Order-disorder phase transition in black-hole star clusters -- III. A
  mono-energetic cluster
  Supermassive black holes at the centres of galaxies are often surrounded by
dense star clusters. For a wide range of cluster properties and orbital radii
the resonant relaxation times in these clusters are much shorter than the
Hubble time. Since resonant relaxation conserves semimajor axes, these clusters
should be in the maximum-entropy state consistent with the given semimajor axis
distribution. We determine these maximum-entropy equilibria in a simplified
model in which all of the stars have the same semimajor axes. We find that the
cluster exhibits a phase transition from a disordered, spherical,
high-temperature equilibrium to an ordered low-temperature equilibrium in which
the stellar orbits have a preferred orientation or line of apsides. Here
`temperature' is a measure of the non-Keplerian or self-gravitational energy of
the cluster; in the spherical state, temperature is a function of the rms
eccentricity of the stars. We explore a simple two-parameter model of
black-hole star clusters -- the two parameters are semimajor axis and
black-hole mass --- and find that clusters are susceptible to the lopsided
phase transition over a range of ~100 in semimajor axis, mostly for black-hole
masses less than $10^{7.5}$ solar masses.
