Finding the birthplace of HMXBs in the Galaxy using Gaia EDR3: kinematical age determination through orbit integration

High-Mass X-ray Binaries (HMXBs) are produced after the first supernova event in a massive binary. These objects are intrinsically young, and can suffer from a significant natal kick. As such, the progenitors of HMXBs are likely to have formed away from the current location of the X-ray emitting systems. We aim to find the birthplace of the known HMXBs of our Milky Way. Specifically, we want to answer the question whether the formation of HMXBs can be associated to open stellar clusters and/or Galactic spiral structures, and infer from that the time elapsed since the first supernova event. We use astrometric data from the Gaia EDR3 to initialize the position and velocity of each known HMXBs from the Galaxy, and integrate their motion back in time. In parallel, we perform the same calculations on a sample of 1381 open clusters detected by Gaia as well as for four Galactic spiral arms which shape and motion have also been recently modelled using Gaia data. We report on all the encounter candidates between HMXBs and clusters or spiral arms in the past 100 Myr. In our sample of 26 HMXBs, we infer that 7 were born in clusters, 8 were born near a Galactic spiral arm, and conclude that 7 others could have formed isolated. The birthplaces of the remaining 4 HMXBs are still inconclusive due to a combination of great distance, poor astrometric data and lack of known open cluster in the vicinity. We provide the kinematical age since supernova of 15 HMXBs. The astrometry from Gaia and the orbit integration we employ are effective at finding the birthplaces of HMXBs in the Milky Way. By considering the biases in our data and method, we find it is likely that the progenitors of HMXBs preferentially formed alongside other massive stars in open clusters.