Wandering of the central black hole in a galactic nucleus and
  correlation of the black hole mass with the bulge mass
  We investigate a mechanism for a super-massive black hole at the center of a
galaxy to wander in the nucleus region. A situation is supposed in which the
central black hole tends to move by the gravitational attractions from the
nearby molecular clouds in a nuclear bulge but is braked via the dynamical
frictions by the ambient stars there. We estimate the approximate kinetic
energy of the black hole in an equilibrium between the energy gain rate through
the gravitational attractions and the energy loss rate through the dynamical
frictions, in a nuclear bulge composed of a nuclear stellar disk and a nuclear
stellar cluster as observed from our Galaxy. The wandering distance of the
black hole in the gravitational potential of the nuclear bulge is evaluated to
get as large as several 10 pc, when the black hole mass is relatively small.
The distance, however, shrinks as the black hole mass increases and the
equilibrium solution between the energy gain and loss disappears when the black
hole mass exceeds an upper limit. As a result, we can expect the following
scenario for the evolution of the black hole mass: When the black hole mass is
smaller than the upper limit, mass accretion of the interstellar matter in the
circum-nuclear region, causing the AGN activities, makes the black hole mass
larger. However, when the mass gets to the upper limit, the black hole loses
the balancing force against the dynamical friction and starts spiraling
downward to the gravity center. From simple parameter scaling, the upper mass
limit of the black hole is found to be proportional to the bulge mass and this
could explain the observed correlation of the black hole mass with the bulge
mass.
