  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.
