Radiative cooling in disks and its effects on the formation of giant planets via the gravitational instability

Gravitational instability provides a means of rapidly forming giant planets with large orbital radii. For protoplanetary disks to be unstable to gravitational fragmentation, they must (1) have a Toomre Q ≲ 1 and (2) be able to cool the excess energy from a collapsing perturbation in less than the dynamical time (Otcool ≲ 1). We present an analytical technique for calculating this perturbation cooling time for externally illuminated disks and/or disks with internal heating. We compare our analytical technique with a numerical Monte Carlo code, and find good agreement.;We use our analytical technique to test the ability of the gravitational instability to re-create the observed planetary systems of Fomalhaut, HR 8799, and HL Tau. We find that the required disk mass interior to the planet's orbital radius is ∼0.1 M⊙ for Fomalhaut b, the protoplanet orbiting HL Tau, and the outermost planet of HR 8799. The two inner planets of HR 8799 probably could not have formed in situ by disk fragmentation.;The perturbation cooling time can be reduced significantly through the inclusion of geometrical effects, specifically fragmentation originating at a location other than the disk mid-plane, and/or dust settling. In particular, dust settling to one-tenth of the gas scale height can reduce the perturbation cooling below the fragmentation threshold for all surface densities Sigma ≲ 103 g/cm2.;We study the fragmentation criteria and fragment masses produced for a grid of parameters covering pre-main sequence masses ranging from 0.1--5 M⊙ , ages ranging from 0.5--10 Myr, and differing degrees of dust settling. We find that the instability criteria and fragment mass scales with pre-main sequence mass (as expected), while the pre-main sequence age (i.e., luminosity) provides only a modest effect---indicating that disk fragmentation is equally likely at all stages of pre-main sequence evolution, given sufficiently high disk mass. Dust settling can lead to disk fragmentation at orbital radii that are an order of magnitude smaller than in the unsettled case.