Since all level populations are calculated explicitly, line emissivities and cooling rates are calculated as a straightforward product of the population of the line upper level, the spontaneous transition probability and an escape fraction.
Line optical depths may be large in some nebular situations.
Photons emitted near the centers of these lines are likely to be
absorbed by the transition which emitted them and reemitted at a new
frequency. This line scattering will repeat many times until the
photon either escapes the gas, is destroyed by continuum photoabsorption or
collisional deexcitation, or is degraded into longer wavelength photons
which may then escape. Our treatment of resonance line transfer is based on the
assumption of complete redistribution. That is, we assume that there
is no correlation of photon frequencies before and after each
scattering event. This has been shown to be a good approximation for a
wide variety of situations, particularly when the line profile is
dominated by Doppler broadening. In this case, more accurate numerical
simulations (e.g., [Hummer and Rybicki 1971]) have shown that line scattering
is restricted to a small spatial region near the point where the
photons are emitted. Line photons first scatter to a frequency such
that the gas cloud is optically thin and then escape in a single long
flight. The probability of escape per scattering depends on the
optical depth, at the center of the line. For
, the resonant trapping is effectively local. For
, the lines become
optically thick in the damping wings, and the
line escapes as a result of diffusion in both space and frequency.
Since the scattering in the Doppler core is always dominated by
complete redistribution, and since most of the lines in our models are
optically thin in the wings, we assume that all line scattering takes
place in the emission region.
We use the following expression for escape probability ([Kwan and Krolik, 1981]):
where
is the optical depth at line center,
and
.
The rates for line emission and the probabilities for the various resonance line escape and destruction probabilities depend on the state of the gas at each point in the cloud. The cooling function for the gas depends on the line escape probabilities, and the effects of line trapping must be incorporated in the solution for the temperature and ionization of the gas. Once the state of the gas at a given point has been determined, the emission in each line is calculated as the product of the upper level population and the corresponding net decay rate, including the suppression due to multiple scattering.