An upper limit to the velocity dispersion of relaxed stellar systems
  without massive black holes
  Massive black holes have been discovered in all closely examined galaxies
with high velocity dispersion. The case is not as clear for lower-dispersion
systems such as low-mass galaxies and globular clusters. Here we suggest that
above a critical velocity dispersion of roughly 40 km/s, massive central black
holes will form in relaxed stellar systems at any cosmic epoch. This is because
above this dispersion primordial binaries cannot support the system against
deep core collapse. If, as previous simulations show, the black holes formed in
the cluster settle to produce a dense subcluster, then given the extremely high
densities reached during core collapse the holes will merge with each other.
For low velocity dispersions and hence low cluster escape speeds, mergers will
typically kick out all or all but one of the holes due to three-body kicks or
the asymmetric emission of gravitational radiation. If one hole remains, it
will tidally disrupt stars at a high rate. If none remain, one is formed after
runaway collisions between stars, then it tidally disrupts stars at a high
rate. The accretion rate after disruption is many orders of magnitude above
Eddington. If, as several studies suggest, the hole can accept matter at that
rate because the generated radiation is trapped and advected, then it will grow
quickly and form a massive central black hole.
