Analytical expressions for the deprojected Sersic model

TL;DR

This paper derives exact analytical formulas for the deprojected Sersic model's luminosity density, enabling precise calculations of galaxy brightness profiles for all Sersic indices using special functions.

Contribution

It generalizes previous work by providing a universal analytical expression for the Sersic luminosity density using the Fox H function for all indices.

Findings

Exact expression for luminosity density in terms of Fox H function

Simplified formulas using Meijer G function for rational indices

Analysis of asymptotic behavior at small and large radii

Abstract

The Sersic model has become the standard to parametrize the surface brightness distribution of early-type galaxies and bulges of spiral galaxies. A major problem is that the deprojection of the Sersic surface brightness profile to a luminosity density cannot be executed analytically for general values of the Sersic index. Mazure & Capelato (2002) used the Mathematica computer package to derive an expression of the Sersic luminosity density in terms of the Meijer G function for integer values of the Sersic index. We generalize this work using analytical means and use Mellin integral transforms to derive an exact, analytical expression for the luminosity density in terms of the Fox H function for all values of the Sersic index. We derive simplified expressions for the luminosity density, cumulative luminosity and gravitational potential in terms of the Meijer G function for all rational values of the Sersic index and we investigate their asymptotic behaviour at small and large radii. As implementations of the Meijer G function are nowadays available both in symbolic computer algebra packages and as high-performance computing code, our results open up the possibility to calculate the density of the Sersic models to arbitrary precision.