Static Thin Disks with Power-law Density Profiles ^*

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.

Black Hole Encircled by a Thin Disk: Fully Relativistic Solution*

“…Later, found a potential of a disk with a general power-law density profile in terms of infinite series and studied its properties when superposed with a black hole. In a recent revision of this topic in Kotlařík et al (2022), we provided this result in closed form. All these disks are thin and infinite and have an inner rim, where higher derivatives of curvature are singular.…”

Black hole encircled by a thin disk: fully relativistic solution

“…Later, Semerák (2004) found a potential of a disk with a general power-law density profile in terms of infinite series and studied its properties when superposed with a black hole. In a recent revision of this topic in Kotlařík et al (2022), we provided this result in closed-form. All these disks are thin, infinite, and have an inner rim, where higher derivatives of curvature are singular.…”