On the transition to self-gravity in low mass AGN and YSO accretion   discs
  The equations governing the vertical structure of a stationary keplerian accretion disc are presented. The model is based on the alpha-viscosity, includes self-gravity, convective transport and turbulent pressure. A few properties of the model are discussed for circumstellar and AGN discs.   We show the strong sensitivity of the disc structure to the viscous energy deposition towards the vertical axis, specially when entering inside the self-gravitating part of the disc. The local version of the alpha-prescription leads to a "singular" behavior which is also predicted by the vertically averaged model. With respect, a much softer transition is observed with the   "alpha-P" formalism. Turbulent pressure is important only for alpha > 0.1. It lowers vertical density gradients, significantly thickens the disc, tends to wash out density inversions and pushes the self-gravitating region to slightly larger radii. Curves localizing the inner edge of the self-gravitating disc as functions of the viscosity parameter and accretion rate are given. The lower alpha, the closer to the center the self-gravitating regime, and the sensitivity to the accretion rate is generally weak, except for alpha < 0.1. This study suggests that models aiming to describe T-Tauri discs beyond about a few to a few tens astronomical units from the central protostar using the alpha-theory should consider vertical self-gravity. The Primitive Solar Nebula was probably a bit (if not strongly) self-gravitating at the actual orbit of giant planets. Alpha-discs hosted by active galaxies are self-gravitating beyond about a thousand Schwarzchild radii. The inferred surface density remains too high to lower the accretion time scale. More efficient mechanisms driving accretion are required. 
Jean-Marc Hure (DAEC/Observatoire de Meudon & Universite Paris 7)