Detecting Galactic MACHOs with VERA through Astrometric Microlensing of Distant Radio Sources

In this paper we investigate the properties of astrometric microlensing of distant radio sources (QSOs and radio galaxies) due to MACHOs, and discuss their implications for VERA (VLBI Exploration of Radio Astrometry). First we show that in case of astrometric microlensing of distant sources, the event duration is only a function of the lens mass and tangential velocity, but independent of the lens distance, in contrast to the well-known three-fold degeneracy for photometric microlensing. Moreover, the lens mass $M$ is scaled by the tangential velocity $v_\perp$ as $M\propto v_\perp$, rather than $M\propto v_\perp^2$ which is the case for photometric microlensing. Thus, in astrometric microlensing the dependence of the lens mass on the unknown parameter $v_\perp$ is weaker, indicating that the duration of astrometric microlensing event is a better quantity to study the mass of lensing objects than that of photometric microlensing. We also calculate the optical depth and event rate, and show that within 20$^\circ$ of the galactic center a typical event rate for 10 $\mu$as-level shift is larger than $2.5 \times 10^{-4}$ event per year, assuming that a quarter of the halo is made up with MACHOs. This indicates that if one monitors a few hundred sources for $\sim$20 years, such an astrometric microlensing event is detectable. Since a typical event duration is found to be fairly long (5 to 15 years), the frequency of the monitoring observation can be relatively low, i.e., once per six months, which is rather reasonable for practical observations. We discuss practical strategy for observing astrometric microlensing with VERA, and argue that an astrometric microlensing event due to MACHOs can be detected by VERA within a few decades.