  Self-gravitating accretion discs in a gravitoturbulent state, including
radiation and gas pressures, are studied using a set of new analytical
solutions. While the Toomre parameter of the disc remains close to its critical
value for the onset of gravitational instability, the dimensionless stress
parameter is uniquely determined from the thermal energy reservoir of the disc
and its cooling rate. Our solutions are applicable to the accretion discs with
dynamically important radiation pressure like in the quasars discs. We show
that physical quantities of a gravitoturbulent disc in the presence of
radiation are significantly modified compared to solutions with only gas
pressure. We show that the dimensionless stress parameter is an increasing
function of the radial distance so that its steepness strongly depends on the
accretion rate. In a disc without radiation its slope is 4.5, however, we show
that in the presence of radiation, it varies between 2 and 4.5 depending on the
accretion rate and the central mass. As for the surface density, we find a
shallower profile with an exponent -2 in a disc with sub-Eddington accretion
rate compared to a similar disc, but without radiation, where its surface
density slope is -3 independent of the accretion rate. We then investigate
gravitational stability of the disc when the stress parameter reaches to its
critical value. In order to self-consistently determine the fragmentation
boundary, however, it is shown that the critical value of the stress parameter
is a power-law function of the ratio of gas pressure and the total pressure and
its exponent is around 1.7. We also estimate the maximum mass of the central
black hole using our analytical solutions.
