FUV Irradiated Disk Atmospheres: Ly$\alpha$ and the Origin of Hot H$_2$ Emission

Protoplanetary disks are strongly irradiated by a stellar FUV spectrum that is dominated by Ly$\alpha$ photons. We investigate the impact of stellar Ly$\alpha$ irradiation on the terrestrial planet region of disks ($\lesssim 1$AU) using an updated thermal-chemical model of a disk atmosphere irradiated by stellar FUV and X-rays. The radiative transfer of Ly$\alpha$ is implemented in a simple approach that includes scattering by H I and absorption by molecules and dust. Because of their non-radial propagation path, scattered Ly$\alpha$ photons deposit their energy deeper in the disk atmosphere than the radially propagating FUV continuum photons. We find that Ly$\alpha$ has a significant impact on the thermal structure of the atmosphere. Photochemical heating produced by scattered Ly$\alpha$ photons interacting with water vapor and OH leads to a layer of hot (1500 - 2500 K) molecular gas. The temperature in the layer is high enough to thermally excite the H$_2$ to vibrational levels from which they can be fluoresced by Ly$\alpha$ to produce UV fluorescent H$_2$ emission. The resulting atmospheric structure may help explain the origin of UV fluorescent H$_2$ that is commonly observed from classical T Tauri stars.