Nonlinear outcome of gravitational instability in disks with realistic cooling

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 «τ_c»Ω ∼ 1, where «τ _c» 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 τ_c0 for a uniform disk with Q = 1 can differ by orders of magnitude from «τ_c» in the nonlinear outcome. The difference is caused by sharp variations in the opacity with temperature. The condition τ_c0Ω ∼ 1 therefore does not necessarily indicate where fragmentation will occur. (3) The largest difference between «τ_c» and τ_c0 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 ≲ 1.4. Finally, we discuss some physical processes not included in our model and find that most are likely to make disks more susceptible to fragmentation. We conclude that disks with «τ_c»Ω ≲ 1 do not exist.