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

TL;DR

This paper derives a mathematically justified, local prescription for the softening length in 2D simulations of self-gravitating gaseous discs, improving physical realism and computational accuracy.

Contribution

It provides an analytical formula for the softening length that matches the potential of a thin disc, addressing a gap in prior heuristic approaches.

Findings

LAMBDA is a fraction of the local disc thickness H.

LAMBDA is independent of the numerical mesh.

LAMBDA typically ranges between 0.13 and 0.29 times H.

Abstract

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 LAMBDA is twofold: i) to avoid numerical singularities in the integral representation of the potential (i.e., arising when the relative separation vanishes), and ii) to acount for stratification of matter in the direction perpendicular to the disc mid-plane. So far, most studies have considered LAMBDA 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 LAMBDA such that the gravitational potential of a flat disc coincides at order zero with that of a geometically thin disc of the same surface density. Our prescription for LAMBDA, 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 LAMBDA : 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 < LAMBDA/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).