  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).

Instrumentation and Methods for Astrophysics