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We use the Very Long Baseline Array to conduct high precision astrometry of a sample of 33 compact, flat spectrum, variable radio sources in the direction of the Galactic plane (Becker et al. 2010). Although Becker et al. (2010) ruled out a few potential scenarios for the origin of the radio emission, the study could not rule out that these sources were black hole X-ray binaries (BHXBs). Most known BHXBs are first detected by X-ray or optical emission when they go into an outburst, leaving the larger quiescent BHXB population undiscovered. In this paper, we attempt to identify any Galactic sources amongst the Becker et al. (2010) sample by measuring their proper motions as a first step to finding quiescent BHXB candidates. Amongst the 33 targets, we could measure the proper motion of six sources. We find that G32.7193-0.6477 is a Galactic source and are able to constrain the parallax of this source with a 3σ significance. We found three strong Galactic candidates, G32.5898-0.4468, G29.1075-0.1546, and G31.1494-0.1727, based purely on their proper motions, and suggest that G29.1075-0.1546 is also likely Galactic. We detected two resolved targets for multiple epochs (G30.1038+0.3984 and G29.7161-0.3178). We find six targets are only detected in one epoch and have an extended structure. We cross-match our VLBA detections with the currently available optical, infrared, and X-ray surveys, and did not find any potential matches. We did not detect 19 targets in any VLBA epochs and suggest that this could be due to limited uv-coverage, drastic radio variability, or faint, extended nature of the sources.

We use very long baseline interferometry to measure the proper motions of three black hole X-ray binaries (BHXBs). Using these results together with data from the literature and Gaia DR2 to collate the best available constraints on proper motion, parallax, distance, and systemic radial velocity of 16 BHXBs, we determined their three-dimensional Galactocentric orbits. We extended this analysis to estimate the probability distribution for the potential kick velocity (PKV) a BHXB system could have received on formation. Constraining the kicks imparted to BHXBs provides insight into the birth mechanism of black holes (BHs). Kicks also have a significant effect on BH–BH merger rates, merger sites, and binary evolution, and can be responsible for spin–orbit misalignment in BH binary systems. 75 per cent of our systems have potential kicks $\gt 70\, \rm {km\,s^{-1}}$. This suggests that strong kicks and hence spin–orbit misalignment might be common among BHXBs, in agreement with the observed quasi-periodic X-ray variability in their power density spectra. We used a Bayesian hierarchical methodology to analyse the PKV distribution of the BHXB population, and suggest that a unimodal Gaussian model with a mean of 107 $\pm \,\,16\, \rm {km\,s^{-1}}$ is a statistically favourable fit. Such relatively high PKVs would also reduce the number of BHs likely to be retained in globular clusters. We found no significant correlation between the BH mass and PKV, suggesting a lack of correlation between BH mass and the BH birth mechanism. Our python code allows the estimation of the PKV for any system with sufficient observational constraints.