Discerning Parallax Amplitude in Astrometric Microlensing

Gravitational microlensing is a powerful method for discovering isolated stellar-mass black holes (ISMBHs). These objects make long-duration microlensing events. To characterize these lensing objects by fully resolving the microlensing degeneracy, measurements of parallax and astrometric deflections are necessary. Microlensing events due to ISMBHs have considerable astrometric deflections but small parallax amplitudes, ${\pi }_{{\rm{E}}}\propto 1/\sqrt{{M}_{{\rm{l}}}}$, where Ml is the lens mass. We numerically investigate the possibility of inferring parallax amplitude from astrometric deflection in microlensing events due to ISMBHs. The parallax amplitude in astrometric deflections is proportional to the relative parallax πrel, which means that it (i) does not strongly depend on Ml and (ii) increases in microlensing observations toward the Magellanic Clouds. We assume that these events will be potentially detected in upcoming microlensing surveys such as (1) the Roman observations of the Galactic bulge (GB) and (2) the LSST observations of the Large Magellanic Cloud (LMC) and that the Extremely Large Telescope (ELT) will follow up on them with one data point every 10 days. We evaluate the probability of inferring parallax amplitude from these observations by calculating the Fisher and covariance matrices. For the GB, the efficiencies for discerning parallax amplitudes with a relative error of <4% through astrometric and photometric observations are 3.8% and 29.1%, respectively. For observations toward the LMC, these efficiencies are 41.1% and 23.0%, respectively. Measuring parallax amplitude through astrometric deflections is plausible in GB events with a lens distance of ≲2.7 kpc and in LMC halo lensing. By monitoring long-duration microlensing events, the ELT can detect astrometric deflections and their parallax-induced deviations.