Evolution Of Binary Supermassive Black Holes In Rotating Nuclei
  Interaction of a binary supermassive black hole with stars in a galactic
nucleus can result in changes to all the elements of the binary's orbit,
including the angles that define its orientation. If the nucleus is rotating,
the orientation changes can be large, causing large changes in the binary's
orbital eccentricity as well. We present a general treatment of this problem
based on the Fokker-Planck equation for f, defined as the probability
distribution for the binary's orbital elements. First- and second-order
diffusion coefficients are derived for the orbital elements of the binary using
numerical scattering experiments, and analytic approximations are presented for
some of these coefficients. Solutions of the Fokker-Planck equation are then
derived under various assumptions about the initial rotational state of the
nucleus and the binary hardening rate. We find that the evolution of the
orbital elements can become qualitatively different when we introduce nuclear
rotation: 1) the orientation of the binary's orbit evolves toward alignment
with the plane of rotation of the nucleus; 2) binary orbital eccentricity
decreases for aligned binaries and increases for counter-aligned ones. We find
that the diffusive (random-walk) component of a binary's evolution is small in
nuclei with non-negligible rotation, and we derive the time-evolution equations
for the semimajor axis, eccentricity and inclination in that approximation. The
aforementioned effects could influence gravitational wave production as well as
the relative orientation of host galaxies and radio jets.
