The Newtonian potential of thin disks

TL;DR

This paper extends a 1D ODE for the gravitational potential of thin disks to full space, incorporating thickness effects via a softening length, enabling efficient numerical solutions for various astrophysical disk scenarios.

Contribution

It introduces a softened ODE that accurately models the potential of thick disks and can be efficiently solved numerically for diverse astrophysical applications.

Findings

The softened ODE matches the thin disk potential well.

Solutions are regular at edges and have correct long-range behavior.

Method allows fast grid generation for gravitational force calculations.

Abstract

The one-dimensional, ordinary differential equation (ODE) by Hur\'e & Hersant (2007) 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" $\lambda$. 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 $\lambda \rightarrow 0$. 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.