Three periodically variable stars have recently been discovered (V407 Vul, P=9.5 min; ES Cet, P=10.3 min; RX J0806.3+1527, P=5.3 min) with properties that suggest that their photometric periods are also their orbital periods, making them the most compact binary stars known. If true, this might indicate that close, detached, double white dwarfs are able to survive the onset of mass transfer caused by gravitational wave radiation and emerge as the semi-detached, hydrogen-deficient stars known as the AM CVn stars. The accreting white dwarfs in such systems are large compared to the orbital separations. This has two effects: first it makes it likely that the mass transfer stream can hit the accretor directly, and second it causes a loss of angular momentum from the orbit which can destabilise the mass transfer unless the angular momentum lost to the accretor can be transferred back to the orbit. The effect of the destabilisation is to reduce the number of systems which survive mass transfer by as much as one hundred-fold. In this paper we analyse this destabilisation and the stabilising effect of a dissipative torque between the accretor and the binary orbit. We obtain analytic criteria for the stability of both disc-fed and direct impact accretion, and carry out numerical integrations to assess the importance of secondary effects, the chief one being that otherwise stable systems can exceed the Eddington accretion rate. We show that to have any effect upon survival rates, the synchronising torque must act on a timescale of order 1000 years or less. If synchronisation torques are this strong, then they will play a significant role in the spin rates of white dwarfs in cataclysmic variable stars as well.
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