Relating Peak Optical Luminosity and Orbital Period of Stellar-Mass Black Holes in X-ray Binaries

We compare the peak optical luminosity with the orbital period for a sample of 22 stellar-mass black hole candidates with good measurements of both quantities. We find that the peak absolute magnitude for the outbursts follows a linear relation with $M_{V,peak}=3.48(\pm 0.85)-3.89(\pm 0.91)\mathrm{l}\mathrm{o}\mathrm{g}{P}_{orb}$, which corresponds to a $L_V\propto P_{orb}^{1.56\pm 0.36}$ power law relation. Excluding V4641 Sgr which is a strong outlier and not likely to have outbursts produced by the standard disc instability model, in addition to BW Cir and V821 Ara – both of which have highly uncertain distances; the new correlation for the 19 sources is found to be $M_{V,peak}=3.01(\pm 0.93)-3.21(\pm 1.04)\mathrm{l}\mathrm{o}\mathrm{g}{P}_{orb}$, which corresponds to $L_V\propto P_{orb}^{1.28\pm 0.42}$. This is an analogous relationship to the “Warner relation” between orbital period and peak luminosity found for cataclysmic variables. We discuss the implications of these results for finding black hole X-ray binaries in other galaxies and in our own Galaxy with the Large Synoptic Survey Telescope and other future large time domain surveys.