Potential-density pairs for a family of finite disks

Abstract

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

13 pages 2 figures Accepted by ApJ

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