Investigating fragmentation conditions in self-gravitating accretion discs

The issue of fragmentation in self-gravitating gaseous accretion discs has implications both for the formation of stars in discs in the nuclei of active galaxies, and for the formation of gaseous planets or brown dwarfs in circumstellar discs. It is now well established that fragmentation occurs if the disc is cooled on a time-scale smaller than the local dynamical time-scale, while for longer cooling times the disc reaches a quasi-steady state in thermal equilibrium, with the cooling rate balanced by the heating due to gravitational stresses. We investigate here how the fragmentation boundary depends on the assumed equation of state. We find that the cooling time required for fragmentation increases as the specific heat ratio γ decreases, exceeding the local dynamical time-scale for γ= 7/5 . This result can be easily interpreted as a consequence of there being a maximum stress (in units of the local disc pressure) that can be sustained by a self-gravitating disc in quasi-equilibrium. Fragmentation occurs if the cooling time is such that the stress required to reach thermal equilibrium exceeds this value, independent of γ. This result suggests that a quasi-steady, self-gravitating disc can never produce a stress that results in the viscous α parameter exceeding ∼0.06.