Viscosity prescription for gravitationally unstable accretion disks

TL;DR

This paper develops an analytical model for gravitationally unstable accretion disks assuming a constant Toomre Q parameter, providing a robust and versatile framework for understanding disk evolution in various astrophysical contexts.

Contribution

The authors introduce a new formalism based on a fixed Toomre Q, improving upon ad hoc viscosity prescriptions for modeling gravitoturbulent disks.

Findings

The Q=Q_0 approach offers a flexible and straightforward method for disk evolution modeling.

Analytical calculations demonstrate the model's application to snow line and dead zone locations.

The formalism remains valid with additional angular momentum transport mechanisms and external irradiation.

Abstract

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.