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 ${\alpha }_{\mathrm{A}}$ (sum over recombinations to all hydrogen states) or the Case B recombination coefficient ${\alpha }_{\mathrm{B}}$ (sum over all hydrogen states except the ground state). If the cloud is optically thick to ionizing photons, ${\alpha }_{\mathrm{B}}$ 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 ${\alpha }_{\mathrm{A}}$ to ${\alpha }_{\mathrm{B}}$ as the optical thickness increases. The results can be readily implemented in numerical simulations and semi-analytical calculations.