Escape of Resonantly Scattered Ly$β$ and H$α$ from Hot and Optically Thick Media
We investigate the escape of Ly$β$ from emission nebulae with a significant population of excited hydrogen atoms in the level $n=2$, rendering them optically thick in H$α$. The transfer of Ly$β$ line photons in these optically thick regions is complicated by the presence of another scattering channel leading to re-emission of H$α$, alternating their identities between Ly$β$ and H$α$. In this work, we develop a Monte Carlo code to simulate the transfer of Ly$β$ line photons incorporating the scattering channel into H$α$. Both H$α$ and Ly$β$ lines are formed through diffusion in frequency space, where a line photon enters the wing regime after a fairly large number of resonance scatterings with hydrogen atoms. Various line profiles of H$α$ and Ly$β$ emergent from our model nebulae are presented. It is argued that the electron temperature is a critical parameter which controls the flux ratio of emergent Ly$β$ and H$α$. Specifically for $T=3 \times 10^4{\rm\ K}$ and H$α$ line center optical depth $τ_α=10$, the number flux ratio of emergent Ly$β$ and H$α$ is $\sim 49$ percent, which is quite significant. We propose that the leaking Ly$β$ can be an interesting source for the formation of H$α$ wings observed in many symbiotic stars and active galactic nuclei. Similar broad H$α$ wings are also expected in Ly$α$ emitting halos found in the early universe, which can be potentially probed by the {\it James Webb Telescope} in the future.