Inversion of the Abel equation for toroidal density distributions

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.