/0/1">Zacharias Roupas</a>
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 star clusters. The same analysis
performed on an unbounded phase space accounts for continuous rigid tops.
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 star clusters. The same analysis
performed on an unbounded phase space accounts for continuous rigid tops.
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