An accurate and simple, asymptotically matched deprojection of the Sersic law

Abstract

The Sersic law reproduces very well the surface brightness profile of early-type galaxies, and therefore is routinely used in observational and theoretical works. Unfortunately, its deprojection can not 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. We searched for a very simple, accurate, and theoretically motivated deprojection formula of the Sersic law, without free parameters, not based on fits of the numerical deprojection, and holding for generic $n > 1$. The formula has been found by requiring it to reproduce the analytical expressions for the inner and outer asymptotic expansions of the deprojected Sersic law of given $n$, and by matching the two expansions at intermediate radii with the request that the total luminosity coincides with that of the original Sersic profile of same $n$. The resulting formula is algebraically very simple, by construction its inner and outer parts are the exact (asymptotic) deprojection of the Sersic law, and it depends on two coefficients that are analytical functions of $n$ of immediate evaluation. The accuracy of the formula over the whole radial range is very good and increases for increasing $n$, with maximum relative deviations from the true numerical deprojection of $\simeq 8\,10^{-3}$ for the de Vaucouleurs profile. In the Appendix, the extension of the proposed formula to profiles with $n <1$ is also presented and discussed. The formula obtained is a useful tool of simple use in the modeling of early-type galaxies. Its ellipsoidal generalization is immediate.