The Newtonian potential of thin disks

¹ Université de Bordeaux, OASU, 2 rue de l’Observatoire, BP 89, 33271 Floirac Cedex, France
² CNRS, UMR 5804, LAB, 2 rue de l’Observatoire, BP 89, 33271 Floirac Cedex, France
e-mail: jean-marc.hure@obs.u-bordeaux1.fr

Received: 1 October 2010
Accepted: 30 March 2011

Abstract

The one-dimensional, ordinary differential equation (ODE) that satisfies the midplane gravitational potential of truncated, flat power-law disks is extended to the whole physical space. It is shown that thickness effects (i.e. non-flatness) can be easily accounted for by implementing an appropriate “softening length” λ. The solution of this “softened ODE” has the following properties: i) it is regular at the edges (finite radial accelerations); ii) it possesses the correct long-range properties; iii) it matches the Newtonian potential of a geometrically thin disk very well; and iv) it tends continuously to the flat disk solution in the limit λ → 0. As illustrated by many examples, the ODE, subject to exact Dirichlet conditions, can be solved numerically with efficiency for any given colatitude at second-order from center to infinity using radial mapping. This approach is therefore particularly well-suited to generating grids of gravitational forces in order to study particles moving under the field of a gravitating disk as found in various contexts (active nuclei, stellar systems, young stellar objects). Extension to non-power-law surface density profiles is straightforward through superposition. Grids can be produced upon request.