Resolution requirements for Smoothed Particle Hydrodynamics simulations
  of self-gravitating accretion discs
  Stimulated by recent results by Meru and Bate (2010a,b), we revisit the issue
of resolution requirements for simulating self-gravitating accretion discs with
Smoothed Particle Hydrodynamics (SPH). We show that the results by Meru and
Bate (2010a) are consistent with those of Meru and Bate (2010b) if they are
both interpreted as driven by resolution effects, therefore implying that the
resolution criterion for cooling gaseous discs is a function of the imposed
cooling rate. We discuss two possible numerical origins of such dependence,
which are both consistent with the limited number of available data. Our
results tentatively indicate that convergence for current simulations is being
reached for a number of SPH particles approaching 10 millions (for a disc mass
of order 10 per cent of the central object mass), which would set the critical
cooling time for fragmentation at about $15\Omega^{-1}$, roughly a factor two
larger than previously thought. More in general, we discuss the extent to which
the large number of recent numerical results are reliable or not. We argue that
those results that pertain to the dynamics associated with gravitational
instabilities (such as the locality of angular momentum transport, and the
relationship between density perturbation and induced stress) are robust, while
those pertaining to the thermodynamics of the system (such as the determination
of the critical cooling time for fragmentation) can be affected by poor
resolution.
