Case A or Case B? The effective recombination coefficient in gas clouds of arbitrary optical thickness
In calculations of the ionization state, one is often forced to choose between the Case A recombination coefficient \(\alpha_{\rm A}\) (sum over recombinations to all hydrogen states) or the Case B recombination coefficient \(\alpha_{\rm B}\) (sum over all hydrogen states except the ground state). I...
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Veröffentlicht in: | arXiv.org 2023-05 |
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Sprache: | eng |
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Zusammenfassung: | In calculations of the ionization state, one is often forced to choose between the Case A recombination coefficient \(\alpha_{\rm A}\) (sum over recombinations to all hydrogen states) or the Case B recombination coefficient \(\alpha_{\rm B}\) (sum over all hydrogen states except the ground state). If the cloud is optically thick to ionizing photons, \(\alpha_{\rm B}\) is usually adopted on the basis of the "on-the-spot" approximation, wherein recombinations to the ground state are ignored because they produce ionizing photons absorbed nearby. In the opposite case of an optically thin cloud, one would expect the Case A recombination coefficient to better describe the effective recombination rate in the cloud. In this paper, I derive an analytical expression for the effective recombination coefficient in a gas cloud of arbitrary optical thickness which transitions from \(\alpha_{\rm A}\) to \(\alpha_{\rm B}\) as the optical thickness increases. The results can be readily implemented in numerical simulations and semi-analytical calculations. |
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ISSN: | 2331-8422 |
DOI: | 10.48550/arxiv.2305.05764 |