The density and temperature dependence of the cooling timescale for fragmentation of self-gravitating disks

Abstract

The purpose of this paper is to explore the influences of cooling timescale on fragmentation of self-gravitating protoplanetary disks. We assume the cooling timescale, expressed in terms of the dynamical timescale Ω tcool, has a power-law dependence on temperature and density, Ω tcool ∝ Σ−aT−b, where a and b are constants. We use this cooling timescale in a simple prescription for the cooling rate, du/dt = −u/tcool, where u is the internal energy. We perform our simulations using the smoothed particle hydrodynamics method. The simulations demonstrate that the disk is very sensitive to the cooling timescale, which depends on density and temperature. Under such a cooling timescale, the disk becomes gravitationally unstable and clumps form in the disk. This property even occurs for cooling timescales which are much longer than the critical cooling timescale, Ω tcool ≳ 7. We show that by adding the dependence of a cooling timescale on temperature and density, the number of clumps increases and the clumps can also form at smaller radii. The simulations imply that the sensitivity of a cooling timescale to density is more than to temperature, because even for a small dependence of the cooling timescale on density, clumps can still form in the disk. However, when the cooling timescale has a large dependence on temperature, clumps form in the disk. We also consider the effects of artificial viscosity parameters on fragmentation conditions. This consideration is performed in two cases, where Ω tcool is a constant and Ω tcool is a function of density and temperature. The simulations consider both cases, and results show the artificial viscosity parameters have rather similar effects. For example, using too small of values for linear and quadratic terms in artificial viscosity can suppress the gravitational instability and consequently the efficiency of the clump formation process decreases. This property is consistent with recent simulations of self-gravitating disks. We perform simulations with and without the Balsara form of artificial viscosity. We find that in the cooling and self-gravitating disks without the Balsara switch, the clumps can form more easily than those with the Balsara switch. Moreover, in both cases where the Balsara switch is present or absent, the simulations show that the cooling timescale strongly depends on density and temperature.