Static Thin Disks with Power-law Density Profiles * * Dedicated to our teacher, Professor Jiří Bičák, on the occasion of his 80th birthday. For O.S., Jiří's interest in thin disks was, 30 yr ago (Bičák et al. ), the first motivation to study the topic.

Abstract

The task of finding the potential of a thin circular disk with power-law radial density profile is revisited. The result, given in terms of infinite Legendre-type series in the above reference, has now been obtained in closed form thanks to the method of Conway employing Bessel functions. Starting from a closed-form expression for the potential generated by the elementary density term ρ 2l , we cover more generic—finite solid or infinite annular—thin disks using superposition and/or inversion with respect to the rim. We check several specific cases against the series-expansion form by numerical evaluation at particular locations. Finally, we add a method to obtain a closed-form solution for finite annular disks whose density is of “bump” radial shape, as modeled by a suitable combination of several powers of radius. Density and azimuthal pressure of the disks are illustrated on several plots, together with radial profiles of free circular velocity.