A numerical study of self-gravitating protoplanetary disks

Abstract

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/τ cool , that u and τ cool are internal energy and cooling timescale, respectively. The ratio of local cooling to dynamical timescales Ωτ cool is assumed as a constant and also as a function of local temperature. The solutions for protoplanetary discs show that in situation of Ωτ cool = constant, the disc does not show any gravitational instability in small radii for a typically mass accretion rate, M = 10 −6 M ⊙ yr −1 , while by choosing Ωτ 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 α, 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 ⊙ yr −1 , the ratio of the specific heats γ = 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.

A two-dimensional study for cooling and self-gravitating accretion discs

“…However, the study on self-gravitating discs has received more attention recently in numerous publications (e.g. Nelson et al 1998Nelson et al , 2000Gammie 2001;Rice et al 2003Rice et al , 2005Pickett and Durisen 2007;Kley and Nelson 2008;Durisen et al 2008;Cossins et al 2010;Faghei 2012Faghei , 2013Kimura and Tsuribe 2012;Zhang et al 2014). Observational evidence has confirmed the existence of selfgravitating accretion discs in AGNs (Greenhill and Gwinn 1997;Lodato and Bertin 2003;Kondratko et al 2005) and in protostellar and protoplanetary discs (Larson 1984;Lin and Pringle 1987;Beltrán et al 2004;Chini et al 2004;Rodríguez et al 2005;Zhou et al 2005).…”

“…But the critical values in their simulations significantly differ from each other. Faghei (2012) studied the influence of the cooling rate on the radial structure and stability of the one-dimensional self-gravitating discs. The heated disc due to dissipative processes was assumed to be cooled with the β cooling.…”

“…Faghei (2012) applied a monopole approximation for the radial gravitational force due to the self-gravity of disc and ignored its vertical influences. By assuming a balance between the heating and cooling rates, Faghei (2012) found that the structure and stability of a self-gravitating accretion disc strongly depends on the magnitudes of the cooling timescale. In this paper, we want to extend the solution by Faghei (2012) and to improve some simplifications applied in his model.…”

“…By assuming a balance between the heating and cooling rates, Faghei (2012) found that the structure and stability of a self-gravitating accretion disc strongly depends on the magnitudes of the cooling timescale. In this paper, we want to extend the solution by Faghei (2012) and to improve some simplifications applied in his model. Thus, we study a two-dimensional disc in the presence of radial and vertical gravitational forces due to the self-gravity of the disc.…”

“…Thus, we study a two-dimensional disc in the presence of radial and vertical gravitational forces due to the self-gravity of the disc. Unlike to the solutions by Faghei (2012), here, we assume that the cooling rate is not in balance with the heating rate. This property provides conditions to study the influences of heating and cooling rates on the structure and stability of the disc.…”