Implementation of a radial disk ionization profile in the relxill_nk model

Very steep reflection emissivity profiles in the inner part of accretion disks are commonly found in the analysis of x-ray observations of black hole binaries and active galactic nuclei, but there is some debate about their exact origin. While steep reflection emissivity profiles can be naturally produced by compact coronae close to black holes, the measured radial emissivity parameter can be further increased by the radial disk ionization profile when the theoretical model assumes a disk with constant ionization. In this paper, we implement the possibility of a radial disk ionization profile in the reflection model relxill_nk, which is a package designed to calculate reflection spectra of "deformed" Kerr black holes. We analyze a NuSTAR observation of the black hole binary EXO 1846-031, which was previously found to have a very high inner emissivity index. We find that the model with a radial disk ionization profile improves the fit, but the impact on the estimate of the black hole spin parameter and on the constraint of the deformation parameter is modest. However, we show that the analysis of future observations of Athena and eXTP will necessarily require models with a radial disk ionization profile to have accurate constraints of the deformation parameters.