Constraining X-ray Coronal Size with Transverse Motion of AGN Ultra-Fast Outflows

One of the canonical physical properties of ultra-fast outflows (UFOs) seen in a diverse population of active galactic nuclei (AGNs) is its seemingly very broad width (i.e. $\Delta v \sim 10,000$ km~s$^{-1}$) , a feature often required for X-ray spectral modeling. While unclear to date, this condition is occasionally interpreted and justified as internal turbulence within the UFOs for simplicity. In this work, we exploit a transverse motion of a three-dimensional accretion-disk wind, an essential feature of non-radial outflow morphology unique to magnetohydrodynamic (MHD) outflows. We argue that at least part of the observed line width of UFOs may reflect the degree of transverse velocity gradient due to Doppler broadening around a putative compact X-ray corona in the proximity of a black hole. In this scenario, line broadening is sensitive to the geometrical size of the corona, $R_c$. We calculate the broadening factor as a function of coronal radius $R_c$ and velocity smearing factor $f_{\rm sm}$ at a given plasma position. We demonstrate, as a case study of the quasar, PDS~456, that the spectral analysis favors a compact coronal size of $R_c /R_g \lesssim 10$ where $R_g$ is gravitational radius. Such a compact corona is long speculated from both X-ray reverberation study and the lamppost model for disk emission also consistent with microlensing results. Combination of such a transverse broadening around a small corona can be a direct probe of a substantial rotational motion perhaps posing a serious challenge to radiation-driven wind viewpoint.