  Observations support the hypothesis that gas disk gravitational instability
might explain the formation of massive or wide-orbit gas giant exoplanets. The
situation with regard to Jupiter-mass exoplanets orbiting within $\sim$ 20 au
is more uncertain. Theoretical models yield divergent assessments often
attributed to the numerical handling of the gas thermodynamics. Boss (2019)
used the $\beta$ cooling approximation to calculate three dimensional
hydrodynamical models of the evolution of disks with initial masses of 0.091
$M_\odot$ extending from 4 to 20 au around 1 $M_\odot$ protostars. The models
considered a wide range (1 to 100) of $\beta$ cooling parameters and started
from an initial minimum Toomre stability parameter of $Q_i = 2.7$
(gravitationally stable). The disks cooled down from initial outer disk
temperatures of 180 K to as low as 40 K as a result of the $\beta$ cooling,
leading to fragmentation into dense clumps, which were then replaced by virtual
protoplanets (VPs) and evolved for up to $\sim$ 500 yr. The present models test
the viability of replacing dense clumps with VPs by quadrupling the spatial
resolution of the grid once dense clumps form, sidestepping in most cases VP
insertion. After at least $\sim$ 200 yr of evolution, the new results compare
favorably with those of Boss (2019): similar numbers of VPs and dense clumps
form by the same time for the two approaches. The results imply that VP
insertion can greatly speed disk instability calculations without sacrificing
accuracy.
