Issue 
A&A
Volume 649, May 2021



Article Number  A165  
Number of page(s)  7  
Section  Astrophysical processes  
DOI  https://doi.org/10.1051/00046361/202040005  
Published online  02 June 2021 
Full non–LTE spectral line formation
I. Setting the stage
^{1}
Université de Toulouse, UPSObservatoire MidiPyrénées, Cnrs, Cnes, Irap, Toulouse, France
email: frederic.paletou@univtlse3.fr
^{2}
Cnrs, Institut de Recherche en Astrophysique et Planétologie, 14 av. E. Belin, 31400 Toulouse, France
^{3}
Université de Toulouse, Faculté des Sciences et de l’Ingénierie, 31062 Toulouse cedex 9, France
Received:
27
November
2020
Accepted:
15
March
2021
Radiative transfer out of local thermodynamic equilibrium (NLTE) has been increasingly adressed, mostly numerically, for about six decades now. However, the standard NLTE problem most often refers to the only deviation of the distribution of photons from their equilibrium, that is to say a Planckian distribution. Hereafter we revisit after Oxenius (1986, Kinetic theory of particles and Photons – Theoretical Foundations of non–LTE Plasma Spectroscopy, Springer) the socalled full NLTE problem, which considers coupling and therefore solving self–consistently for deviations from equilibrium distributions of photons as well as for massive particles constituting the atmospheric plasma.
Key words: radiation mechanisms: general / line: formation / radiative transfer / line: profiles / plasmas
© F. Paletou and C. Peymirat 2021
Open Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
The “standard” nonlocal thermodynamic equilibrium (NLTE) radiation transfer problem considers that the distribution of photons in a given “atmosphere”, that is, more generally, whatever sample of celestial body material where light–matter interactions are taking place, may depart from the equilibrium distribution described by the Planck law (Planck 1900).
It is a routinely solved problem in astrophysics which, after about six decades of constant endeavor, nowadays also considers both complex atomic models and atmospheric structures (see e.g., the monograph of Hubeny & Mihalas 2014). However, the vast majority of these problems still rely on what became a somewhat implicit assumption that the distribution of atoms both responsible for, and experiencing light–matter interactions, remain a priori known and characterized by the Maxwell–Boltzmann equilibrium distribution. Another issue related to the standard NLTE radiation transfer problem concerns the redistribution of photons after scattering onto these very massive particles, at least in frequencies if we restrict ourselves to isotropic scattering. It is also important to realize that, in the vast majority of the cases, and even though the problem of partial frequency redistribution (hereafter PRD) has been addressed quite early in the 1960s (e.g., Avrett & Hummer 1965; Auer 1968 for pioneering works), the standard NLTE problem also remains limited to the frame of complete redistribution (CRD) in most cases.
The solution of a full(er) NLTE radiation transfer problem would consist in assuming that the velocity distribution functions (VDF) of massive particles may also depart from an equilibrium distribution, as it is now commonly assumed for the photons. Therefore the solution to this problem would consist in the self–consistent resolution of a set of kinetic equations both for massive particles, potentially including free electrons, and the photons.
This general issue has already been discussed and summarized, mainly by Oxenius (1986; see also Oxenius & Simonneau 1994). However, so far and because of the inherent difficulty of the problem, only a few studies have already been conducted in the past (see e.g., Borsenberger et al. 1986, 1987).
We believe that the last decades of evolution of numerical methods (see e.g., Lambert et al. 2016 and references therein, as well as Noebauer & Sim 2019 for a recent review on Monte–Carlo radiative transfer) for radiation transfer should allow one to reconsider this issue, and evaluate up to which level of difficulty one could reasonably be able to achieve nowadays. This new evaluation should also give us new hints about which kind of astrophysical problems should be revisited, using full NLTE.
In this study, we set again the various equations governing this problem, using standard notations as much as possible. We also adopt several physical assumptions, which have already been discussed in past studies, in order to be able to set up initial numerical experiments. We discuss partial versus complete frequency redistribution, as seen from this description of the problem. We do indeed present some effects expected on departures of the VDF of excited atoms versus the Maxwell–Boltzmann distribution, as well as significative differences between computed and radiationdependent emission profiles, and an a priori given absorption profile for a simple twolevel atom model. We finally discuss future work in that somewhat forgotten physical frame for the unpolarized radiation transfer problem.
2. A simplified Boltzmann equation for massive particles
We write the evolution equation for a general distribution F(t, r, v) as follows:
where L is the Liouville operator and C is the collision operator.
Therefore, the lefthand side of Eq. (1) has the general expression:
where after the time evolution term, we find an advection one and finally a force term, where we used K to avoid any confusion with the distribution F.
The righthand side of Eq. (1) describes the various origins of collisions, which may induce modifications to the distribution. It is usually expressed as the sum of three contributions as follows:
As a matter of fact, two main mechanisms are able to produce changes in a distribution F. One of them is the socalled streaming of the massive particles, which move on their trajectories, and a second one is due to collisions between particles.
In the frame of this kinetic description, and even though the classical and academic “twolevel atom” case is the simplest one we may consider, the general problem would consist in the selfconsistent solution of four coupled kinetic equations for the atoms in their ground state (F_{1}), the atoms in their excited state (F_{2}), electrons (F_{e}), and finally the photons (I), respectively.
However, and in order to be able to make the first steps in resolving the extremely complex and cumbersome full NLTE radiation transfer problem, we shall proceed with several simplifying assumptions. First, we shall consider that only the VDF of excited atoms, that is to say F_{2}, may depart from a MaxwellBoltzmann distribution. As previously discussed by Oxenius (1979), we shall consider that the velocity distribution of the groundstate atoms and the free electrons remain Maxwellian with the same temperature T. The latter is also supposed to be low enough, that is kT ≪ hν_{0}, where ν_{0} is the central frequency of the transition between the two atomic levels of our model, so that: (i) the number density of excited atoms is much smaller than the one of groundstate atoms, that is n_{2} ≪ n_{1}, and (ii) stimulated emission may be further neglected.
Then we shall consider a timeindependent problem for photons as well as a stationary and forcefree problem for atoms. And finally, following the debate between Oxenius (1965, 1979) and Hubený (1981), we shall also neglect the potential streaming (i.e., v.∂_{r} ≡ 0) of these excited atoms. In that frame, the Boltzmann equation for atoms can be summarized as follows:
where C is the collision operator.
3. The kinetic equation of excited atoms
Each of the three contributions to the collision operator can be explained now, following Oxenius (1986).
The first one involves the radiation field and can be expressed as:
where stimulated emission has been neglected, and A_{21} and B_{12} are the usual Einstein coefficients for spontaneous emission and radiative absorption, respectively. Here, we also introduced the normalized to unity f_{i} distribution, such as F_{i}(v) = n_{i}f_{i}(v).
The second one involves inelastic collisions and can be written as follows:
where the C_{ij} are collisional (de)excitation coefficients and n_{e} is the electron density. We note that one may also consider that the electrons are not necessarily characterized by a Maxwell–Boltzmann velocity distribution.
Finally, and according to Oxenius (1986; see also Bhatnagar et al. 1954), a good approximation for the elastic collision term is:
where Q_{2} is a velocitychanging elastic collision rate. After Borsenberger et al. (1986), one may consider Q_{2} = n_{1}v_{th.}σ_{el.}, where v_{th.} is the “most probable” velocity of the atoms, and σ_{el.} is an “average cross section” for this class of elastic collision.
The scattering integral J_{12}, which appears in Eq. (5), is defined as:
where α_{12} is the atomic absorption profile and I_{ν}(z, Ω) is the usual specific intensity. At this stage we omit the potentiel dependence with depth z of J_{12} on purpose, since we shall stay away from the frame of radiation transfer in this study. The specific intensity (see Sect. 4) is also dependent on the direction Ω of the radiation. All integrals dΩ mean an angular integration over all possible directions of propagation of light, as is usual in the field of radiation transfer. It is more important now to keep in mind that this scattering integral is a function of the velocity of the massive particles. Indeed, we have to take into account the Fizeau–Doppler relationship between frequency ξ in the atomic frame, and ν in the observer’s frame:
where v is the velocity of the atoms and Ω is the direction of propagation of the photons. This latter coupling and the explicit dependence of J_{12} on the velocity of these “scattering centers,” which constitute the atoms present in the atmospheric plasma, also naturally lead to the role that will be played by their velocity distribution.
The atomic absorption profile α_{12} is known a priori, and we may consider either coherent scattering, that is:
where ξ is the incoming photon frequency, or radiation damping which is characterized by a Lorentzian profile such that:
and where, in both cases, ν_{0} is the central wavelength of the transition we consider. In the remainder of this study, we shall assume coherent scattering in the atomic frame, together with “infinitely sharp” energy levels for the model spectral line at ν_{0}.
It is finally important to keep in mind that at this stage the atomic emission profile η_{21} may differ from α_{12}, in general. This, combined with possible deviations from a Maxwell–Boltzmann VDF for excited atoms will “naturally” lead to nonstandard as well as radiationdependent redistribution in frequency for the photons.
4. The kinetic equation of the photons
The radiation transfer equation (RTE) is also the kinetic equation governing the distribution, or specific intensity I, of the photons. The correspondance between these two “forms” is nicely described in Sect. 3 of Oxenius (1986).
Assuming the very usual 1D plane–parallel geometry, the timeindependent RTE can be written^{1} as:
where κ_{ν}(z, Ω) is the absorption coefficient and S_{ν}(z, Ω) is the socalled source function, which a priori depends on depth z in the atmosphere, frequency ν, and direction Ω.
The absorption coefficient is usually defined as:
where B_{21} is the Einstein coefficient for stimulated emission, such as g_{1}B_{12} = g_{2}B_{21} assuming that the statistical weights g_{i} and the (line) source function is, by definition, the ratio between the emissivity:
and the absorption coefficient. And finally, the more usual RTE introduces the optical depth τ_{ν}(Ω) = − κ_{ν}(z, Ω)dz in order to substitute the geometrical length for another quantity more relevant to what photons effectively experience during multiple light–matter interactions.
Should we completely neglect stimulated emission, that is to say going beyond the (very usually) adopted socalled “weak radiation field regime” for which stimulated emission is treated as “negative” absorption (as in Eq. (13)), then the source function can be simply written as:
This assumption is also consistent with the low–temperature assumption, which is also compatible with n_{1} ≫ n_{2}.
Furthermore, the respective (and potentially also depthdependent) emission and absorption profiles are defined as
and
In the latter expressions appear explicitly the respective VDFs for these atoms on the ground state, f_{1}, and for the ones in their first excited state, f_{2}.
In general, atomic profiles η_{21} and α_{12} are not identical. Moreover, the emission profile η_{21} depends on the radiation field. We return to this issue in Sect. 8.
5. A two–distribution, twolevel atom problem
Hereafter, we shall therefore simplify the problem, considering that only f_{2} and the specific intensity I may depart from equilibrium distributions.
We shall first reassess earlier results which may still be quite largely unknown to the astronomical community by considering coherent scattering, that is α_{12}(ξ) = δ(ξ − ν_{0}), where ν_{0} is the central wavelength of the spectral line associated with our twolevel atom.
We therefore start with the additional simplifications of a stationary, forcefree, and no advection (or streaming) case for which
Under these assumptions, we can derive the following expression for f_{2} and, while adopting a Maxwellian velocity distribution for electrons, using the more classic formalism such that C_{ij} ← n_{e}C_{ij} (see also Eq. (9.51) of Hubeny & Mihalas 2014 for instance), we have
Should we assume that f_{1, 2} are both Maxwellian, which is a common assumption even for NLTE radiation transfer problems, and should we integrate Eq. (19) over all velocities, we recover the classic form of the equation of statistical equilibrium (ESE) for a twolevel atom case, neglecting stimulated emission:
We shall come back to the integral term after Eq. (30). We note here that because this particular contribution vanishes after integration over velocities, the potential effects of velocitychanging elastic collisions can be considered explicitly and reasonably discussed, although only by using such a kinetic description.
Now from Eq. (19), we are able to derive an explicit expression for the nonequilibrium VDF f_{2}, assuming that all atoms in their fundamental state of energy follow a Maxwellian VDF, that is to say f_{1} ≡ f^{M}. As in Borsenberger et al. (1986), we shall normalize the specific intensity to the “low temperature limit” of the Planck function, that is the Wien function:
The use of this expression, instead of the Planck function, is also consistent with our assumption of a negligible rate of stimulated emission. We shall also define the ratio
as well as use the following relationships, first between Einstein coefficients
with g_{1}B_{12} = g_{2}B_{21}, and where g_{1, 2} are the statistical weights of each level.
In addition, collisional rates are such that
where are the LTE values of the respective densities of population, satisfying the Boltzmann law:
We shall therefore now introduce the normalized population, which allows for some measurement of deviations from LTE: .
We now introduce:
which is another parameter otherwise traducing the amount of “departure from LTE” in a given atmosphere, and the less common:
which characterizes the amount of elastic collisions. Assuming that and f_{1} ≡ f^{M}, we may now rewrite the following:
This expression is indeed identical to Eq. (2.44) of Borsenberger et al. (1986) when one neglects streaming of particles (his η = 0).
Now, integrating Eq. (28) over all velocities, we find that:
where we defined
which is equivalent to the common scattering integral used in the standard approach. We finally recover the following:
This is the central relationship allowing us to quantify departures from “Maxwellianity” for f_{2} and potential differences between macroscopic emission and absorption profiles, depending on the nature of the radiation field characterized by I.
6. Two comments on the standard theory
Complete redistribution in frequency relies on the a priori assumption that the “macroscopic” emission and absorption profile are identical. Then the line source function reduces to the simple ratio S = n_{2}A_{21}/n_{1}B_{12}, that is to say a quantity independent of the frequency after Eq. (15). It is also a considerable simplification of the NLTE radiation transfer problem. Assuming, as we do here, that the atomic profiles are identical, that is, α_{12} ≡ η_{21}, CRD also assumes the identity implicitly as follows:
whose effects therefore result from expressions given in Eqs. (16) and (17). In that frame, we also recover that the normalized source function is such that
This expression is indeed consistent with the classical (CRD) “twolevel atom” case described in any radiation transfer textbook following the publication of Avrett & Hummer (1965).
However, in eliminating the case of little astrophysical interest of an isotropic “white” radiation field, Eq. (32) is clearly inconsistent with the more general Eq. (31), unless setting ζ = 0 and ε = 1, that is assuming LTE. Therefore NLTE should always come together with some partial frequency redistribution and no complete redistribution a priori.
Additionally, even when “standard” partial frequency redistribution is used, with a priori known redistribution functions, in most cases a “branching ratio” usually defined as
is introduced (see Omont et al. 1972; Heinzel & Hubený 1982; Heinzel et al. 1987) to form a linear combination of R_{II} and R_{III} redistribution such as
following the nomenclature of Hummer (1962), and where x′ and x are the usual incoming and outgoing reduced frequencies, respectively, which are usually defined as:
with the Doppler width Δν_{D} = (ν_{0}/c)v_{th.} and the most probable velocity as usual.
The elastic collision rate Q_{E} which appears in the branching ratio γ should not be confused with the Q_{2} elastic collision rate introduced in Eq. (7). The latter is indeed, and more precisely, the rate of these velocitychanging elastic collisions, that is a fraction of the total amount Q_{E} of elastic collisions. Bommier (2016a), for instance, provides a synthetic but accurate and valuable explanation about the different classes of elastic collisions. Moreover, as mentioned in this discussion, Landi Degl’Innocenti & Landolfi (2004; see their Sect. 13.2) estimate that Q_{2} should be very small in solar–like atmospheres, so that the socalled Boltzmann elastic collision term may be practically neglected for radiative modeling under such physical conditions.
Nevertheless, quantifying any significant effet due to elastic collisions in the NLTE problem for other atmospheres with physical conditions characterized by a more important amount of velocitychanging collisions than in the solar–like case would thus require using the more general theoretical frame we adopt here (see also Belluzzi et al. 2013 for instance). Indeed, using velocity integrated expressions such as Eqs. (20) or (29), which is the case in the standard NLTE approach, and even with standard PRD, the elastic collision term introduced in Eq. (7) just vanishes.
7. Quantifying departures from equilibrium
The stage is now set for the first evaluation of potential departures from “Maxwellianity,” owing to the nature of the radiation field we may adopt.
For a first exploration, we shall first assume pure Doppler broadening for the spectral line, that is to say coherent scattering in the atom’s frame and
where both atomic profiles are also assumed to be isotropic. Should we moreover assume no elastic collisions so far, that is ζ = 0, then after Eq. (31) we end up with
where we also introduced the normalized velocity u = v/v_{th.}.
NLTE implies that ε < 1, which in turn necessarily implies that f_{2} differs from being Maxwellian in general. Should these differences be significant enough, then after Eq. (17) and despite the equality between respective atomic profiles, we however expect differences between macroscopic absorption and emission profiles, unlike what assumes, ab initio, the widely adopted CRD.
The abovementioned assumptions lead to
Then, since f_{1} ≡ f^{M}, using this classical expression for the (normalized to unity) macroscopic absorption profile
we can also establish that
which finally allows us to evaluate departures of f_{2} from the equilibrium distribution depending on the thus far given (and normalized to unity) radiation field I(x).
According to these expressions, one can notice that, with coherent scattering in the atom’s frame, it remains however an ultimate option for “saving” the CRD case. Indeed, assuming isotropic “white” light illumination, that is a frequency independent radiation field such as I(x) = I_{0}, is the only condition left to us in order to keep f_{2} ≡ f^{M} and therefore ψ_{x} ≡ φ_{x}.
Further illustrative computations have been made using several incident radiation profiles, as displayed in Fig. 1. Our selection covers distinct possibilities of central reversal amplitudes, that is to say I_{max.}/I(0) as well as peaks located at different and increasing positions away from line center. These may mimic some observed and wellresolved resonance line profiles, similar to one of the good candidates for future modeling such as Lyα of H I.
Fig. 1.
Isotropic and normalized intensity profiles I vs. the reduced frequency x, used for the evaluation of departures from the equilibrium velocity distribution for the VDF of these excited atoms, f_{2}. We adopted different values for (i) the location of the (symmetric) peaks, and (ii) of the central reversal of each profile. These variations can also mimic some resonance lines observations that may form under different astrophysical conditions. 

Open with DEXTER 
Then, we computed ratios of f_{2}/f^{M} according to Eq. (38), and further adopting ε = 0. Our results for f_{2}/f^{M} as a function of the normalized velocity u are displayed in Fig. 2, for which we used the same linestyle convention as in Fig. 1.
Fig. 2.
Ratios f_{2}/f^{M} as a function of the normalized velocity u, for different radiation field profiles, and for ε = 0 and ζ = 0. The same linestyle convention as in Fig. 1 was adopted. The magnitude of these departures from a Maxwellian velocity distribution depends on both (i) the distance of the peaks of I vs. line center, and (ii) the amount of its central reversal characterized by I_{max.}/I(0). 

Open with DEXTER 
Departures from a Maxwellian distribution show up at almost all normalized velocities u. These departures tend to increase both with (i) an increasing distance of the peaks of the incident radiation profile away from line center, as well as with (ii) an increase of the central reversal of the incident profile, which is characterized by the ratio between the maximum value of the profile I_{max} and its central value I(0).
Finally, it is also possible to compute the ratio between the macroscopic emission profile ψ_{x} and the (Gaussian) absorption profile φ_{x}. Indeed, after Eqs. (38) and (17), and with ε = 0, one can derive (see also Appendix A) that
for the case of coherent scattering in the atom’s frame.
We also note that, at this stage, with the knowledge of the radiationdependent emission profile, we are also able to compute the source function, following Eq. (15).
Significant differences between emission and absorption profiles are put in evidence from our relatively simple computations. They are displayed in Fig. 3 which shows macrocopic profile ratios as a function of x for the same set of incident radiation profiles as before.
Fig. 3.
Ratios ψ_{x}/φ_{x} as a function of the reduced frequency x, for the different radiation field profiles displayed in Fig. 1, using also the same linestyle convention. We observe here, for the case of coherent scattering in the atom’s frame, similar tendencies as the ones displayed in Fig. 2 for the ratios between VDFs. 

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For these ratios and the special case of coherent scattering in the atom’s frame, we observe the same tendancies as the ones already observed for the ratio between VDFs. It essentially shows the very limit of the theoretical framework of CRD and its intrinsic inconsistency with NLTE.
8. Discussion
So far, the radiation field has been imposed a priori using a simple I(x) function. Potential anisotropies of the radiation field could be taken into account within this formalism of course. Also, more realistic line profiles will be adopted further, as we did for the computation of the ratio f_{2}/f^{M} displayed in Fig. 4. In that case, we used the profile of the H I Lyα resonance line deduced from SoHO/Sumer solar observations (Gunár et al. 2020, in particular their Fig. 5). We adopted a temperature of 10^{4} K, leading to a I(x) profile extending up to x ∼ 30. Then we fixed ζ = 0 and ε = 10^{−7}, that is a value which is quite typical of conditions usually met in solar prominences for instance (see e.g., Paletou et al. 1993). Departures from a Maxwellian distribution mainly appear significant around the peak of the line profile at x ∼ 5 in our case, and in the line wings.
Fig. 4.
Ratio f_{2}/f^{M} vs. the normalized velocity u for ε = 10^{−7} (ζ = 0), and a realistic H I Lyα illumination profile as observed by SoHO/Sumer. The peak of the line profile is located at x ∼ 5, and the central selfreversal is about 0.7 for this resonance line (see Fig. 5 of Gunár et al. 2020). 

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But the most important thing to address further will be the self–consistent computation of the two distributions f_{2} and I. This should be possible within an iterative cycle involving the following: an initial guess for the source function S assuming for instance both LTE and CRD, then a formal solution for the RTE, allowing for the computation of and 𝒥_{12}. Then f_{2} should be estimated, as well as ψ_{x}, making it possible to update the source function further, start another iteration, and so forth, until reaching convergence for both f_{2} and I.
We believe that stateoftheart iterative methods presently used for NLTE radiative transfer (see e.g., Lambert et al. 2016, and references therein) will allow one to explore all of the potential of this alternative and a more detailed theoretical frame, beyond what could be achieved at that time by Borsenberger et al. (1986, 1987) and Atanackovič et al. (1987).
In parallel with this numerical setup, we shall also consider the more realistic case of “natural broadening” of the upper level of the atomic transition. The alternative atomic absorption profile is now a Lorentzian profile, whose expression was given in Eq. (11). However, in such a case, atomic absorption and emission profiles will be different since the atomic emission profile will now depend explicitly on the radiation field. Expressions for this emission profile η_{21} are discussed in Appendix B of Oxenius (1986); see also Hubený et al. (1983; their Sect. 4.3). The derivation of the latter expressions remains, however, questionable and it more likely leads to erroneous coefficients (V. Bommier, priv. comm.). Therefore, the modified emissivity consisting in the sum of a “classical” (order2) term with an extra “order4” contribution proposed by Bommier (1997) will be preferred. In that frame, based on sound physical grounds, it appears that only the order2 term emissivity will be associated with the iterative modifications of the VDF of the excited atoms. Indeed, the socalled order4 coefficient introduced by Bommier (1997) involves the lower level atomic state (see her Eq. (92) for the unpolarized case). However, as well as the order2 contribution (via the statistical equilibrium equation), the order4 contribution^{2} to emissivity depends on the radiation field, so that will also be need in order to be “runned to convergence” together with the distributions I and f_{2}. The associated numerical burden will mainly amount to recomputing Voigtlike functions (see e.g., Paletou et al. 2020) at every iteration, according to changes in f_{2} and I, and at every depth in the atmosphere.
Finally, any significant effect of these velocitychanging elastic collisions, which are “by construction” not taken into account in the standard NLTE approach, could be properly discussed for different astrophysical situations using the present formalism.
9. Conclusion
We have revisited and rewritten basic elements of Oxenius’ kinetic approach for NLTE spectral line formation, using more conventional notations for. We also brought clarification to some technical parts which were not always clearly explained in his textbook and some other subsequent, but still remaining “little acknowledged” studies. We note that similar elements of this approach have been recently discussed again in the more complete, but more likely less accessible frame of polarized radiation transfer, using the density matrix formalism (see e.g., Belluzzi et al. 2013).
Here we fully agree with the statement about “the necessity of resolving the statistical equilibrium equations for each velocity class” expressed by Bommier (2016b; see her Sect. 5.3). Therefore, in a next step, we shall implement an iterative numerical scheme, based on preexisting practices in astrophysical radiation transfer (see e.g., Lambert et al. 2016 and references therein) in order to compute self–consistently both for the specific intensity (or distribution of the photons) and, in a first stage, for the velocity distribution of these excited atoms present in the atmosphere. As a matter of fact, directly tackling the computation of the VDF of excited atoms also naturally leads to a generalization of the PRD theoretical frame, somewhat, without using any a priori defined redistribution function.
This will allow us to identify further and explore in more detail plausible astrophysical situations for which this “full–NLTE” approach for the very general issue of spectral line formation will prove to be indispensable.
As a matter of fact, for the respective kinetic equations of atoms and photons to be equivalent, while being further solved self–consistently, the issue of the streaming of particles described by a spatial derivative term, which remains further in the RTE but in the kinetic equation for the excited atoms, has to be discussed carefully (see again Hubený 1981, and references therein).
This very contribution is also reminiscent of (R_{II} − R_{III}), which clearly appears looking at Eq. (102) of Bommier (1997) for the special case of an infinitely sharp lower atomic level.
Acknowledgments
We are grateful to Dr Stanislav Gunár (Astronomical Institute of the Czech Academy of Sciences) who very kindly made his data available to us. We also thank an anonymous referee, and Dr Véronique Bommier (Cnrs and Observatoire de Paris, France) for valuable comments and encouragement. Finally Frédéric Paletou is grateful to his radiative transfer Sensei, Dr. L. H. “Larry” Auer, with whom we started discussing about these issues long time ago.
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Appendix A: Derivation of the macroscopic emission profile ψ_{x}
We find it quite unfortunate that in his monograph, Oxenius (1986) did not make the derivation of the macroscopic emission profile, whose expression is given in Eq. (42), explicit. This is why we propose the following steps.
We shall first assume that
where H is the usual Heaviside or unit step function. Using such an angular integration for the combination of our Eqs. (17) and (38) when ε = 0 can be written as follows:
First, the angular integration will transform the Dirac distribution modeling coherent scattering, into the Heaviside function.
Then, using spherical coordinates in the velocity space for the integration in u, that is
where θ and φ are the usual, respectively, polar angle and azimuth, one gets
which finally leads to Eq. (42) using the property of the H function which is then different from 0 for u ≥ x.
All Figures
Fig. 1.
Isotropic and normalized intensity profiles I vs. the reduced frequency x, used for the evaluation of departures from the equilibrium velocity distribution for the VDF of these excited atoms, f_{2}. We adopted different values for (i) the location of the (symmetric) peaks, and (ii) of the central reversal of each profile. These variations can also mimic some resonance lines observations that may form under different astrophysical conditions. 

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In the text 
Fig. 2.
Ratios f_{2}/f^{M} as a function of the normalized velocity u, for different radiation field profiles, and for ε = 0 and ζ = 0. The same linestyle convention as in Fig. 1 was adopted. The magnitude of these departures from a Maxwellian velocity distribution depends on both (i) the distance of the peaks of I vs. line center, and (ii) the amount of its central reversal characterized by I_{max.}/I(0). 

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In the text 
Fig. 3.
Ratios ψ_{x}/φ_{x} as a function of the reduced frequency x, for the different radiation field profiles displayed in Fig. 1, using also the same linestyle convention. We observe here, for the case of coherent scattering in the atom’s frame, similar tendencies as the ones displayed in Fig. 2 for the ratios between VDFs. 

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In the text 
Fig. 4.
Ratio f_{2}/f^{M} vs. the normalized velocity u for ε = 10^{−7} (ζ = 0), and a realistic H I Lyα illumination profile as observed by SoHO/Sumer. The peak of the line profile is located at x ∼ 5, and the central selfreversal is about 0.7 for this resonance line (see Fig. 5 of Gunár et al. 2020). 

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In the text 
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