Resolution requirements for smoothed particle hydrodynamics calculations with self-gravity

We determine a new resolution requirement for the smoothed particle hydrodynamics (SPH) numerical method when self-gravity is included. Comparison calculations between an SPH code and an Eulerian grid code are performed. The calculations are of a computationally demanding molecular cloud collapse and fragmentation problem. We demonstrate that the results given by the two different hydrodynamic methods are in good agreement, so long as the minimum resolvable mass in the SPH calculations is always less than the Jeans mass. If this criterion is not satisfied, SPH may give incorrect results, with the stability of near-Jeans-mass clumps depending on the details of how the SPH code is implemented rather than on physical processes. We give examples, from the literature, of problems that have been encountered in simulations of star, galaxy and cosmological-structure formation where this resolution requirement is ignored.