Generation of potential/surface density pairs in flat disks Power law distributions

TL;DR

This paper introduces a simple, analytical method to generate potential and surface density pairs in finite flat disks, significantly improving convergence and accuracy over classical multipole expansions, especially for power law distributions.

Contribution

The authors present a novel analytical approach that decomposes potential/density pairs into a homogeneous component and a residual series with cubic convergence, enhancing computational efficiency.

Findings

Residual series converges cubically inside the source.

Low order truncation yields potential errors below a few percent.

Method outperforms classical multipole expansion in convergence speed.

Abstract

We report a simple method to generate potential/surface density pairs in flat axially symmetric finite size disks. Potential/surface density pairs consist of a ``homogeneous'' pair (a closed form expression) corresponding to a uniform disk, and a ``residual'' pair. This residual component is converted into an infinite series of integrals over the radial extent of the disk. For a certain class of surface density distributions (like power laws of the radius), this series is fully analytical. The extraction of the homogeneous pair is equivalent to a convergence acceleration technique, in a matematical sense. In the case of power law distributions, the convergence rate of the residual series is shown to be cubic inside the source. As a consequence, very accurate potential values are obtained by low order truncation of the series. At zero order, relative errors on potential values do not exceed a few percent typically, and scale with the order N of truncation as 1/N**3. This method is superior to the classical multipole expansion whose very slow convergence is often critical for most practical applications.