Detections of nearly bias-free core shifts with 5-30 micro-arcsecond precisions at 8-43 GHz in BL Lacertae

This repository contains a single, fully self-contained Jupyter notebook, along with all required CSV and TXT files, to reproduce the figures presented in the accepted manuscript, *"Detections of nearly bias-free core shifts with 5–30 micro-arcsecond precisions at 8–43 GHz in BL Lacertae"*. The notebook provides a complete and reproducible analysis workflow. For convenience, a HTML version—generated with `jupyter-nbconvert`—is included for direct viewing. Abstract 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 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 $\mathrm{\mu as}$, significant at the $>3\sigma$ level. The core shift between 8.4 and 43.2 GHz reaches $250\,\mathrm{\mu as}$. The apparent core shifts scale with frequency as $\nu^{-1/k_r}$, implying the presence of an optically thick region upstream in the jet. The derived core-shift index, $k_r = 1.18^{+0.59}_{-0.34}$, is consistent, within uncertainties, with the canonical value $k_r = 1$ expected under energy equipartition between the particle and magnetic-field energy densities in the jet, while still allowing for modest deviations because BL Lacertae was observed in a flaring state.