The gravitational force and potential of the finite Mestel disk

TL;DR

This paper derives exact solutions for the gravitational potential and field of the finite Mestel disk, a realistic galaxy model with a cuspy inner and exponential outer mass distribution, requiring less mass to match observed velocities.

Contribution

It provides the first closed-form solutions for the potential and gravitational field of the finite Mestel disk across all space.

Findings

Exact solutions for potential and field are obtained.

Finite Mestel disk requires less mass than other models.

The disk's mass distribution is cuspy in the center and exponential outward.

Abstract

Mestel determined the surface mass distribution of the finite disk for which the circular velocity is constant in the disk and found the gravitational field for points in the $z=0$ plane. Here we find the exact closed form solutions for the potential and the gravitational field of this disk in cylindrical coordinates over all the space. The Finite Mestel Disk (FMD) is characterized by a cuspy mass distribution in the inner disk region and by an exponential distribution in the outer region of the disk. The FMD is quite different from the better known exponential disk or the untruncated Mestel disk which, being infinite in extent, are not realistic models of real spiral galaxies. In particular, the FMD requires significantly less mass to explain a measured velocity curve.