On the fragmentation criteria of self-gravitating protoplanetary discs
  We investigate the fragmentation criterion in massive self-gravitating discs.
We present new analysis of the fragmentation conditions which we test by
carrying out global three-dimensional numerical simulations. Whilst previous
work has placed emphasis on the cooling timescale in units of the orbital
timescale, \beta , we find that at a given radius the surface mass density
(i.e. disc mass and profile) and star mass also play a crucial role in
determining whether a disc fragments or not as well as where in the disc
fragments form. We find that for shallow surface mass density profiles (p<2,
where \Sigma \propto R^{-p}), fragments form in the outer regions of the disc.
However for steep surface mass density profiles (p \gtrsim 2), fragments form
in the inner regions of a disc. In addition, we also find that the critical
value of the cooling timescale in units of the orbital timescale found in
previous simulations is only applicable to certain disc surface mass density
profiles and for particular disc radii and is not a general rule for all discs.
We find an empirical fragmentation criteria between the cooling timescale in
units of the orbital timescale, \beta , the surface mass density, the star mass
and the radius.
