We carry out a resolution study on the fragmentation boundary ofself-gravitating discs. We perform three-dimensional Smoothed ParticleHydrodynamics simulations of discs to determine whether the critical value ofthe cooling timescale in units of the orbital timescale, beta_{crit}, convergeswith increasing resolution. Using particle numbers ranging from 31,250 to 16million (the highest resolution simulations to date) we do not findconvergence. Instead, fragmentation occurs for longer cooling timescales as theresolution is increased. These results suggest that at the very least, thecritical value of the cooling timescale is longer than previously thought.However, the absence of convergence also raises the question of whether or nota critical value exists. In light of these results, we caution against usingcooling timescale or gravitational stress arguments to deduce whethergravitational instability may or may not have been the formation mechanism forobserved planetary systems.
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