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We investigate a mechanism for a super-massive black hole at the center of agalaxy to wander in the nucleus region. A situation is supposed in which thecentral black hole tends to move by the gravitational attractions from thenearby molecular clouds in a nuclear bulge but is braked via the dynamicalfrictions by the ambient stars there. We estimate the approximate kineticenergy of the black hole in an equilibrium between the energy gain rate throughthe gravitational attractions and the energy loss rate through the dynamicalfrictions, in a nuclear bulge composed of a nuclear stellar disk and a nuclearstellar cluster as observed from our Galaxy. The wandering distance of theblack hole in the gravitational potential of the nuclear bulge is evaluated toget 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 theequilibrium solution between the energy gain and loss disappears when the blackhole mass exceeds an upper limit. As a result, we can expect the followingscenario for the evolution of the black hole mass: When the black hole mass issmaller than the upper limit, mass accretion of the interstellar matter in thecircum-nuclear region, causing the AGN activities, makes the black hole masslarger. However, when the mass gets to the upper limit, the black hole losesthe balancing force against the dynamical friction and starts spiralingdownward to the gravity center. From simple parameter scaling, the upper masslimit of the black hole is found to be proportional to the bulge mass and thiscould explain the observed correlation of the black hole mass with the bulgemass.

Wandering of the central black hole in a galactic nucleus and correlation of the black hole mass with the bulge mass