  In this paper, the effect of self-gravity on the protoplanetary discs is
investigated. The mechanisms of angular momentum transport and energy
dissipation are assumed to be the viscosity due to turbulence in the accretion
disc. The energy equation is considered in situation that the released energy
by viscosity dissipation is balanced with cooling processes. The viscosity is
obtained by equality of dissipation and cooling functions, and is used for
angular momentum equation. The cooling rate of the flow is calculated by a
prescription, $d u/d t=-u/\tau_{cool}$, that $u$ and $\tau_{cool}$ are internal
energy and cooling timescale, respectively. The ratio of local cooling to
dynamical timescales $\Omega \tau_{cool}$ is assumed as a constant and also as
a function of local temperature. The solutions for protoplanetary discs show
that in situation of $\Omega \tau_{cool} = constant$, the disc does not show
any gravitational instability in small radii for a typically mass accretion
rate, $\dot{M} = 10^{-6} M_{\odot} yr^{-1}$, while by choosing $\Omega
\tau_{cool}$ as a function of temperature, the gravitational instability for
this amount of mass accretion rate or even less can occur in small radii. Also,
by study of the viscous parameter $\alpha$, we find that the strength of
turbulence in the inner part of self-gravitating protoplanetary discs is very
low. These results are qualitatively consistent with direct numerical
simulations of protoplanetary discs.
