A Multi-Species Atmospheric Escape Model with Excited Hydrogen and Helium: Application to HD209458b

Atmospheric escape shapes exoplanet evolution and star-planet interactions, with He I 10830 \AA\ absorption serving as a key tracer of mass loss in hot gas giants. However, transit depths vary significantly across observed systems for reasons that remain poorly understood. HD209458b, the archetypal hot-Jupiter, exhibits relatively weak He I 10830 \AA\ and H$\alpha$ absorption, which has been interpreted as evidence for a high H/He ratio (98/2), possibly due to diffusive separation. To investigate this possibility and other processes that control these transit depths, we reassess excitation and de-excitation rates for metastable helium and explore the impact of diffusion processes, stellar activity, and tidal forces on the upper atmosphere and transit depths using a model framework spanning the whole atmosphere. Our model reproduces the observed He I transit depth and H$\alpha$ upper limit, showing strong diffusive separation. We match the observations assuming a photoelectron efficiency of 20-40\%, depending on the composition of the atmosphere, corresponding to mass-loss rates of $1.9-3\times10^{10}$ g/s. We find that the He I 10830 \AAtransit depth is sensitive to both stellar activity and diffusion processes, while H$\alpha$ is largely unaffected due to its strong dependence on Lyman-$\alpha$ excitation. These differences may help explain the system-to-system scatter seen in population-level studies of the He I line. While He I data alone may not tightly constrain mass-loss rates or temperatures, they do confirm atmospheric escape and help narrow the viable parameter space when interpreted with physically motivated models. Simultaneous observations of He I, H$\alpha$, and stellar activity indicators provide powerful constraints on upper atmosphere dynamics and composition, even in the absence of full transmission spectra.