Nonlinear Outcome of Gravitational Instability in Disks with Realistic Cooling

Abstract

We consider the nonlinear outcome of gravitational instability in optically thick disks with a realistic cooling function. We use a numerical model that is local, razor thin, and unmagnetized. External illumination is ignored. Cooling is calculated from a one-zone model using analytic fits to low-temperature Rosseland mean opacities. The model has two parameters: the initial surface density Σ0 and the rotation frequency Ω. We survey the parameter space and find the following. (1) The disk fragments when <<τc>>Ω~1, where <<τc>> is an effective cooling time defined as the average internal energy of the model divided by the average cooling rate. This is consistent with earlier results that used a simplified cooling function. (2) The initial cooling time τc0 for a uniform disk with Q=1 can differ by orders of magnitude from <<τc>> in the nonlinear outcome. The difference is caused by sharp variations in the opacity with temperature. The condition τc0Ω~1 therefore does not necessarily indicate where fragmentation will occur. (3) The largest difference between <<τc>> and τc0 is near the opacity gap, where dust is absent and hydrogen is largely molecular. (4) In the limit of strong illumination the disk is isothermal; we find that an isothermal version of our model fragments for Q <<τc>>Ω