An accurate and simple asymptotically matched deprojection of the Sérsic law

¹ Department of Physics and Astronomy, University of Bologna, Via Gobetti 93/2, 40129 Bologna, Italy
² INAF – Osservatorio di Astrofisica e Scienza dello Spazio di Bologna, Via Gobetti 93/3, 40129 Bologna, Italy
³ International PhD College – Collegio Superiore, University of Bologna, Bologna, Italy

^⋆ Corresponding author; luca.ciotti@unibo.it

Received: 11 October 2024
Accepted: 2 January 2025

Abstract

Context. The Sérsic law reproduces very well the surface brightness profile of early-type galaxies, and therefore it is routinely used in observational and theoretical works. Unfortunately, its deprojection cannot be expressed in terms of elementary functions for generic values of the shape parameter n. Over the years, different families of approximate deprojection formulae have been proposed, generally based on fits of the numerical deprojection over some radial range.

Aims. We searched for a very simple, accurate, and theoretically motivated deprojection formula of the Sérsic law without free parameters, not based on fits of the numerical deprojection, and holding for generic n > 1.

Methods. We found the formula by requiring it to reproduce the analytical expressions for the inner and outer asymptotic expansions of the deprojected Sérsic law of a given n and by matching the two expansions at intermediate radii with the request that the total luminosity coincides with that of the original Sérsic profile of the same n.

Results. The resulting formula is algebraically very simple. By construction, its inner and outer parts are the exact (asymptotic) deprojection of the Sérsic law, and it depends on two coefficients that are analytical functions of the n of immediate evaluation. The accuracy of the formula over the whole radial range is very good and increases for increasing n, with a maximum of relative deviations from the true numerical deprojection of ≃8 10⁻³ for the de Vaucouleurs profile. In the appendix, the extension of the proposed formula to profiles with n < 1 is also presented and discussed.

Conclusions. The formula we obtained is a simple and useful tool that can be used in the modeling of early-type galaxies, and its ellipsoidal generalization is immediate.