Order-Disorder Phase Transition in Black-Hole Star Clusters

The centers of most galaxies contain massive black holes surrounded by dense star clusters. The structure of these clusters determines the rate and properties of observable transient events, such as flares from tidally disrupted stars and gravitational-wave signals from stars spiraling into the black hole. Most estimates of these rates enforce spherical symmetry on the cluster. Here we show that, in the course of generic evolutionary processes, a star cluster surrounding a black hole can undergo a robust phase transition from a spherical thermal equilibrium to a lopsided equilibrium, in which most stars are on high-eccentricity orbits with aligned orientations. The rate of transient events is expected to be much higher in the ordered phase. Better models of cluster formation and evolution are needed to determine whether clusters should be found in the ordered or disordered phase.

Figure 1

Lopsided phase transition in a maximum-entropy cluster of stars. The cluster stars have a single semimajor axis and zero total angular momentum. The vertical axis shows the magnitude of the mean eccentricity vector. The horizontal axes show the self-gravitational energy, as well as the rms eccentricity $e_{\mathrm{rms}}^{\mathrm{sph}}$ and inverse temperature $\beta$ of the spherical system with the same energy. Open black squares show the system described in the section on maximum-entropy states of a monoenergetic cluster, while filled blue circles with error bars are from the section on MCMC simulations. The units of $E$ and ${\beta }^{-1}$ are $G{M}_{\star }^2/a_0$.

Figure 2

Secular evolution of a cluster of stars. The initial cluster is composed of $N_0=2048$ stars with a single semimajor axis $a_0$ and zero total angular momentum. The evolution is followed with an $N$-wire code [13]. Stars passing within $0.01a_0$ of the central black hole are removed, so the number $N$ of stars declines with time (black curve). The energy $E$ in units of $G{M}_{\star }^2(t)/a_0$ also declines with time (red curve), and the mean eccentricity vector $|⟨\mathbf{e}⟩|$ grows (blue curve). The bottom axis shows the time in units of $t_{\mathrm{dyn}}{M}_{•}/m$, the dynamical time multiplied by the ratio of black-hole to stellar mass, which apart from dimensionless constants equals the resonant relaxation time when relativistic precession is ignored. The red cross marks the critical energy for the phase transition, $E=-0.38$, according to the MCMC simulation of Fig. 1.