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.
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