Potential-density pairs for a family of finite disks

TL;DR

This paper derives exact analytical potential-density pairs for three specific finite disk models with different surface density profiles, providing closed-form solutions useful for astrophysical applications.

Contribution

It introduces new closed-form solutions for the gravitational potential and field of three finite disks with specific surface densities, expanding analytical tools in astrophysics.

Findings

Closed-form solutions for potential and gravitational field.

Analytical expressions for three finite disk models.

Enhanced results for the n=0 disk potential.

Abstract

Exact analytical solutions are given for the three finite disks with surface density $\Sigma_n=\sigma_0 (1-R^2/\alpha^2)^{n-1/2} \textrm{with} n=0, 1, 2$. Closed-form solutions in cylindrical co-ordinates are given using only elementary functions for the potential and for the gravitational field of each of the disks. The n=0 disk is the flattened homeoid for which $\Sigma_{hom} = \sigma_0/\sqrt{1-R^2/\alpha^2}$. Improved results are presented for this disk. The n=1 disk is the Maclaurin disk for which $\Sigma_{Mac} = \sigma_0 \sqrt{1-R^2/\alpha^2}$. The Maclaurin disk is a limiting case of the Maclaurin spheroid. The potential of the Maclaurin disk is found here by integrating the potential of the n=0 disk over $\alpha$, exploiting the linearity of Poisson's equation. The n=2 disk has the surface density $\Sigma_{D2}=\sigma_0 (1-R^2/\alpha^2)^{3/2}$. The potential is found by integrating the potential of the n=1 disk.