  Accretion disks that become gravitationally unstable can fragment into
stellar or sub-stellar companions. The formation and survival of these
fragments depends on the precarious balance between self-gravity, internal
pressure, tidal shearing, and rotation. Disk fragmentation depends on two key
factors (1) whether the disk can get to the fragmentation boundary of Q=1, and
(2) whether fragments can survive for many orbital periods. Previous work
suggests that to reach Q=1, and have fragments survive, a disk must cool on an
orbital timescale. Here we show that disks heated primarily by external
irradiation always satisfy the standard cooling time criterion. Thus even
though irradiation heats disks, and makes them more stable in general, once
they reach the fragmentation boundary, they fragment more easily. We derive a
new cooling criterion that determines fragment survival, and calculate a
pressure modified Hill radius, which sets the maximum size of
pressure-supported objects in a Keplerian disk. We conclude that fragmentation
in protostellar disks might occur at slightly smaller radii than previously
thought, and recommend tests for future simulations that will better predict
the outcome of fragmentation in real disks.
