• accretion, accretion discs;
  • gravitation;
  • hydrodynamics;
  • instabilities;
  • planets and satellites: formation;
  • protoplanetary discs


We carry out simulations of gravitationally unstable discs using a smoothed particle hydrodynamics (SPH) code and a grid-based hydrodynamics code, fargo, to understand the previous non-convergent results reported by Meru & Bate. We obtain evidence that convergence with increasing resolution occurs with both SPH and fargo and in both cases we find that the critical cooling time-scale is larger than previously thought. We show that SPH has a first-order convergence rate, while fargo converges with a second-order rate. We show that the convergence of the critical cooling time-scale for fragmentation depends largely on the numerical viscosity employed in both SPH and fargo. With SPH, particle velocity dispersion may also play a role. We show that reducing the dissipation from the numerical viscosity leads to larger values of the critical cooling time at a given resolution. For SPH, we find that the effect of the dissipation due to the numerical viscosity is somewhat larger than had previously been appreciated. In particular, we show that using a quadratic term in the SPH artificial viscosity (βSPH) that is too low appears to lead to excess dissipation in gravitationally unstable discs, which may affect any results that sensitively depend on the thermodynamics, such as disc fragmentation. We show that the two codes converge to values of the critical cooling time-scale, βcrit > 20 (for a ratio of specific heats of γ = 5/3), and perhaps even as large as βcrit ≈ 30. These are approximately three to five times larger than has been found by most previous studies. This is equivalent to a maximum gravitational stress that a disc can withstand without fragmenting of αGI, crit ≈ 0.013 − 0.02, which is much smaller than the values typically used in the literature. It is therefore easier for self-gravitating discs to fragment than has been concluded from most past studies.