  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.
