Solutions of the axi-symmetric Poisson equation from elliptic integrals

Abstract

Abstract. In a series of two papers, we present numerical, integral-based methods to compute accurately the self-gravitating field and potential induced by tri-dimensional, axially symmetric fluids, with a special regard for tori, discs and rings. This first article is concerned with a fully numerical approach. Complex shapes, small/large aspect ratios, important density gradients and compact/extended systems can be accounted for. Loop singularities in the Poisson integrals are carefully treated from kernel splitting and/or density splitting. Field components are obtained from density splitting: the local density field is separated into a vertically homogeneous contribution for which the integrable singularity is known in a closed form, plus a "residual" contribution which yields a regular integrand. This technique is exact in the vertically homogeneous limit. The potential is computed from double splitting: kernel splitting (to isolate explicitly the singular function), followed by density splitting. In each two directions, numerical quadratures are performed using a dynamical mesh (very efficient for flat and extended systems), combined with a high-order, irregular-spacing scheme. Global performances are demonstrated through a few test-configurations. The accuracy is potentially very high, and can reach the computer precision (for simple geometries and smooth density profiles). In terms of computing time to precision ratio, present methods are asymptotically more efficient than classical finite-difference methods. These can be employed either as a "Poisson-solver" or only to compute a sub-set of Dirichlet/Neumann-type boundary conditions in order to initialize spectral/finite-difference/finite element methods. As an example of an astrophysical application, we determine the structure of an uniformly rotating polytrope belonging to the compressible, Dyson-Wong sequence. In a second paper (Paper II), we show that singularities are integrable analytically for geometrically thin systems where the density is of the form ρ ∝ z n . Improvements and extensions (such as the non-axi-symmetric case) are possible and discussed.

On the rotation curves for axially symmetric disc solutions of the Vlasov–Poisson system

“…Since this equation involves only a one-dimensional integration we can handle this fact by using a sufficiently high resolution in the radial variable and replacing the cases s = r by s = r + dr. In a fully three dimensional, axially symmetric situation more sophisticated methods will be required as discussed in (Huré 2005). …”

Modeling the NIR-silhouette massive disk candidate in M 17

“…The precise knowledge of the radial density profile with a power index close to −1 together with the meridional crosssection of the disk enable however a more quantitative discussion. We have numerically computed the gravitational field due to the disk, using the density splitting method as described in Huré (2005), which yields accurate solutions of the Poisson equation in axially symmetric 3D-systems, for various shapes and density profiles. Figure 12 displays the ratio of the central proto-star gravitational acceleration g z to the disk's gravity g disk z in the (r, z)-plane, assuming that the mass contained in the M 17 silhouette and the central mass are equal.…”

The Newtonian potential of thin disks

“…24 with the Newtonian potential of a geometrically thin disk with the same edges, same surface density (and mass), and semithickness h ∝ a (precisely ǫ = 0.1) (hereafter thin disk configuration A). This reference is computed from the splitting method (Huré 2005), which is very accurate. A typical result, limited to the vicinity of the inner edge, is shown in Fig.…”