Spectroastrometry and Reverberation Mapping: the Mass and Geometric
  Distance of the Supermassive Black Hole in the Quasar 3C 273
  The quasar 3C 273 has been observed with infrared spectroastrometry (SA) on
broad Pa$\alpha$ line and optical reverberation mapping (RM) on broad H$\beta$
line. SA delivers information about the angular size and structure of the
Pa$\alpha$ broad-line region (BLR), while RM delivers information about the
physical size and structure of the H$\beta$ BLR. Based on the fact that the two
BLRs share the mass of the supermassive black hole (SMBH) and viewing
inclination, a combination of SA and velocity-resolved RM (SARM) thereby allows
us to simultaneously determine the SMBH mass and geometric distance through
dynamically modeling the two BLRs. We construct a suite of dynamical models
with different geometric configurations and apply a Bayesian approach to obtain
the parameter inferences. Overall the obtained masses and distances are
insensitive to specific BLR configurations but more or less depend on
parameterizations of the vertical distributions. The most probable model,
chosen in light of the Bayes factor, yields an angular-size distance of
$\log\,(D_{\rm A}/{\rm Mpc}) = 2.83_{-0.28}^{+0.32}$ and SMBH mass of
$\log\,(M_\bullet/M_\odot)=9.06_{-0.27}^{+0.21}$, which agrees with the
relationships between SMBH masses and bulge properties. The BLRs have an
inclination of $5_{-1}^{+1}$ degrees, consistent with that of the large-scale
jet in 3C 273. Our approach reinforces the capability of SARM analysis to
measure SMBH mass and distance of AGNs even though SA and RM observations are
undertaken with different emission lines and/or in different periods.
