Potential-density pairs for a family of finite disks

Exact analytical solutions are given for the three finite disks with surface density ${\mathrm{\Sigma }}_n={\sigma }_0(1-R^2\mathrm{/}{\alpha }^2{)}^{n-1\mathrm{/}2}\text{with}n=0,1,2$. 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.