Self-similar Evolution of Self-Gravitating Viscous Accretion Discs
  A new one-dimensional, dynamical model is proposed for geometrically thin,
self-gravitating viscous accretion discs. The vertically integrated equations
are simplified using the slow accretion limit and the monopole approximation
with a time-dependent central point mass to account for self-gravity and
accretion. It is shown that the system of partial differential equations can be
reduced to a single non-linear advection diffusion equation which describes the
time evolution of angular velocity.
  In order to solve the equation three different turbulent viscosity
prescriptions are considered. It is shown that for these parametrizations the
differential equation allows for similarity transformations depending only on a
single non-dimensional parameter. A detailed analysis of the similarity
solutions reveals that this parameter is the initial power law exponent of the
angular velocity distribution at large radii. The radial dependence of the
self-similar solutions is in most cases given by broken power laws. At small
radii the rotation law always becomes Keplerian with respect to the current
central point mass. In the outer regions the power law exponent of the rotation
law deviates from the Keplerian value and approaches asymptotically the value
determined by the initial condition. It is shown that accretion discs with
flatter rotation laws at large radii yield higher accretion rates.
  The methods are applied to self-gravitating accretion discs in active
galactic nuclei. Fully self-gravitating discs are found to evolve faster than
nearly Keplerian discs. The implications on supermassive black hole formation
and Quasar evolution are discussed.
