A key-formula to compute the gravitational potential of inhomogeneous discs in cylindrical coordinates

Abstract

We have established the exact expression for the gravitational potential of a homogeneous polar cell—an elementary pattern used in hydrodynamical simulations of gravitating discs. This formula, which is a closed-form, works for any opening angle and radial extension of the cell. It is valid at any point in space, i.e. in the plane of the distribution (inside and outside) as well as off-plane, thereby generalizing the results reported by Durand (Electrostatique. Vol. I. Les Distributions, 1953) for the circular disc. The three components of the gravitational acceleration are given. The mathematical demonstration proceeds from the incomplete version of Durand’s formula for the potential (based on complete elliptic integrals). We determine first the potential due to the circular sector (i.e. a pie-slice sheet), and then deduce that of the polar cell (from convenient radial scaling and subtraction). As a by-product, we generate an integral theorem stating that “the angular average of the potential of any circular sector along its tangent circle is ${\frac{2}{\pi}}$ times the value at the corner”. A few examples are presented. For numerical resolutions and cell shapes commonly used in disc simulations, we quantify the importance of curvature effects by performing a direct comparison between the potential of the polar cell and that of the Cartesian (i.e. rectangular) cell having the same mass. Edge values are found to deviate roughly like ${2 \times 10^{-3}\times N/256}$ in relative (N is the number of grid points in the radial direction), while the agreement is typically four orders of magnitude better for values at the cell’s center. We also produce a reliable approximation for the potential, valid in the cell’s plane, inside and close to the cell. Its remarkable accuracy, about ${5\times 10^{-4}\times N/256}$ in relative, is sufficient to estimate the cell’s self-acceleration.