Abstract The effect of self-gravity on protoplanetary disks is investigated. The mechanisms of angular momentum transport and energy dissipation are assumed to be the viscosity due to turbulence in the accretion disk. The energy equation is considered in a situation where 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 to derive the angular momentum equation. The cooling rate of the flow is calculated by a prescription, du/dt = −u/τcool , where u and τcool are the internal energy and cooling timescale, respectively. The ratio of local cooling to dynamical timescales Ωτcool is assumed to be a constant and also a function of the local temperature. The solutions for protoplanetary disks show that in the case of Ωτcool = constant, the disk does not exhibit any gravitational instability over small radii for a typical mass accretion rate, = 10−6M⊙ yr−1 , but when choosing Ωτcool to be a function of temperature, gravitational instability can occur for this value of mass accretion rate or even less in small radii. Also, by studying the viscosity parameter α, we find that the strength of turbulence in the inner part of self-gravitating protoplanetary disks is very low. These results are qualitatively consistent with direct numerical simulations of protoplanetary disks. Also, in the case of cooling with temperature dependence, the effect of physical parameters on the structure of the disk is investigated. These solutions demonstrate that disk thickness and the Toomre parameter decrease by adding the ratio of disk mass to central object mass. However, the disk thickness and the Toomre parameter increase by adding mass accretion rate. Furthermore, for typical input parameters such as mass accretion rate 10−6M⊙ yr−1 , the ratio of the specific heat γ = 5/3 and the ratio of disk mass to central object mass q = 0.1, gravitational instability can occur over the whole radius of the disk excluding the region very near the central object.

Keywords accretion, accretion disks — planetary systems: protoplanetary disks — planetary systems: formation

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