Nonlinear Outcome of Gravitational Instability in Disks with Realistic Cooling

We consider the nonlinear outcome of gravitational instability in opticallythick 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 Σo and the rotation frequency Ω. We survey the parameter space and find: (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 τco 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 τcoΩ ∼ 1 therefore does not necessarily indicate where fragmentation will occur. (3) The largest difference between 〈〈τc〉〉 and τco 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 〈〈τc〉〉Ω . 1 do not exist. Subject headings: accretion, accretion disks, solar system: formation, galaxies: nuclei