Non-LTE Radiative Transfer of Spectral Lines
in a Plane-Parallel Medium

The non-LTE source function in a multiple-scattering/ absorbing plane-parallel atmosphere is generally determined by the integral equation.

(1)       S(τ) = S₀(τ) + ₀∫^τ_v dτ' S(τ') ^. K₁(|τ'-τ|, |t'(τ')-t(τ)|)

where τ is the vertical optical depth variable for the scatterers (τ_v being the total optical depth), t the corresponding optical depth for the absorbers, S₀(τ) the initial source function, and the kernel function K₁ gives the probability for a photon to traverse unscattered and unabsorbed between τ' and τ and be (isotropically) scattered at the latter level. In case of a continuous (i.e. frequency-independent) scattering/absorption cross section, K₁ is simply determined by the usual exponential absorption law (i.e. for a plane-parallel atmosphere by the first exponential integral E₁), but for the case of a spectral line it is given by

(2)       K₁(τ,t)= 1/2/√π^._-∞∫^∞dx Φ²(x)^.E₁(τ^.Φ(x)+t) ,

where Φ(x) is the line profile function (which is normalized such that the integral over it (from -∞ to +∞) is equal to √π (e.g. in case of a pure Gaussian (Doppler broadened) profile Φ(x)=e^-x2, or in case of combined Doppler and natural broadening Φ(x)=H(a,x) (the Voigt function)).
This form for K₁ implies that both the emission and the absorption profile are given by Φ(x) (complete frequency redistribution) which is furthermore assumed to be independent of τ here (the latter implies for instance that the atmosphere is isothermal). Both of these assumptions are usually reasonably good approximations if one is merely interested in the line-integrated intensities and source functions rather than the details of the spectral profile.

The method of choice here is an iterative solution of Eq.(1). First of all, Eq.(1) is discretized by assuming the source function to be constant within a given sub-interval j and integrating over τ'. This yields

                                    _N
(3)       S(τ_j) = S₀(τ_j) + Σ S(τ_k) ^.[ | K₂(|τ_k+1-τ_j| , |t_k+1-t_j| , α_k) - K₂(|τ_k-τ_j|, |t_k-t_j| , α_k) | ]     ,
                                   ^k=1

where

(4)       K₂(τ,t,α) = ₀∫^τ dτ' K₁(τ',t'(τ')) = 1/2/√π^._-∞∫^∞dx Φ²(x)/(Φ(x)+α) ^.E₂(τ^.Φ(x)+t)   .

with E₂ the second exponential integral, and the optical depth of the absorbers assumed to change proportionally to the optical depth of the scatterers in each interval, i.e.

(5)       t'(τ') = t_k + α_k^.(τ'- τ_k)   ;   τ_k≤τ'≤τ_k+1

Although Eq.(3) could be iterated as such, it is better to strictly solve it for S(τ_j) (which also appears on the right hand side of the equation (for k=j)). This yields

                                      _N
(6)       S(τ_j) = [ S₀(τ_j) + Σ S(τ_k) ^.dK₂(j,k) ] / [1-dK₂(j,j) ]     ,
                                    ^k=1(≠j)

where

(7)       dK₂(j,k) = |   K₂(|τ_k+1-τ_j| , |t_k+1-t_j| , α_k) - K₂(|τ_k-τ_j|, |t_k-t_j| , α_k)   |   .

Eq.(6) represents now a system of N equations for the N unknowns S(τ_j), which can be solved by iteration, using the given initial source function S₀(τ_j) as starting values.

The source functions are of course not directly observable, but only the intensities, which for 'up-looking' directions (looking into directions of decreasing τ) under an angle η to the normal are here determined by (using μ=cos(η))

(8)       I_up(τ_j,μ) = 1/μ^. ₀∫^τ_j dτ' S(τ') ^. T( (τ_j-τ')/μ , (t_j-t')/μ ) )

where

(9)       T(τ,t) = 1/√π^._-∞∫^∞dx Φ(x)^.e^{-(τ.Φ(x)+t)} :

Discretization and integration of Eq.(8) yields then

                               _j-1
(10)       I_up(τ_j,μ) = Σ S(τ_k) ^.[ Z(τ_j-τ_k)/μ , (t_j-t_k)/μ , α_k) - Z(τ_j-τ_k+1)/μ , (t_j-t_k+1)/μ , α_k) ]     ,
                              ^k=1

where

(11)       Z(τ,t,α) = ₀∫^τ dτ' T(τ',t'(τ')) = 1/√π^._-∞∫^∞dx Φ(x)/(Φ(x)+α)^.[e^-(t-ατ) -e^{-(τ.Φ(x)+t)}] .

Analogously one obtains for the 'down-looking' intensities at level τ_j

                                   _N
(12)       I_down(τ_j,μ) = Σ S(τ_k) ^.[ Z(τ_k+1-τ_j)/μ , (t_k+1-t_j)/μ , α_k) - Z(τ_k-τ_j)/μ , (t_k-t_j)/μ , α_k) ]     .
                                  ^k=j

On the page Radiative Transfer Code for Multiple Scattering in a Plane Parallel Atmosphere I have translated this formalism into a corresponding C/C++ program (this can also be used for the case of continuum- (monochromatic scattering) as the corresponding kernel functions are simply obtained from the above by leaving out the frequency integration and setting Φ(x)=1 (resulting thus in straight exponential functions and exponential integrals))
(note that contrary to the above formulation, the program for the intensity uses the line-of-sight optical depth variable rather than the vertical optical depth; this makes it suitable for non-plane-parallel geometries as well).

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