Stable counteralignment of a circumbinary disc

In general, when gas accretes on to a supermassive black hole (SMBH) binary, it is likely to have no prior knowledge of the binary angular momentum. Therefore, a circumbinary disc forms with a random inclination angle ? to the binary. It is known that for ? < 90 degrees the disc will coalign with respect to the binary. If ? > 90 degrees, the disc wholly counteraligns if it satisfies cos ? < -Jd/2Jb, where Jd and Jb are the magnitudes of the disc and binary angular momentum vectors, respectively. If, however, ? > 90 degrees and this criterion is not satisfied, the same disc may counteralign its inner regions and, on longer time-scales, coalign its outer regions. I show that for typical disc parameters, describing an accretion event on to an SMBH binary, a misaligned circumbinary disc is likely to wholly coalign or counteralign with the binary plane. This is because the binary angular momentum dominates the disc angular momentum. However, with extreme parameters (binary mass ratio M2/M1 << 1 or binary eccentricity e similar to 1), the same disc may simultaneously coalign and counteralign. 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.