2025PASJ...77..678K

We carry out astrometric Very Long Baseline Interferometry (VLBI) observations for Cygnus X-3, a high-mass X-ray binary (HMXB), with the Korean-Japanese joint VLBI array (KaVA) from 2020 to 2021. The observations have been optimized for the subdued phase when Cygnus X-3 is relatively faint, sidestepping a giant flare between 2020 and 2021. The position of the core in Cygnus X-3 was measured through the observations. Based on accumulating core positions from the available archival data, obtained over $\sim$38 yr with the Very Large Array (VLA), the Very Long Baseline Array (VLBA), and the European VLBI Network (EVN), in addition to our KaVA observations, we estimated the proper motion components to be $\mu _{\alpha }\cos \delta = -2.720\pm 0.019$ mas yr$^{-1}$ and $\mu _\delta = -3.693 \pm 0.019$ mas yr$^{-1}$ in RA and Dec, and $\mu _l \cos b = -4.587 \pm 0.027$ mas yr$^{-1}$ and $\mu _b = -0.001 \pm 0.004$ mas yr$^{-1}$ in galactic longitude and galactic latitude, respectively. The proper motion results are in good agreement with previous results. Furthermore, the uncertainties we obtained are reduced by a factor of 3. The proper motion measurement allows us to constrain the peculiar velocity of the system in three-dimensional space to be $292\pm 127$ km s$^{-1}$ with $1\sigma$ error. Based on the correlation between the orbital period and the peculiar velocity in HMXBs, a peculiar velocity with an orbital period of 0.2 d, like that of Cygnus X-3, is expected to be higher than 100 km s$^{-1}$ within $1\sigma$ error. This suggests the possibility that the compact star probably received a substantial kick during its birth, most likely caused by a supernova explosion rather than a direct collapse.