Detections of nearly bias-free core shifts with 5-30 $\mu$as precisions at 8-43 GHz in BL Lacertae
When a radio jet is partially optically thick in the launching region, its apparent compact core may display frequency-dependent positional shifts. High-precision astrometric measurements of core shifts enable astronomers to pinpoint the jet's origin and place tight constraints on the magnetic field. BL Lacertae, the archetypal BL Lac object, hosts a highly variable and well-collimated jet. To independently constrain its innermost core shifts, we conducted very long baseline interferometric (VLBI) observations at 8.4, 12.4, 15.2, 23.6, and 43.2 GHz. By exploiting a nearby (13.3 arcmin) steep-spectrum calibrator (NVSS J220340+420839) through inverse phase-referencing VLBI astrometry, we detect nearly unbiased two-dimensional core shift measurements with state-of-the-art precisions of 5-30 $\mu$as, which are significant at $>3\sigma$ confidence. The core shift between 8.4 and 43.2 GHz reaches 250 $\mu$as. The apparent core shifts scale with frequency as $\nu^{-1/k_r}$, implying the existence of an optically thick region in the upstream of jet. The derived core-shift index, $k_r\!=\!1.18^{+0.59}_{-0.34}$, is consistent, within uncertainties, with the canonical $k_r\!=\!1$ expected under energy equipartition between the jet particle and magnetic field energy densities, while allowing for modest deviations given that BL Lacertae was captured in a flaring state.