The lamppost model: effects of photon trapping, the bottom lamp and disc truncation

We study the lamppost model, in which the primary X-ray sources in accreting black-hole systems are located symmetrically on the rotation axis on both sides of the black hole surrounded by an accretion disc. We show the importance of the emission of the source on the opposite side to the observer. Due to gravitational light bending, its emission can increase the direct (i.e., not re-emitted by the disc) flux by as much as an order of magnitude. This happens for near to face-on observers when the disc is even moderately truncated. For truncated discs, we also consider effects of emission of the top source gravitationally bent around the black hole. We also present results for the attenuation of the observed radiation with respect to that emitted by the lamppost as functions of the lamppost height, black-hole spin and the degree of disc truncation. This attenuation, which is due to the time dilation, gravitational redshift and the loss of photons crossing the black-hole horizon, can be as severe as by several orders of magnitude for low lamppost heights. We also consider the contribution to the observed flux due to re-emission by optically-thick matter within the innermost stable circular orbit.