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Line Emission and Escape

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, $\tau_0$ at the center of the line. For $1\leq\tau_0\leq 10^6$, the resonant trapping is effectively local. For $\tau_0\geq 10^6$, 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]):

\begin{displaymath}P_{esc., line}(\tau_{line})={{1}\over{\tau_{line}\sqrt{\pi}(1.2+b)}} (\tau_{line}\geq 1) \eqno{(6)} \end{displaymath}

\begin{displaymath}P_{esc., line}(\tau_{line})={{1-e^{-2 \tau_{line}}}\over{2 \tau_{line}}} (\tau_{line}\leq 1) \eqno{(7)} \end{displaymath}


\begin{displaymath}b={{\sqrt{{\rm log}(\tau_{line})}}\over{1+\tau_{line}/\tau_w}} \eqno{(8)} \end{displaymath}


$\tau_{line}$ is the optical depth at line center, and $\tau_w=10^5$.

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.

next up previous contents
Next: Continuum Emission Up: Recombination Continuum Emission and Previous: Recombination Continuum Emission and   Contents
Tim Kallman 2018-12-14