Inferring the iron K emissivity profiles of accretion discs irradiated by extended coronae

One of the most promising methods to measure the spin of an accreting black hole is fitting the broad iron K$\alpha$ line in the X-ray spectrum. The line profile also depends on the geometry of the hard X-ray emitting corona. To put constraints on the black hole spin and corona geometry, it is essential to understand how do they affect the iron K$\alpha$ line emissivity profile. In this work, we present calculations of the illumination and the iron K$\alpha$ emissivity profiles performed with the Monte-Carlo GR radiative transfer code Monk. We focus on distinction between the illumination and emissivity profiles, which is in most previous studies neglected. We show that especially for the case of black hole X-ray binaries (BHXRBs), the difference is very large. For active galactic nuclei (AGNs), the emissivity profile has a more similar shape as the illumination profile, but it is notably steeper in the innermost region within a few gravitational radii. We find out that the different behavior between AGN and black hole X-ray binary discs is due to the different energy spectra of the illuminating radiation. This suggests that the emissivity profile of the iron K$\alpha$ line cannot be determined by black hole spin and corona geometry alone and the energy spectrum of the illuminating radiation has to be taken into account. We also examined the effect of including the self-irradiation, and find it to be more important than the corona emission in BHXRBs.