Rayleigh and Raman scattering cross-sections and phase matrices of the ground-state hydrogen atom, and their astrophysical implications

Abstract

We present explicit expressions for Rayleigh and Raman scattering cross-sections and phase matrices of the ground $1s$ state hydrogen atom based on the Kramers-Heisenberg dispersion formula. The Rayleigh scattering leaves the hydrogen atom in the ground-state while the Raman scattering leaves the hydrogen atom in either $ns$ ($n\ge 2$; $s$-branch) or $nd$ ($n\ge 3$; $d$-branch) excited state, and the Raman scattering converts incident ultraviolet (UV) photons around the Lyman resonance lines into optical-infrared (IR) photons. We show that this Raman wavelength conversion of incident flat UV continuum in dense hydrogen gas with a column density of $N_{\text{H}}>10^{21}{\text{cm}}^{-2}$ can produce broad emission features centred at Balmer, Paschen, and higher-level lines, which would mimic Doppler-broadened hydrogen lines with the velocity width of $\gtrsim 1,000\text{km}{\text{s}}^{-1}$ that could be misinterpreted as signatures of Active Galactic Nuclei, supernovae, or fast stellar winds. We show that the phase matrix of the Rayleigh and Raman $s$-branch scatterings is identical to that of the Thomson scattering while the Raman $d$-branch scattering is more isotropic, thus the Paschen and higher-level Raman features are depolarized compared to the Balmer features due to the flux contribution from the Raman $d$-branch. We argue that observations of the line widths, line flux ratios, and linear polarization of multiple optical/IR hydrogen lines are crucial to discriminate between the Raman-scattered broad emission features and Doppler-broadened emission lines.

32 pages, 8 figures, 2 tables, accepted for publication in MNRAS (the journal version is more abbreviated than the arXiv version)

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• Lyman-$\alpha$ feedback prevails at Cosmic Dawn: Implications for the first galaxies, stars, and star clusters
• Impact of Resonance, Raman, and Thomson Scattering on Hydrogen Line Formation in Little Red Dots