Escape of Resonantly Scattered Ly$\beta$ and H$\alpha$ from Hot and Optically Thick Media.

We investigate the escape of Ly$\beta$ from emission nebulae with a significant population of excited hydrogen atoms in the level $n=2$, rendering them optically thick in H$\alpha$. The transfer of Ly$\beta$ line photons in these optically thick regions is complicated by the presence of another scattering channel leading to re-emission of H$\alpha$, alternating their identities between Ly$\beta$ and H$\alpha$. In this work, we develop a Monte Carlo code to simulate the transfer of Ly$\beta$ line photons incorporating the scattering channel into H$\alpha$. Both H$\alpha$ and Ly$\beta$ 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$\alpha$ and Ly$\beta$ 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$\beta$ and H$\alpha$. Specifically for $T=3 \times 10^4{\rm\ K}$ and H$\alpha$ line center optical depth $\tau_\alpha=10$, the number flux ratio of emergent Ly$\beta$ and H$\alpha$ is $\sim 49$ percent, which is quite significant. We propose that the leaking Ly$\beta$ can be an interesting source for the formation of H$\alpha$ wings observed in many symbiotic stars and active galactic nuclei. Similar broad H$\alpha$ wings are also expected in Ly$\alpha$ emitting halos found in the early universe, which can be potentially probed by the {\it James Webb Telescope} in the future.