Spatially Resolving the Kinematics of the $\lesssim 100\mu$as Quasar Broad-line Region Using Spectroastrometry II. The First Tentative Detection in a Luminous Quasar at $z=2.3$

Direct measurements of the masses of supermassive black holes (SMBHs) are key to understanding their growth and constrain their symbiotic relationship to their host galaxies. However, current methods used to directly measure black hole masses in active quasars become challenging or impossible beyond $z\gtrsim 0.2$. Spectroastrometry (SA) measures the spatial centroid of an object’s spectrum as a function of wavelength, delivering angular resolution far better than the point-spread function (PSF) for high signal-to-noise ratio observations. We observed the luminous quasar SDSS J212329.47–005052.9 at $z=2.279$ with the aim of resolving its $\sim 100\mu \mathrm{a}\mathrm{s}$ H$\alpha$ broad emission-line region (BLR), and present the first SA constraints on the size and kinematic structure of the BLR. Using a novel pipeline to extract the SA signal and reliable uncertainties, we achieved a centroiding precision of $\simeq 100\mu \mathrm{a}\mathrm{s}$, or $>2000\times$ smaller than the $K$-band AO-corrected PSF, yielding a tentative $3.2\sigma$ detection of an SA signal from the BLR. Modeling the BLR emission as arising from an inclined rotating disk with a mixture of coherent and random motions we constrain $r_{\mathrm{B}\mathrm{L}\mathrm{R}}=454_{-162}^{+565}\mu \mathrm{a}\mathrm{s}$ ($3.71_{-1.28}^{+4.65}\mathrm{p}\mathrm{c}$), providing a $95\mathrm{\%}$ confidence upper limit on the black hole mass $M_$\mathrm{B}\mathrm{H}$,\sin^2,i \leq1.8 \times10^9,$\mathrm{M}$\odot$\mathrm{.}Ourresultsagreewiththe$r$\mathrm{B}\mathrm{L}\mathrm{R}$-L$relationmeasuredforlower-$z$ quasars, but expands its dynamic range by an order of magnitude in luminosity. We did not detect the potentially stronger SA signal from the narrow-line region, but discuss in detail why it may be absent. Already with existing instrumentation, SA can deliver $\sim 6\times$ smaller uncertainties ($\sim 15\mu \mathrm{a}\mathrm{s}$) than achieved here, enabling $\sim 10\mathrm{\%}$ measurements of SMBH masses in high-$z$ quasars.