A Precise Analytical Approximation for the Deprojection of the S\'ersic Profile

The S\'ersic model is known to fit well the surface brightness (or surface density) profiles of elliptical galaxies and galaxy bulges, and possibly for dwarf spheroidal galaxies and globular clusters. The deprojected density and mass profiles are important for many astrophysical applications, in particular for mass-orbit modeling of these systems. However, the exact deprojection formula for the S\'ersic model employs special functions not available in most computer languages. We show that all previous analytical approximations to the 3D density profile are imprecise at low S\'ersic index ($n \lesssim 1.5$). We have derived a more precise analytical approximation to the deprojected S\'ersic density profile by fitting two-dimensional 10th-order polynomials to the differences of the logarithms of the numerical deprojection and of the analytical approximation by Lima Neto et al. (1999, LGM) of the density profile on one hand and of the mass profile on the other. Our LGM-based polynomial fits have typical relative precision better than $0.2\%$ for both density and mass profiles, for S\'ersic indices $0.5 \leq n \leq 10$ and radii $0.001 < r/R_{\rm e} < 1000$. Our approximation is much more precise than those of LGM, Simonneau & Prada (1999, 2004), Trujillo et al. (2002) for non-half-integer values of the index, and of Emsellem & van de Ven (2008) for non-one-tenth-integer values with $n \lesssim 3$, and are nevertheless more than $0.2\%$ precise for larger S\'ersic indices, for both density and mass profiles. An appendix compares the deprojected S\'ersic profiles with those of the popular simple models from Plummer (1911), Jaffe (1983), Hernquist (1990), Navarro et al. (1996), and Einasto (1965).