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

Lyman-$\alpha$ (Ly$\alpha$) is the strongest emission line in UV spectra from T-Tauri stars. Due to its resonant nature, Ly$\alpha$ emission carries information about the physical properties of the H I medium via the scattering process. This work presents spatially resolved 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 radiative transfer simulation in a wind-disk geometry. Successfully reproducing the observed spectra requires scattering contributions from both the wind and the H I disk. We constrain the properties of the wind, the H I column density ($\sim 10^{20} \rm cm^{-2}$) and the outflow velocity ($\sim 200 \rm km s^{-1}$). To reproduce the observed spatial distribution of Ly$\alpha$, we find that the wind must cover the H I disk when viewed face-on. Furthermore, to explore the effect of Ly$\alpha$ radiative transfer in T-Tauri stars, we compute the radiation field within the scattering medium and reveal that the wind reflection causes more Ly$\alpha$ photons to penetrate the disk. We also show the dependence between the disk inclination angle and the spatially resolved Ly$\alpha$ spectra. Understanding the role of Ly$\alpha$ emission in T-Tauri stars is pivotal for decoding the complex interactions between the winds, protoplanetary disks, and surrounding environments, which can significantly impact the chemistry in the protoplanetary disk. Our observation and modeling of spatially resolved Ly$\alpha$ show the necessity of spatially resolved Ly$\alpha$ observation of a broad range of targets.