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.