  Observational evidence exists for the formation of gas giant planets on wide
orbits around young stars by disk gravitational instability, but the roles of
disk instability and core accretion for forming gas giants on shorter period
orbits are less clear. The controversy extends to population synthesis models
of exoplanet demographics and to hydrodynamical models of the fragmentation
process. The latter refers largely to the handling of radiative transfer in
three dimensional (3D) hydrodynamical models, which controls heating and
cooling processes in gravitationally unstable disks, and hence dense clump
formation. A suite of models using the $\beta$ cooling approximation is
presented here. The initial disks have masses of 0.091 $M_\odot$ and extend
from 4 to 20 AU around a 1 $M_\odot$ protostar. The initial minimum Toomre
$Q_i$ values range from 1.3 to 2.7, while $\beta$ ranges from 1 to 100. We show
that the choice of $Q_i$ is equal in importance to the $\beta$ value assumed:
high $Q_i$ disks can be stable for small $\beta$, when the initial disk
temperature is taken as a lower bound, while low $Q_i$ disks can fragment for
high $\beta$. These results imply that the evolution of disks toward low $Q_i$
must be taken into account in assessing disk fragmentation possibilities, at
least in the inner disk, i.e., inside about 20 AU. The models suggest that if
low $Q_i$ disks can form, there should be an as yet largely undetected
population of gas giants orbiting G dwarfs between about 6 AU and 16 AU.
