  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 Sigma_0 and the rotation frequency Omega. We survey the parameter space
and find: (1) The disk fragments when t_c,eff Omega = 1, where t_c,eff 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 t_c0 or a
uniform disk with Q = 1 can differ by orders of magnitude from t_c,eff in the
nonlinear outcome. The difference is caused by sharp variations in the opacity
with temperature. The condition t_c0 Omega = 1 therefore does not necessarily
indicate where fragmentation will occur. (3) The largest difference between
t_c,eff and t_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 t_c,eff Omega < 1 do not exist.
