The Speed of Cooling Fronts and the Functional Form of the Dimensionless Viscosity in Accretion Disks

We examine the speed of inward traveling cooling fronts in accretion disks. We show that their speed is determined by the rarefaction wave that precedes them and is approximately $\alpha_F c_{F} (H/r)^q$, where $\alpha_F$ is the dimensionless viscosity, $c_{F}$ is the sound speed, $r$ is the radial coordinate, $H$ is the disk thickness, and all quantities are evaluated at the cooling front. The scaling exponent $q$ lies in the interval $[0,1]$, depending on the slope of the $(T,\Sigma)$ relation in the hot state. For a Kramer's law opacity and $\alpha\propto (H/r)^n$, where $n$ is of order unity, we find that $q\sim 1/2$. This supports the numerical work of Cannizzo, Chen and Livio (1995) and their conclusion that $n\approx3/2$ is necessary to reproduce the exponential decay of luminosity in black hole X-ray binary systems. Our results are insensitive to the structure of the disk outside of the radius where rapid cooling sets in. In particular, the width of the rapid cooling zone is a consequence of the cooling front speed rather than its cause. We conclude that the exponential luminosity decay of cooling disks is probably compatible with the wave-driven dynamo model. It is not compatible with models with separate, constant values of $\alpha$ for the hot and cold states.