The equations governing the vertical structure of a stationary keplerian accretion disc supporting an Eddington atmosphere are presented .
The model is based on the \alpha -prescription for turbulent viscosity ( two versions are tested ) , includes the disc vertical self-gravity , convective transport and turbulent pressure .
We use an accurate equation of state and wide opacity grids which combine the Rosseland and Planck absorption means through a depth-dependent weighting function .
The numerical method is based on single side shooting and incorporates algorithms designed for stiff initial value problems .
A few properties of the model are discussed for a circumstellar disc around a sun-like star and a disc feeding a 10 ^ { 8 } M _ { \odot } central black hole .
Various accretion rates and \alpha -parameter values are considered .

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 : there is an extremely violent density and surface density runaway , a rapid disc collapse and a temperature plateau .
With respect , a much softer transition is observed with the “ \alpha \cal { P } -formalism ” .
Turbulent pressure is important only for \alpha \gtrsim 0.1 .
It lowers vertical density gradients , significantly thickens the disc ( increases its flaring ) , tends to wash out density inversions occurring in the upper layers 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 \gtrsim 0.1 .

This study suggests that models aiming to describe T-Tauri discs beyond about a few to a few tens astronomical units ( depending on the viscosity parameter ) from the central protostar using the \alpha -theory should consider vertical self-gravity , but additional heating mechanisms are necessary to account for large discs .
The Primitive Solar Nebula was probably a bit ( if not strongly ) self-gravitating at the actual orbit of giant planets .
In agreement with vertically averaged computations , \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 as requested to fuel steadily active nuclei for a few hundred millions years .
More efficient mechanisms driving accretion are required .