Order-disorder phase transition in black-hole star clusters. II. A
  scale-free cluster
  The supermassive black holes found at the centers of galaxies are often
surrounded by dense star clusters. The ages of these clusters are generally
longer than the resonant-relaxation time and shorter than the two-body
relaxation time over a wide range of radii. We explore the thermodynamic
equilibria of such clusters using a simple self-similar model. We find that the
cluster exhibits a phase transition between a high-temperature spherical
equilibrium and a low-temperature equilibrium in which the stars are on
high-eccentricity orbits with nearly the same orientation. In the absence of
relativistic precession, the spherical equilibrium is metastable below the
critical temperature and the phase transition is first-order. When relativistic
effects are important, the spherical equilibrium is linearly unstable below the
critical temperature and the phase transition is continuous. A similar phase
transition has recently been found in a model cluster composed of stars with a
single semimajor axis. The presence of the same phenomenon in two quite
different cluster models suggests that lopsided equilibria may form naturally
in a wide variety of black-hole star clusters.
