An accretion disc instability induced by a temperature sensitive $\alpha$ parameter

In the standard thin disc formalism the dimensionless $\alpha$ parameter is usually assumed to be constant. However, there are good theoretical reasons for believing, as well as evidence from simulations, that $\alpha$ is dependent on intrinsic disc properties. In this paper we analyse the conditions for the stability of a thin accretion disc in which $\alpha$ is a function of the magnetic Prandtl number, the ratio of collisional viscosity to resistivity. In the inner disc, where the free electron opacity and radiation viscosity dominate, the disc is unstable if $\alpha$ is proportional to the magnetic Prandtl number with an exponent, $n$, and $6/13<n<10/3$. This is within the range of values for the power-law index found in MHD simulations with simple energetics. We calculate the evolution of the unstable disc within the $\alpha$ formalism and show that the physically accessible solutions form a limit cycle, analogous to the behaviour seen in recurrent dwarf novae. It is noteworthy that the time-dependent global behaviour of the instability results in cyclic heating of the inner section of the disc, when parameters appropriate for an X-ray binary system are used. We calculate a model spectrum of the disc in the flaring and quiescent states and show that the behaviour is compatible with X-ray observations of the thermal accretion disc in flaring X-ray binary systems.