Self-similar, self-gravitating viscous disks

We find a self-similar steady disk solution for the gravitational potentials that gives equilibrium angular velocities as power-law functions of the radii, Omega proportional to R(-delta), with delta being a free parameter. In particular, when a disk is totally self-gravitating in the radial as well as perpendicular directions, we find delta = 1, leading to a flat rotation, constant temperature, and surface density proportional to R(-1). The disk structure is characterized by three parameters: the viscosity parameter, alpha; the ratio of disk thickness to radius; and the mass-flow rate. The solution shows that the nonviscous, isothermal rotating disk modeled by Mestel can be extended to a viscous accretion disk with a constant mass inflow.