Astrometric Microlensing: Centroid Shifts as the Missing Half of the Mass Measurement

This paper computes how the displacement of the light centroid during an event, measurable with space astrometry, supplies the Einstein radius that photometry alone usually cannot. For a representative lens of 1.0 solar masses at 2.0 kiloparsecs, in front of a source at 8.0 kiloparsecs, the angular Einstein radius is 1.75 milliarcseconds, the event timescale is 106 days at a relative proper motion of 6.0 milliarcseconds per year, the microlensing parallax is 0.215, and the physical Einstein radius projected at the lens is 3.5 astronomical units. The central finding is that the maximum centroid shift is a fixed fraction of the Einstein radius, so a single well-sampled astrometric event delivers the angular scale directly. All quantities are computed from stated assumptions and published survey values used as comparison points; the study is simulated and computed, not measured. Where the computation shows that an inference commonly drawn from such events is not supported at the stated precision, that negative result is reported in full.