  Protoplanetary disks fragment due to gravitational instability when there is
enough mass for self-gravitation, described by the Toomre parameter, and when
heat can be lost at a rate comparable to the local dynamical timescale,
described by t_c=beta Omega^-1. Simulations of self-gravitating disks show that
the cooling parameter has a rough critical value at beta_crit=3. When below
beta_crit, gas overdensities will contract under their own gravity and fragment
into bound objects while otherwise maintaining a steady state of
gravitoturbulence. However, previous studies of the critical cooling parameter
have found dependence on simulation resolution, indicating that the simulation
of self-gravitating protoplanetary disks is not so straightforward. In
particular, the simplicity of the cooling timescale t_c prevents fragments from
being disrupted by pressure support as temperatures rise. We alter the cooling
law so that the cooling timescale is dependent on local surface density
fluctuations, a means of incorporating optical depth effects into the local
cooling of an object. For lower resolution simulations, this results in a lower
critical cooling parameter and a disk more stable to gravitational stresses
suggesting the formation of large gas giants planets in large, cool disks is
generally suppressed by more realistic cooling. At our highest resolution
however, the model becomes unstable to fragmentation for cooling timescales up
to beta = 10.
