Properties of vertically self-gravitating accretion discs with a   dissipative corona
  The steady-state structure of a disc with a corona is analyzed when the vertical component of the gravitational force due to the self-gravity of the disc is considered. For the energy exchange between the disc and the corona, we assume a fraction f of the dissipated energy inside the accretion disc is transported to the corona via the magnetic tubes. Analytical solutions corresponding to a prescription for f (in which this parameter directly depends on the ratio of the gas pressure to the total pressure) or free f are presented and their physical properties are studied in detail. We show that the existence of the corona not only decreases the temperature of the disc, but also increases the surface density.The vertical component of the gravitational force due to the self-gravity of the disc decreases the self-gravitating radius and the mass of the fragments at this radius. However, as more energy is transported from the disc to the corona, the effect of the vertical component of the gravitational force due to the self-gravity of the disc on the self-gravitating radius becomes weaker, though the mass of the fragments is reduced irrespective of the amount of the energy exchange from the disc to the corona. 
Fazeleh Khajenabi, Peter Duffy