  As a formation route for objects such as giant planets and low-mass stars in
protostellar discs (as well as stars in AGN discs), theories of
self-gravitating disc fragmentation need to be able to predict the initial
masses of fragments. We describe a means by which the local Jeans mass inside
the spiral structure of a self-gravitating disc can be estimated. If such a
self-gravitating disc satisfies the criteria for disc fragmentation, this
estimate provides a lower limit for the initial mass of any fragments formed.
We apply this approach to a series of self-gravitating protostellar disc
models, to map out the typical masses of fragments produced by this formation
mode. We find a minimum fragment mass of around 3 Jupiter masses, which is
insensitive to the stellar mass, and that - within the parameter space surveyed
- fragments with masses between 10 and 20 Jupiter masses are the most common.
We also describe how the Jeans mass allows us to derive a more general
criterion for disc fragmentation, which accounts for the processes of viscous
heating, radiative cooling, accretion and the disc's thermal history. We
demonstrate how such a criterion can be determined, and show that in limiting
cases it recovers several fragmentation criteria that have been posited in the
past, including the minimum cooling time/maximum stress criterion.
