Non-convergence of the critical cooling timescale 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 (SPH) simulations of discs to determine whether the critical
value of the cooling timescale in units of the orbital timescale, 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 timescales
as the resolution is increased. These results certainly suggest that
beta_{crit} is larger than previously thought. However, the absence of
convergence also questions whether or not a critical value exists. In light of
these results, we caution against using cooling timescale or gravitational
stress arguments to deduce whether gravitational instability may or may not
have been the formation mechanism for observed planetary systems.
