  Using a local model Gammie (2001) has shown that accretion discs with cooling
times t_cool <= 3/Omega fragment into gravitationally bound objects, while
those with cooling times t_cool > 3/Omega evolve into a quasi-steady state. We
use three-dimensional smoothed particle hydrodynamic simulations of
protoplanetary accretion discs to test if the local results hold globally. We
find that for disc masses appropriate for T Tauri discs, the fragmentation
boundary still occurs at a cooling time close to t_cool = 3/Omega. For more
massive discs, which are likely to be present at an earlier stage of the star
formation process, fragmentation occurs for longer cooling times, but still
within a factor of two of that predicted using a local model. These results
have implications not only for planet formation in protoplanetary discs and
star formation in AGN discs, but also for the redistribution of angular
momentum which could be driven by the presence of relatively massive objects
within the accretion disc.
