A local prescription for the softening length in self-gravitating gaseous discs

In 2D-simulations of self-gravitating gaseous discs, the potential is often computed in the framework of “softened gravity” initially designed for N-body codes. In this special context, the role of the softening length λ is twofold: i) to avoid numerical singularities in the integral representation of the potential (i.e., arising when the separation |r -r'| → 0); and ii) to account for stratification of matter in the direction perpendicular to the disc mid-plane. So far, most studies have considered λ as a free parameter and various values or formulae have been proposed without much mathematical justification. In this paper, we demonstrate by means of a rigorous calculus that it is possible to define λ such that the gravitational potential of a flat disc coincides at order zero with that of a geometrically thin disc of the same surface density. Our prescription for λ, valid in the local, axisymmetric limit, has the required properties i) and ii). It is mainly an analytical function of the radius and disc thickness, and is sensitive to the vertical stratification. For mass density profiles considered (namely, profiles expandable over even powers of the altitude), we find that λ: i) is independant of the numerical mesh, ii) is always a fraction of the local thickness H; iii) goes through a minimum at the singularity (i.e., at null separation); and iv) is such that 0.13 ⪉ λ/H ⪉ 0.29 typically (depending on the separation and on density profile). These results should help us to improve the quality of 2D- and 3D-simulations of gaseous discs in several respects (physical realism, accuracy, and computing time).