Abstract
Abstract
In calculations of the ionization state, one is often forced to choose between the Case A recombination coefficient α
A (sum over recombinations to all hydrogen states) or the Case B recombination coefficient α
B (sum over all hydrogen states except the ground state). If the cloud is optically thick to ionizing photons, α
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 α
A to α
B as the optical thickness increases. The results can be readily implemented in numerical simulations and semi-analytical calculations.
Publisher
American Astronomical Society