Probing the jet size of two Black hole X-ray Binaries in the hard state

Using multi-frequency Very Long Baseline Interferometer (VLBI) observations, we probe the jet size in the optically thick hard state jets of two black hole X-ray binary (BHXRB) systems, MAXI J1820+070 and V404 Cygni. Due to optical depth effects, the phase referenced VLBI core positions move along the jet axis of the BHXRB in a frequency dependent manner. We use this "core shift" to constrain the physical size of the hard state jet. We place an upper limit of $0.3$\,au on the jet size measured between the 15 and 5 GHz emission regions of the jet in MAXI J1820+070, and an upper limit of $1.0$\,au between the $8.4$ and $4.8$\,GHz emission regions of V404 Cygni. Our limit on the jet size in MAXI J1820+070 observed in the low-hard state is a factor of $5$ smaller than the values previously observed in the high-luminosity hard state (using time lags between multi-frequency light curves), thus showing evidence of the BHXRB jet scaling in size with jet luminosity. We also investigate whether motion of the radio-emitting region along the jet axis could affect the measured VLBI parallaxes for the two systems, leading to a mild tension with the parallax measurements of Gaia. Having mitigated the impact of any motion along the jet axis in the measured astrometry, we find the previous VLBI parallax measurements of MAXI J1820+070 and V404 Cygni to be unaffected by jet motion. With a total time baseline of $8$ years, due to having incorporated fourteen new epochs in addition to the previously published ones, our updated parallax measurement of V404 Cygni is $0.450 \pm 0.018$\,mas ($2.226 \pm 0.091$\,kpc).