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 _ { o } and the rotation frequency \Omega .
We survey the parameter space and find : ( 1 ) The disk fragments when { { \langle } { \langle } \tau _ { c } { \rangle } { \rangle } } \Omega \sim 1 , where { { \langle } { \langle } \tau _ { c } { \rangle } { \rangle } } 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 { \tau _ { co } } for a uniform disk with Q = 1 can differ by orders of magnitude from { { \langle } { \langle } \tau _ { c } { \rangle } { \rangle } } in the nonlinear outcome .
The difference is caused by sharp variations in the opacity with temperature .
The condition { \tau _ { co } } \Omega \sim 1 therefore does not necessarily indicate where fragmentation will occur .
( 3 ) The largest difference between { { \langle } { \langle } \tau _ { c } { \rangle } { \rangle } } and { \tau _ { co } } 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 \lesssim 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 { { \langle } { \langle } \tau _ { c } { \rangle } { \rangle } } \Omega \lesssim 1 do not exist .