Effects of Massive Central Black Holes on Dense Stellar Systems

Abstract

We discuss some dynamical and astrophysical consequences of the presence of a massive black hole, of mass Mh in a dense stellar system, applying our results to the cores of globular clusters and to galactic nuclei. The black hole would dominate the dynamics of stars out to a radius |${r}_\text{h}\,\simeq \,{GM}_\text{h}/{\upsilon}_c\,^{2},\,{\upsilon }_{c}$| being the velocity dispersion in the core. Within rh, the stellar velocity dispersion is proportional to r–1/2 and the stellar density n(r) may be enhanced. A quasisteady state can be established, involving a steady influx of stars which are swallowed or disrupted near the hole. We define and calculate a ‘critical radius’ rcrit such that most stars on orbits with |$r\,\lesssim\,{r}_\text{crit}$| diffuse into lowangular momentum ‘loss-cone ’ orbits (and are swallowed) in the ‘reference time ’ tR; whereas outside rcrit, loss-cone effects are negligible and a typical star diffuses inward into a more tightly-bound orbit on a time scale |$\sim {t}_\text{R}({r}/{r}_\text{crit})$|⁠. The density profile in the cusp is |$n\text(r)\,\propto \,{r}^{-7/4}\,\text{for}\,{r}_{crit}\,\lesssim\,r\,\lesssim\,{r}_\text{h};$| and somewhat flatter inside rcrit. Generally rcrit is larger than either the tidal radius rT of the hole or the ‘collision radius ’ rcoll at which |${GM}_\text{h}/{r}_\text{coll}$| equals the binding energy per unit mass of a typical star: indeed, in some cases |${r}_\text{crit}\,\gtrsim\,{r}_\text{h}.$| The swallowing or disruption rate of stars varies as M4/3 when |${r}_\text{crit}\,\gtrsim\,{r}_\text{h},$|M61/27 when |${r}_\text{h}\,\lesssim\,{r}_\text{crit}\,\lesssim{r}_\text{coll}$| and M3 when |${r}_\text{crit}\,\lesssim\,{r}_\text{coll.}$| We discuss some consequences of stellar disruption and tidal capture by black holes of |${10}^{3-}\,{10}^{4}\,{M}_{\odot}$| in globular clusters: X-ray emission, possible optical or ultraviolet ‘flares ’, and the likelihood of there being a captured star in orbit near rT. Finally, we briefly apply our considerations to Hills' quasar model, which invokes |$\gtrsim\,{10}^{7}\,{M}_{\odot}$| black holes in galactic nuclei.