  We present a new framework to explain the link between cooling and
fragmentation in gravitationally unstable protostellar discs. This framework
consists of a simple model for the formation of spiral arms, as well as a
criterion, based on the Hill radius, to determine if a spiral arm will
fragment. This detailed model of fragmentation is based on the results of
numerical simulations of marginally stable protostellar discs, including those
found in the literature, as well as our new suite of 3-D radiation
hydrodynamics simulations of an irradiated, optically-thick protostellar disc
surrounding an A star. Our set of simulations probes the transition to
fragmentation through a scaling of the physical opacity. This model allows us
to directly calculate the critical cooling time of Gammie (2001), with results
that are consistent with those found from numerical experiment. We demonstrate
how this model can be used to predict fragmentation in irradiated protostellar
discs. These numerical simulations, as well as the model that they motivate,
provide strong support for the hypothesis that gravitational instability is
responsible for creating systems with giant planets on wide orbits.
