Self‐similar Collapse of a Self‐gravitating Viscous Disk

Abstract

A self-similar solution for time evolution of isothermal, self-gravitating viscous disks is found under the condition that α' ≡ α(H/r) is constant in space (where α is the viscosity parameter and H/r is the ratio of a half-thickness to the radius of the disk). This solution describes a homologous collapse of a disk via self-gravity and viscosity. The disk structure and evolution are 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 according to Σ ∝ 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, a strong central concentration being thereby produced; surface density varies according to Σ ∝ 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.