Theoretical modelling of the AGN iron-line vs continuum time-lags in the lamp-post geometry

Context: Theoretical modelling of time-lags between variations in the Fe K$\alpha$ emission and the X-ray continuum might shed light on the physics and geometry of the X-ray emitting region in active galaxies (AGN) and X-ray binaries. We here present the results from a systematic analysis of time-lags between variations in two energy bands ($5-7$ vs $2-4\mathrm{k}\mathrm{e}\mathrm{V}$) for seven X-ray bright and variable AGN. Aims: We estimate time-lags as accurately as possible and fit them with theoretical models in the context of the lamp-post geometry. We also constrain the geometry of the X-ray emitting region in AGN. Methods: We used all available archival $\mathit{\text{XMM-Newton}}$ data for the sources in our sample and extracted light curves in the $5-7$ and $2-4\mathrm{k}\mathrm{e}\mathrm{V}$ energy bands. We used these light curves and applied a thoroughly tested (through extensive numerical simulations) recipe to estimate time-lags that have minimal bias, approximately follow a Gaussian distribution, and have known errors. Using traditional ${\chi }^2$ minimisation techniques, we then fitted the observed time-lags with two different models: a phenomenological model where the time-lags have a power-law dependence on frequency, and a physical model, using the reverberation time-lags expected in the lamp-post geometry. The latter were computed assuming a point-like primary X-ray source above a black hole surrounded by a neutral and prograde accretion disc with solar iron abundance. We took all relativistic effects into account for various X-ray source heights, inclination angles, and black hole spin values.