Potential-density pairs for a family of finite disks

Abstract

Exact analytical solutions are given for the three finite disks with surface density $Σ_n=σ_0 (1-R^2/α^2)^{n-1/2} \textrm{with} n=0, 1, 2$. 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 $Σ_{hom} = σ_0/\sqrt{1-R^2/α^2}$. Improved results are presented for this disk. The n=1 disk is the Maclaurin disk for which $Σ_{Mac} = σ_0 \sqrt{1-R^2/α^2}$. 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 $α$, exploiting the linearity of Poisson's equation. The n=2 disk has the surface density $Σ_{D2}=σ_0 (1-R^2/α^2)^{3/2}$. The potential is found by integrating the potential of the n=1 disk.