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.本仓库包含一份完整独立的Jupyter笔记本，以及全部所需的CSV与TXT文件，用于复现已接收学术论文《BL Lacertae天体在8–43 GHz频段实现5–30微角秒高精度近乎无偏的核心位移探测》中的全部图表。 该笔记本提供了一套完整且可复现的分析流程。为便于使用，我们还附带了通过`jupyter-nbconvert`生成的HTML版本，可直接浏览查看。 摘要 当射流在喷流启动区域存在部分光学厚区域时，其视在紧致核心可能表现出与频率相关的位置偏移。对核心位移开展高精度天体测量，能够帮助天文学家准确定位喷流起源，并对磁场施加严格约束。BL Lacertae是典型的BL Lac天体，拥有高度可变且准直性极佳的喷流。为独立约束其最内侧核心位移，我们在8.4、12.4、15.2、23.6与43.2 GHz频段开展了甚长基线干涉测量（Very Long Baseline Interferometry, VLBI）观测。通过利用角距为13.3角分的邻近陡谱校准源NVSS J220340+420839，采用反相位参考VLBI天体测量技术，我们以5–30 微角秒（micro-arcsecond, μas）的当前顶尖精度，探测得到近乎无偏的二维核心位移测量结果，其显著性高于3σ。8.4 GHz与43.2 GHz频段间的核心位移可达250 $mathrm{mu as}$。视在核心位移随频率按$ u^{-1/k_r}$规律缩放，表明喷流上游存在光学厚区域。我们推导得到的核心位移指数$k_r = 1.18^{+0.59}_{-0.34}$，在误差范围内与喷流中粒子与磁场能量密度满足能量均分条件下的经典值$k_r=1$相符；考虑到本次观测时BL Lacertae处于耀发状态，该结果也允许存在适度偏差。