Modelling the broad Fe K\alpha\ reverberation in the AGN NGC 4151

The recent detection of X-ray reverberation lags, especially in the Fe K\alpha\ line region, around Active Galactic Nuclei (AGN) has opened up the possibility of studying the time-resolved response (reflection) of hard X-rays from the accretion disk around supermassive black holes. Here, we use general relativistic transfer functions for reflection of X-rays from a point source located at some height above the black hole to study the time lags expected as a function of frequency and energy in the Fe K\alpha\ line region. We explore the models and the dependence of the lags on key parameters such as the height of the X-ray source, accretion disk inclination, black hole spin and black hole mass. We then compare these models with the observed frequency and energy dependence of the Fe K\alpha\ line lag in NGC 4151. Assuming the optical reverberation mapping mass of $4.6\times10^7~M_\odot$ we get a best fit to the lag profile across the Fe K\alpha\ line in the frequency range $(1-2)\times10^{-5}$ Hz for an X-ray source located at a height $h = 7^{+2.9}_{-2.6}~R_G$ with a maximally spinning black hole and an inclination $i < 30^\circ$.