Self-Similar Collapse of a Self-Gravitating Viscous Disk

A self-similar solution for time evolution of isothermal, self-gravitating viscous disks is found under the condition that $\alpha' \equiv \alpha (H/r)$ is constant in space (where $\alpha$ is the viscosity parameter and $H/r$ is the ratio of a half-thickness to radius of the disk). This solution describes a homologous collapse of a disk via self-gravity and viscosity. The disk structure and evolution is distinct in the inner and outer parts. There is a constant mass inflow in the outer portions so that the disk has flat rotation velocity, constant accretion velocity, and surface density decreasing outward as $\Sigma \propto r^{-1}$. In the inner portions, in contrast, mass is accumulated near the center owing to the boundary condition of no radial velocity at the origin, thereby a strong central concentration being produced; surface density varies as $\Sigma \propto r^{-5/3}$. Moreover, the transition radius separating the inner and outer portions increases linearly with time. The consequence of such a high condensation is briefly discussed in the context of formation of a quasar black hole