A parallax distance to 3C 273 through spectroastrometry and
  reverberation mapping
  Distance measurements for extragalactic objects are a fundamental problem in
astronomy and cosmology. In the era of precision cosmology, we urgently need
better measurements of cosmological distances to observationally test the
increasing $H_{0}$ tension of the Hubble constant measured from different
tools. Using spectroastrometry, GRAVITY at The Very Large Telescope
Interferometer successfully revealed the structure, kinematics and angular
sizes of the broad-line region (BLR) of 3C 273 with an unprecedentedly high
spatial resolution. Fortunately, reverberation mapping (RM) of active galactic
nuclei (AGNs) reliably provides linear sizes of their BLRs. Here we report a
joint analysis of spectroastrometry and RM observations to measure AGN
distances. We apply this analysis to 3C 273 observed by both GRAVITY and an RM
campaign,and find an angular distance of $551.5_{-78.7}^{+97.3}\, {\rm Mpc}$
and $H_{0}=71.5_{-10.6}^{+11.9}\,{\rm km\,s^{-1}\,Mpc^{-1}}$. Advantages of the
analysis are 1) its pure geometrical measurements and 2) it simultaneously
yields mass of the central black hole in the BLR. Moreover, we can conveniently
repeat measurements of selected AGNs to efficiently reduce the statistical and
systematic errors. Future observations of a reasonably sized sample ($\sim 30$
AGNs) will provide distances of the AGNs and hence a new way of measuring
$H_{0}$ with a high precision $\left(\lesssim 3\%\right)$ to test the $H_{0}$
tension.
