Non-convergence of the critical cooling time-scale for fragmentation of self-gravitating discs

We carry out a resolution study on the fragmentation boundary of self-gravitating discs. We perform three-dimensional Smoothed Particle Hydrodynamics simulations of discs to determine whether the critical value of the cooling time-scale in units of the orbital time-scale, beta(crit), converges with increasing resolution. Using particle numbers ranging from 31 250 to 16 million (the highest resolution simulations to date) we do not find convergence. Instead, fragmentation occurs for longer cooling time-scales as the resolution is increased. These results suggest that at the very least, the critical value of the cooling time-scale is longer than previously thought. However, the absence of convergence also raises the question of whether or not a critical value exists. In light of these results, we caution against using cooling time-scale or gravitational stress arguments to deduce whether gravitational instability may or may not have been the formation mechanism for observed planetary systems.