First Detection and Modeling of Spatially Resolved Ly$\alpha$ in TW Hya

Lyman-$\alpha$ (Ly$\alpha$) is the strongest emission line in the accretion-generated UV spectra from T-Tauri stars and, as such, plays a critical role in regulating chemistry within the surrounding protoplanetary disks. Due to its resonant nature, the scattering of Ly$\alpha$ photons along the line-of-sight encodes information about the physical properties of the intervening H I medium. In this work, we present the first spatially resolved spectral images of Ly$\alpha$ emission across a protoplanetary disk in the iconic face-on T-Tauri star TW Hya, observed with HST-STIS at spatial offsets 0$''$, $\pm 0.2''$, and $\pm 0.4''$. To comprehensively interpret these Ly$\alpha$ spectra, we utilize a 3D Monte-Carlo Ly$\alpha$ radiative transfer simulation considering the H I wind and protoplanetary disk. From the simulation, we constrain the wind's properties: the H I column density $\sim 10^{20}\, \rm cm^{-2}$ and the outflow velocity $\sim 200\, \rm km\, s^{-1}$. Our findings indicate that successfully interpreting the observed spectra necessitates scattering contributions in the H I layer within the disk. Furthermore, to explore the effect of Ly$\alpha$ radiative transfer on protoplanetary disk chemistry, we compute the radiation field within the scattering medium and reveal that the wind reflection causes more Ly$\alpha$ photons to penetrate the disk. Our results show the necessity of spatially resolved Ly$\alpha$ observations of a broad range of targets, which will decode the complex interactions between the winds, protoplanetary disks, and surrounding environments.