Spin Evolution of Supermassive Black Holes and Galactic Nuclei
  The spin angular momentum S of a supermassive black hole (SBH) precesses due
to torques from orbiting stars, and the stellar orbits precess due to dragging
of inertial frames by the spinning hole. We solve the coupled post-Newtonian
equations describing the joint evolution of S and the stellar angular momenta
Lj, j = 1...N in spherical, rotating nuclear star clusters. In the absence of
gravitational interactions between the stars, two evolutionary modes are found:
(1) nearly uniform precession of S about the total angular momentum vector of
the system; (2) damped precession, leading, in less than one precessional
period, to alignment of S with the angular momentum of the rotating cluster.
Beyond a certain distance from the SBH, the time scale for angular momentum
changes due to gravitational encounters between the stars is shorter than
spin-orbit precession times. We present a model, based on the
Ornstein-Uhlenbeck equation, for the stochastic evolution of star clusters due
to gravitational encounters and use it to evaluate the evolution of S in nuclei
where changes in the Lj are due to frame dragging close to the SBH and to
encounters farther out. Long-term evolution in this case is well described as
uniform precession of the SBH about the cluster's rotational axis, with an
increasingly important stochastic contribution when SBH masses are small. Spin
precessional periods are predicted to be strongly dependent on nuclear
properties, but typical values are 10-100 Myr for low-mass SBHs in dense
nuclei, 100 Myr - 10 Gyr for intermediate mass SBHs, and > 10 Gyr for the most
massive SBHs. We compare the evolution of SBH spins in stellar nuclei to the
case of torquing by an inclined, gaseous accretion disk.
