Non-Lte Transfer. Revisited

Assuming a semi-infinite atmosphere, a two-level atom and complete frequency redistribution, we give, in the isothermal case, a new derivation of the surface value of the source function based only on the integral equation for the source function. This derivation has a straightforward extension to the case where a uniform velocity gradient is included, to the time-dependent case (thermal source B switched on at time t = 0) and to the determination of the mean number of scatterings undergone by an escaping photon (with Ambartsumian's definition). Our derivation suggests also a simple approximate differential equation for the complete source function $$\partial({P}^{1/2}S)/\partial\tau =\epsilon{P}^{-1/2}\,\partial B/\partial\tau$$ where $$P = \epsilon +(1-\epsilon ){K}^{2}(\tau)$$ and $${K}^{2}(\tau)$$ is the direct escape probability for outwards emitted photons. This approximation is tested both for isothermal and exponential atmosphere. As long as the scale of B is large compared to the photon mean free-path at the centre of the line, our approximation reproduces all the qualitative features and is usually slightly in excess of the exact solution. The approximation is easily extended to non-uniform ϵ and multilevel atoms.