We study the interplay between mass transfer , accretion and gravitational torques onto a black hole binary migrating in a self-gravitating , retrograde circumbinary disc .
A direct comparison with an identical prograde disc shows that : ( i ) because of the absence of resonances , the cavity size is a factor \mathfrak { a } ( 1+ \mathfrak { e } ) smaller for retrograde discs ; ( ii ) nonetheless the shrinkage of a circular binary semi–major axis \mathfrak { a } is identical in both cases ; ( iii ) a circular binary in a retrograde disc remains circular while eccentric binaries grow more eccentric .
For non-circular binaries , we measure the orbital decay rates and the eccentricity growth rates to be exponential as long as the binary orbits in the plane of its disc .
Additionally , for these co-planar systems , we find that interaction ( \sim non–zero torque ) stems only from the cavity edge plus \mathfrak { a } ( 1+ \mathfrak { e } ) in the disc , i.e .
for dynamical purposes , the disc can be treated as a annulus of small radial extent .
We find that simple ’ dust ’ models in which the binary- disc interaction is purely gravitational can account for all main numerical results , both for prograde and retrograde discs .
Furthermore , we discuss the possibility of an instability occurring for highly eccentric binaries causing it to leave the disc plane , secularly tilt and converge to a prograde system .
Our results suggest that there are two stable configurations for binaries in self-gravitating discs : the special circular retrograde case and an eccentric ( \mathfrak { e } \sim 0.6 ) prograde configuration as a stable attractor .