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 , du / dt = - 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 .
Also , in the case of cooling with temperature dependence , the effect of physical parameters on the structure of the disc is investigated .
The solutions represent that disc thickness and Toomre parameter decrease by adding the ratio of disc mass to central object mass .
While , the disc thickness and Toomre parameter increase by adding mass accretion rate .
Furthermore , for typically input parameters such as mass accretion rate 10 ^ { -6 } M _ { \odot } yr ^ { -1 } , the ratio of the specific heats \gamma = 5 / 3 , and the ratio of disc mass to central object mass q = 0.1 , the gravitational instability can occur in whole radii of the discs excluding very near to the central object .