An analysis is presented of a numerical investigation of the dynamics and geometry of accretion discs in binary systems with mass ratios q = M _ { 2 } / M _ { 1 } < 0.1 , applicable to ultra-compact X-ray binaries , AM CVn stars and very short period cataclysmic variables .
The steady-state geometry of the disc in the binary reference frame is found to be quite different from that expected at higher mass ratios .
For q \sim 0.1 , the disc takes on the usual elliptical shape , with the major axis aligned perpendicular to the line of centres of the two stars .
However , at smaller mass ratios the elliptical gaseous orbits in the outer regions of the disc are rotated in the binary plane .
The angle of rotation increases with gas temperature , but is found to vary inversely with q .
At q = 0.01 , the major axis of these orbits is aligned almost parallel to the line of centres of the two stars .
These effects may be responsible for the similar disc structure inferred from Doppler tomography of the AM CVn star GP Com ( Morales-Rueda et al .
2003 ) , which has q = 0.02 .
The steady-state geometry at low mass ratios is not predicted by an inviscid , restricted three-body model of gaseous orbits ; it is related to the effects of tidal-viscous truncation of the disc near the Roche lobe boundary .
Since the disc geometry can be inferred observationally for some systems , it is proposed that this may offer a useful diagnostic for the determination of mass ratios in ultra-compact binaries .