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Convective cooling and fragmentation of gravitationally unstable disks

Gravitationally unstable disks can fragment and form bound objects provided that their cooling time is short. In protoplanetary disks radiative cooling is likely to be too slow to permit formation of planets by fragmentation within several tens of AU from the star. Recently, convection has been suggested as a faster means of heat loss from the disk but here we demonstrate that it is only marginally more efficient than the radiative cooling. The crucial factor is the rate at which energy can be radiated from the disk photosphere, which is robustly limited from above in the convective case by the adiabatic temperature gradient (given a certain midplane temperature). Thus, although vigorous convection is definitely possible in disks, the inefficiency of radiative loss from the photosphere may create a bottleneck limiting the ability of the disk to form self-gravitating objects. Based on this argument we derive a set of analytical constraints which diagnose the susceptibility of an unstable disk to fragmentation and show that the formation of giant planets by fragmentation of protoplanetary disks is unlikely to occur at distances of tens of AU. At the same time these constraints do not preclude the possibility of fragmentation and star formation in accretion disks around supermassive black holes.