Evidence for low kick velocities among high-mass X-ray binaries in the Small Magellanic Cloud from the spatial correlation function

We present the two-point cross-correlation function between high-mass X-ray binaries (HMXBs) in the Small Magellanic Cloud (SMC) and their likely birthplaces (OB Associations: OBAs). This function compares the spatial correlation between the observed HMXB and OBA populations against mock catalogs in which the members are distributed randomly across the sky. A significant correlation (15 sigma) is found for the HMXB and OBA populations when compared with a randomized catalog in which the OBAs are distributed uniformly over the SMC. A less significant correlation (4 sigma) is found for a randomized catalog of OBAs built with a bootstrap method. However, no significant correlation is detected when the randomized catalogs assume the form of a Gaussian ellipsoid or a distribution that reflects the star-formation history from 40 Myr ago. Based on their observed distributions and assuming a range of migration timescales, we infer that the average value of the kick velocity inherited by an HMXB during the formation of its compact object is 2-34 km/s. This is considerably less than the value obtained for their counterparts in the Milky Way hinting that the galactic environment affecting stellar evolution plays a role in setting the average kick velocity of HMXBs.