Solutions of the axi-symmetric Poisson equation from elliptic integrals. II. Semi-analytical approach

International audienceIn a series of two papers, we present numerical integral-based methods to compute accurately the self-gravitating field and potential induced by a tri-dimensional, axially symmetric fluid, with special regard for tori, discs and rings. In this second article, we show that "point mass" singularities are integrable analytically for systems with aspect ratio (H/R)2 ≪ 1. We derive second-order accurate, integral formulae for the field components and potential as well, assuming that the mass density locally expands following powers of the altitude (the parabolic case is treated in detail). These formulae are valid inside the entire system: from the equatorial plane to the surface, and especially at the inner and outer edges where they remain regular, in contrast to those derived in the classical bi-dimensional, "razor-thin" approach. Their relative precision ~(H/R)2 has been checked in many situations by comparison with highly accurate, numerical solutions of the Poisson equation obtained from splitting methods described in Paper I. Time inexpensive and reliable, they offer powerful means to investigate vertically stratified systems where self-gravity plays a role. Three formulae for "one zone" disc models are given