Viscosity prescription for gravitationally unstable accretion disks

Gravitationally unstable accretion disks emerge in a variety of astrophysical contexts - giant planet formation, FU Orioni outbursts, feeding of AGNs, and the origin of Pop III stars. When a gravitationally unstable disk is unable to cool rapidly it settles into a quasi-stationary, fluctuating gravitoturbulent state, in which its Toomre Q remains close to a constant value Q_0~1. Here we develop an analytical formalism describing the evolution of such a disk, which is based on the assumptions of Q=Q_0 and local thermal equilibrium. Our approach works in the presence of additional sources of angular momentum transport (e.g. MRI), as well as external irradiation. Thermal balance dictates a unique value of the gravitoturbulent stress \alpha_{gt} driving disk evolution, which is a function of the local surface density and angular frequency. We compare this approach with other commonly used gravitoturbulent viscosity prescriptions, which specify the explicit dependence of stress \alpha_{gt} on Toomre Q in an ad hoc fashion, and identify the ones that provide consistent results. We nevertheless argue that our Q=Q_0 approach is more flexible, robust, and straightforward, and should be given preference in applications. We illustrate this with a couple of analytical calculations - locations of the snow line and of the outer edge of the dead zone in a gravitoturbulent protoplanetary disk - which clearly show the simplicity and versatility of the Q=Q_0 approach.