The effect of returning radiation on relativistic reflection

We study the effect of returning radiation on the shape of the X-ray reflection spectrum in the case of thin accretion discs. We show that the returning radiation mainly influences the observed reflection spectrum for a large black hole spin (a > 0.9) and a compact primary source of radiation close to the black hole at height h < 5r_g, and that it dominates the reflected flux for extreme values of spin and compactness. The main effect of the returning radiation is to increase the irradiating flux on to the outer parts of the accretion disc, leading to stronger reflection and a flatter overall emissivity profile. By analysing simulated observations we show that neglecting returning radiation in existing studies of reflection-dominated sources has likely resulted in overestimating the height of the corona above the black hole. An updated version of the publicly available RELXILL suite of relativistic reflection models which includes returning radiation is also presented.