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We use Very Long Baseline Interferometry to measure the proper motions ofthree black hole X-ray binaries (BHXBs). Using these results together with datafrom the literature and Gaia-DR2 to collate the best available constraints onproper motion, parallax, distance and systemic radial velocity of 16 BHXBs, wedetermined their three dimensional Galactocentric orbits. We extended thisanalysis to estimate the probability distribution for the potential kickvelocity (PKV) a BHXB system could have received on formation. Constraining thekicks imparted to BHXBs provides insight into the birth mechanism of blackholes (BHs). Kicks also have a significant effect on BH-BH merger rates, mergersites, and binary evolution, and can be responsible for spin-orbit misalignmentin BH binary systems. $75\%$ of our systems have potential kicks$>70\,\rm{km~s^{-1}}$. This suggests that strong kicks and hence spin-orbitmisalignment might be common among BHXBs, in agreement with the observedquasi-periodic X-ray variability in their power density spectra. We used aBayesian hierarchical methodology to analyse the PKV distribution of the BHXBpopulation, and suggest that a unimodal Gaussian model with a mean of$107\pm16\,\rm{km~s^{-1}}$ is a statistically favourable fit. Such relativelyhigh PKVs would also reduce the number of BHs likely to be retained in globularclusters. 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 sufficientobservational constraints.

Potential Kick Velocity distribution of black hole X-ray binaries and implications for natal kicks