Stable counteralignment of a circumbinary disc
  In general, when gas accretes on to a supermassive black hole binary it is
likely to have no prior knowledge of the binary angular momentum. Therefore a
circumbinary disc forms with a random inclination angle, theta, to the binary.
It is known that for theta < 90 degrees the disc will coalign wrt the binary.
If theta > 90 degrees the disc wholly counteraligns if it satisfies cos(theta)
< -J_d/2J_b, where J_d and J_b are the magnitudes of the disc and binary
angular momentum vectors respectively. If however theta > 90 degrees and this
criterion is not satisfied the same disc may counteralign its inner regions
and, on longer timescales, coalign its outer regions. I show that for typical
disc parameters, describing an accretion event on to a supermassive black hole
binary, a misaligned circumbinary disc is likely to wholly co-- or
counter--align with the binary plane. This is because the binary angular
momentum dominates the disc angular momentum. However with extreme parameters
(binary mass ratio M_2/M_1 << 1 or binary eccentricity e ~ 1) the same disc may
simultaneously co- and counter-align. It is known that coplanar prograde
circumbinary discs are stable. I show that coplanar retrograde circumbinary
discs are also stable. A chaotic accretion event on to an SMBH binary will
therefore result in a coplanar circumbinary disc that is either prograde or
retrograde with respect to the binary plane.
