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One of the canonical physical properties of ultra-fast outflows (UFOs) seenin a diverse population of active galactic nuclei (AGNs) is its seemingly verybroad width (i.e. $\Delta v \sim 10,000$ km~s$^{-1}$) , a feature oftenrequired for X-ray spectral modeling. While unclear to date, this condition isoccasionally interpreted and justified as internal turbulence within the UFOsfor simplicity. In this work, we exploit a transverse motion of athree-dimensional accretion-disk wind, an essential feature of non-radialoutflow morphology unique to magnetohydrodynamic (MHD) outflows. We argue thatat least part of the observed line width of UFOs may reflect the degree oftransverse velocity gradient due to Doppler broadening around a putativecompact X-ray corona in the proximity of a black hole. In this scenario, linebroadening is sensitive to the geometrical size of the corona, $R_c$. Wecalculate the broadening factor as a function of coronal radius $R_c$ andvelocity smearing factor $f_{\rm sm}$ at a given plasma position. Wedemonstrate, as a case study of the quasar, PDS~456, that the spectral analysisfavors a compact coronal size of $R_c /R_g \lesssim 10$ where $R_g$ isgravitational radius. Such a compact corona is long speculated from both X-rayreverberation study and the lamppost model for disk emission also consistentwith microlensing results. Combination of such a transverse broadening around asmall corona can be a direct probe of a substantial rotational motion perhapsposing a serious challenge to radiation-driven wind viewpoint.

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