The X-ray light-curves of the recurring outbursts observed in low-mass X-ray binaries provide strong test beds for constraining (still) poorly understood disc-accretion processes. These light-curves act as a powerful diagnostic to probe the physics behind the mechanisms driving mass inflow and outflow in these binary systems. We have thus developed an innovative methodology, combining a foundation of Bayesian statistics, observed X-ray light-curves, and accretion disc theory. With this methodology, we characterize the angular-momentum (and mass) transport processes in an accretion disc, as well as the properties of the X-ray irradiation-heating that regulates the decay from outburst maximum in low-mass X-ray transients. We recently applied our methodology to the Galactic black-hole low-mass X-ray binary population, deriving from their lightcurves the first-ever quantitative measurements of the $alpha$-viscosity parameter in these systems citep{tetarenko2018}. In this paper, we continue the study of these binaries, using Bayesian methods to investigate the X-ray irradiation of their discs during outbursts of strong accretion. We find that the predictions of the disc-instability model, assuming a source of X-ray irradiation proportional to the central accretion rate throughout outburst, do not adequately describe the later stages of BH-LMXB outburst light-curves. We postulate that the complex and varied light-curve morphology observed across the population is evidence for irradiation that varies in time and space within the disc, throughout individual transient outbursts. Lastly, we demonstrate the robustness of our methodology, by accurately reproducing the synthetic model light-curves computed from numerical codes built to simulate accretion flows in binary systems.
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