Numerical requirements for simulations of self-gravitating and non-self-gravitating discs

We define three requirements for accurate simulations that attempt to model circumstellar discs and the formation of collapsed objects (e.g. planets) within them. First, we define a resolution requirement based on the wavelength for neutral stability of self-gravitating waves in the disc, where a Jeans analysis does not apply. For particle-based or grid-based simulations, this criterion takes the form, respectively, of a minimum number of particles per critical (‘Toomre’) mass or maximum value of a ‘Toomre number’, T=δx/λT, where the wavelength, λT, is the wavelength for neutral stability for waves in discs. The requirements are analogues of the conditions for cloud collapse simulations as discussed in Bate & Burkert and Truelove et al., where the required minimum resolution was shown to be twice the number of neighbours per Jeans mass or four–five times the local Jeans wavelength, λJ, for particle or grid simulations, respectively. We apply our criterion to particle simulations of disc evolution and find that in order to prevent numerically induced fragmentation of the disc, the Toomre mass must be resolved by a minimum of six times the average number of neighbour particles used. We investigate the origin of the apparent discrepancy between the number of particles required by the cloud and disc fragmentation criteria and find that it is due largely to ambiguities in the definition of the Jeans mass, as used by different authors. We reconcile the various definitions, and when an identical definition of the Jeans mass is used, the condition that J≲ 1/4 in the Truelove condition is equivalent to requiring about 10–12 times the average number of neighbour particles per Jeans mass in a smoothed particle hydrodynamics (SPH) simulation, reducing the difference between simulations of discs and clouds to about two. While the numbers of particles per critical mass are similar for both the Jeans and Toomre formalisms, the Toomre requirement is more restrictive than the Jeans requirement when the local value of the Toomre stability parameter Q falls below about one half. Second, we require that particle-based simulations with self-gravity use a variable gravitational softening, in order to avoid inducing fragmentation by an inappropriate choice of softening length. We show that using a fixed gravitational softening length for all particles can lead either to artificial suppression or enhancement of structure (including fragmentation) in a given disc, or both in different locations of the same disc, depending on the value chosen for the softening length. Unphysical behaviour can occur whether or not the system is properly resolved by the new Toomre criterion. Third, we require that three-dimensional SPH simulations resolve the vertical structure with at least ∼4 particle smoothing lengths per scaleheight at the disc mid-plane, a value which implies a substantially larger number per vertical column because the disc itself extends over many scaleheights. We suggest that a similar criterion applies to grid-based simulations. We demonstrate that failure to meet this criterion leads to underestimates in the mid-plane density of up to 30–50 per cent at resolutions common in the literature. As a direct consequence, gas pressures will be dramatically underestimated and simulations of self-gravitating systems may artificially and erroneously inflate the likelihood of fragmentation. We outline an additional condition on the vertical resolution in simulations that include radiative transfer in order to ensure a correct description of the cooling, specifically that the temperature structure near the disc photosphere must be well resolved. As an example, we demonstrate that for an isentropic vertical structure, the criterion translates into resolution comparable to H/20 near the disc photosphere, to avoid serious errors in transfer rates of thermal energy in and out of the disc. Finally, we discuss results in the literature that purport to form collapsed objects and conclude that many are likely to have violated one or more of our criteria, and have therefore made incorrect conclusions regarding the likelihood for fragmentation and planet formation.