Lorentz Factors of compact jets in Black hole X-ray binaries

Compact, continuously launched jets in black hole X-ray binaries (BHXBs) produce radio to optical-infrared synchrotron emission. In most BHXBs, an infrared (IR) excess (above the disc component) is observed when the jet is present in the hard spectral state. We investigate why some BHXBs have prominent IR excesses and some do not, quantified by the amplitude of the IR quenching or recovery over the transition from/to the hard state. We find that the amplitude of the IR excess can be explained by inclination dependent beaming of the jet synchrotron emission, and the projected area of the accretion disc. Furthermore, we see no correlation between the expected and the observed IR excess for Lorentz factor 1, which is strongly supportive of relativistic beaming of the IR emission, confirming that the IR excess is produced by synchrotron emission in a relativistic outflow. Using the amplitude of the jet fade and recovery over state transitions and the known orbital parameters, we constrain for the first time the bulk Lorentz factor range of compact jets in several BHXBs (with all the well-constrained Lorentz factors lying in the range of $\Gamma$ = 1.3 - 3.5). Under the assumption that the Lorentz factor distribution of BHXB jets is a power-law, we find that N($\Gamma$) $\propto \Gamma^{ -1.88^{+0.27}_{-0.34}}$. We also find that the very high amplitude IR fade/recovery seen repeatedly in the BHXB GX 339-4 favors a low inclination angle ($< 15^\circ$) of the jet.