Exact Time-dependent Solutions for the Thin Accretion Disc Equation: Boundary Conditions at Finite Radius

TL;DR

This paper derives exact time-dependent Green's-function solutions for thin accretion discs with boundary conditions at finite radius, extending previous models and enabling more realistic descriptions of disc evolution near the central object.

Contribution

It provides the first exact solutions for accretion discs with boundary conditions at a finite inner radius for arbitrary initial profiles and viscosity power-law index.

Findings

Solutions have finite luminosity at finite inner radius.

Solutions match steady-state luminosity at long times.

Applicable to modeling inner disc evolution in astrophysics.

Abstract

We discuss Green's-function solutions of the equation for a geometrically thin, axisymmetric Keplerian accretion disc with a viscosity prescription "\nu ~ R^n". The mathematical problem was solved by Lynden-Bell & Pringle (1974) for the special cases with boundary conditions of zero viscous torque and zero mass flow at the disc center. While it has been widely established that the observational appearance of astrophysical discs depend on the physical size of the central object(s), exact time-dependent solutions with boundary conditions imposed at finite radius have not been published for a general value of the power-law index "n". We derive exact Green's-function solutions that satisfy either a zero-torque or a zero-flux condition at a nonzero inner boundary R_{in}>0, for an arbitrary initial surface density profile. Whereas the viscously dissipated power diverges at the disc center for the previously known solutions with R_{in}=0, the new solutions with R_{in}>0 have finite expressions for the disc luminosity that agree, in the limit t=>infinity, with standard expressions for steady-state disc luminosities. The new solutions are applicable to the evolution of the innermost regions of thin accretion discs.