Inversion of the Abel equation for toroidal density distributions

TL;DR

This paper introduces new methods for Abel inversion in astronomy, enabling the recovery of three-dimensional toroidal emissivity distributions from projected brightness profiles, with applications to Gaussian and multimodal profiles.

Contribution

It presents a novel inversion formula for toroidal emissivity with constant surface brightness and demonstrates its effectiveness with Gaussian and multimodal brightness profiles.

Findings

Inversion formula allows recovery of toroidal distributions from projected brightness.

Gaussian profiles lead to simple, non-negative spatial emissivity.

Multimodal brightness distributions can be deprojected into sums of non-negative toroidal emissivities.

Abstract

In this paper I present three new results of astronomical interest concerning the theory of Abel inversion. 1) I show that in the case of a spatial emissivity that is constant on toroidal surfaces and projected along the symmetry axis perpendicular to the torus' equatorial plane, it is possible to invert the projection integral. From the surface (i.e. projected) brightness profile one then formally recovers the original spatial distribution as a function of the toroidal radius. 2) By applying the above-described inversion formula, I show that if the projected profile is described by a truncated off-center gaussian, the functional form of the related spatial emissivity is very simple and - most important - nowhere negative for any value of the gaussian parameters, a property which is not guaranteed - in general - by Abel inversion. 3) Finally, I show how a generic multimodal centrally symmetric brightness distribution can be deprojected using a sum of truncated off-center gaussians, recovering the spatial emissivity as a sum of nowhere negative toroidal distributions.