# Full non–LTE spectral line formation I. Setting the stage

Frédéric Paletou<EMAIL_ADDRESS>Université de Toulouse,
Observatoire Midi–Pyrénées, Cnrs, Cnes, Irap, F–31400 Toulouse, France
Christophe Peymirat<EMAIL_ADDRESS>Université de Toulouse,
Faculté des Sciences et d’Ingénierie, F–31062 Toulouse cedex 9, France

###### Abstract

Radiative transfer out of local thermodynamic equilibrium (LTE) has been
increasingly adressed, mostly numerically, for about six decades now. However
the standard non–LTE problem most often refers to the _only_ deviation of the
distribution of photons from their equilibrium i.e., Planckian, distribution.
Hereafter we revisit after Oxenius (1986) the so-called “full non–LTE”
problem, which considers to couple and therefore to solve self–consistently
for deviations from equilibrium distributions of photons as well as for
massive particles constituting the atmospheric plasma.

##  I Introduction

The “standard” non–LTE (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 endeavour, nowadays considers also 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 with the standard NLTE radiation transfer problem
concerns the _redistribution_ , at least in frequencies if we restrict
ourselves to isotropic scattering, of photons after scattering onto these very
massive particles. 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 60’s (e.g., Avrett &
Hummer 1965, Auer 1968 for pioneering works), the standard NLTE problem also
remains, in most cases, limited to the frame of _complete_ redistribution
(hereafter, CRD).

The solution of a _full(er)_ non–LTE radiation transfer problem would consist
in assuming that the velocity distribution functions (hereafter VDF) of
massive particles may _also_ depart from an equilibrium distribution, as it is
now commonly assumed for the photons. And therefore the solution of 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 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 as much as possible standard notations. We adopt also several physical
assumptions, already discussed in past studies, in order to be able to set up
initial numerical experiments. We discuss partial vs. complete frequency
redistribution, as seen from this description of the problem. We present
indeed some effects expected on departures of the VDF of excited atoms vs. the
Maxwell–Boltzmann distribution, as well as significative differences between
computed and _radiation dependent_ emission profiles, and _a priori given_
absorption profile, for a simple two-level atom model. We finally discuss
future work in that somewhat forgotten physical frame for the (unpolarized)
radiation transfer problem.

##  II A simplified Boltzmann equation for massive particles

Let us write the evolution equation for a general distribution
$F(t,\vec{r},\vec{v})$ as:

$L[F]=C[F]\,,$ (1)

where $L$ is the _Liouville_ operator, and $C$ the collision operator.

Therefore, the left-hand side of Eq. (1) has the general expression:

$L[F]={{\partial F}\over{\partial t}}+{p\over m}{{\partial F}\over{\partial
r}}+K{{\partial F}\over{\partial p}}\,,$ (2)

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 right-hand side of Eq. (1) describes the various origins of collisions,
which may induce modifications of the distribution. It is usually expressed as
the sum of three contributions such as:

$C[F]=\left({{\delta F}\over{\delta t}}\right)_{\rm rad.}+\left({{\delta
F}\over{\delta t}}\right)_{\rm inel.}+\left({{\delta F}\over{\delta
t}}\right)_{\rm el.}\,.$ (3)

As a matter of fact, two main mechanisms are able to produce changes in a
distribution $F$. One of them is the so-called _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 “two-level atom” case is the simplest one may consider, the general
problem would consist in the self-consistent solution of four coupled kinetic
equations for, respectively, the atoms in their ground state ($F_{1}$), the
atoms in their excited state ($F_{2}$), electrons ($F_{e}$) – and finally the
photons ($I$).

However, and in order to be able to make first steps into the _extremely_
complex and cumbersome full non–LTE radiation transfer problem, we shall
proceed with several simplifying assumptions. First, we shall consider that
_only_ the velocity distribution function (hereafter VDF) of _excited_ atoms
i.e. $F_{2}$, may depart from a Maxwell-Boltzmann distribution. As previously
discussed by Oxenius (1979), we shall consider the velocity distribution of
the ground-state atoms, and of the free electrons to remain Maxwellian with
the same temperature $T$. The latter is also supposed to be low enough i.e.,
$kT\ll h\nu_{0}$, where $\nu_{0}$ is the central frequency of the transition
between the two atomic levels of our model, so that: (i) the number density of
excited atoms if much smaller than the one of ground-state atoms i.e.,
$n_{2}\ll n_{1}$ and (ii) stimulated emission may be further neglected.

Then we shall consider a time independent problem, as for photons, as well as
a _stationnary_ and _force-free_ problem for the atoms. And finally, following
the debate between Oxenius (1965, 1979) and Hubený (1981), we shall also
neglect the potential _streaming_ (i.e., ${v}.\partial_{r}\equiv 0$) of these
excited atoms. In that frame, the Boltzmann equation for atoms resumes to:

$C[F_{2}]=0\,,$ (4)

where $C$ is the collision operator.

##  III The kinetic equation of excited atoms

Each of the three contributions to the collision operator can be explicited
now, following Oxenius (1986).

The first one involves the radiation field, and can be expressed as:

$\left({{\delta F_{2}}\over{\delta t}}\right)_{\rm
rad.}=n_{1}B_{12}J_{12}(\vec{v})f_{1}(\vec{v})-n_{2}A_{21}f_{2}(\vec{v})\,,$
(5)

where stimulated emission has been neglected, and $A_{21}$ and $B_{12}$ are
the usual Einstein coefficients for, respectively, spontaneous emission and
radiative absorption. We also introduced here the _normalized to unity_
$f_{i}$ distribution, such as $F_{i}(\vec{v})=n_{i}f_{i}(\vec{v})$.

The second one involves inelastic collisions and writes:

$\left({{\delta F_{2}}\over{\delta t}}\right)_{\rm
inel.}=n_{1}n_{e}C_{12}f_{1}(\vec{v})-n_{2}n_{e}C_{21}f_{2}(\vec{v})\,,$ (6)

where the $C_{ij}$ are collisional (de)excitation coefficients, and $n_{e}$ is
the electron density. 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:

$\left({{\delta F_{2}}\over{\delta t}}\right)_{\rm el.}=n_{2}Q_{2}\left[f^{\rm
M}(\vec{v})-f_{2}(\vec{v})\right]\,,$ (7)

where $Q_{2}$ is a velocity-changing elastic collision rate. After
Borsenberger et al. (1986), one may consider $Q_{2}=n_{1}v_{\rm
th.}\sigma_{\rm el.}$, where $v_{\rm th.}$ is the “most probable” velocity of
the atoms, and $\sigma_{\rm el.}$ is an “average cross section” for this class
of elastic collision.

The scattering integral $J_{12}$ which appears in Eq. (5) is defined as:

$J_{12}(\vec{v})=\oint{{{d\Omega}\over{4\pi}}}\int{\alpha_{12}(\xi)I_{\nu}(z,\vec{\Omega})d\nu}\,,$
(8)

where $\alpha_{12}$ is the _atomic_ absorption profile, and
$I_{\nu}(z,\vec{\Omega})$ the usual specific intensity. At this stage we ommit
on purpose the potentiel dependence with depth $z$ of $J_{12}$, since we shall
stay away still, in this study, from the frame of radiation transfer. The
specific intensity (see our next §4.) is also dependent on the direction
$\vec{\Omega}$ of the radiation. All integrals $d\Omega$ 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 $\xi$ in the atomic frame, and $\nu$ in the
observer’s frame:

$\nu=\xi+\left(\frac{\nu_{0}}{c}\right)\vec{v}\cdot\vec{\Omega}$ (9)

where $\vec{v}$ is the velocity of the atoms, and $\vec{\Omega}$ 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, lead also
naturally to the role that will be played by their velocity distribution.

The _atomic_ absorption profile $\alpha_{12}$ is known a priori, and we may
consider either coherent scattering, i.e.:

$\alpha_{12}(\xi)=\delta(\xi-\nu_{0})\,,$ (10)

where $\xi$ is the incoming photon frequency, or radiation damping which is
characterized by a Lorentzian profile such that:

$\alpha_{12}(\xi)=\left(a\over\pi\right){1\over{(\xi-\nu_{0})^{2}+a^{2}}}\,,$
(11)

and where, in both cases, $\nu_{0}$ 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 $\nu_{0}$.

It is finally important to keep in mind, at this stage, that the atomic
emission profile $\eta_{21}$ may _differ_ from $\alpha_{12}$, in general.
This, combined with possible deviations from a Maxwell–Boltzmann VDF for
excited atoms will “naturally” lead to _non standard_ as well, and _radiation
dependent_ redistribution in frequency for the photons.

##  IV 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 §3. of Oxenius
(1986).

Assuming the very usual 1D plane–parallel geometry, the _time independent_ RTE
can be written111As 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). as:

$\frac{\partial
I_{\nu}(z,\vec{\Omega})}{dz}=\kappa_{\nu}(z,\vec{\Omega})\left[S_{\nu}(z,\vec{\Omega})-I_{\nu}(z,\vec{\Omega})\right]$
(12)

where $\kappa_{\nu}(z,\vec{\Omega})$ is the absorption coefficient, and
$S_{\nu}(z,\vec{\Omega})$ the so-called source function, which a priori
depends on depth $z$ in the atmosphere, frequency $\nu$ and direction
$\vec{\Omega}$.

The absorption coefficient is usually defined as:

$\kappa_{\nu}(z,\vec{\Omega})=\left({{h\nu_{0}}\over{4\pi}}\right)\left[n_{1}B_{12}\varphi_{\nu}(\vec{\Omega})-n_{2}B_{21}\psi_{\nu}(\vec{\Omega})\right]\,,$
(13)

where $B_{21}$ is the Einstein coefficient for stimulated emission, such as
$g_{1}B_{12}=g_{2}B_{21}$ assuming the statistical weights $g_{i}$, and the
(line) source function is, by definition, the ratio between the emissivity:

$\eta_{\nu}(z,\vec{\Omega})=\left({{h\nu_{0}}\over{4\pi}}\right)n_{2}A_{21}\psi_{\nu}(\vec{\Omega})\,,$
(14)

and the absorption coefficient. And finally, the more usual RTE introduces the
optical depth $\tau_{\nu}(\vec{\Omega})=-\kappa_{\nu}(z,\vec{\Omega})dz$ in
order to substitute the geometrical length for another quantity more relevant
to what photons effectively experience during multiple light–matter
interactions.

Should we neglect _completely_ stimulated emission i.e., going beyond the
(very usually) adopted so-called “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:

$S_{\nu}(z,\vec{\Omega})={{n_{2}A_{21}\psi_{\nu}(\vec{\Omega})}\over{n_{1}B_{12}\varphi_{\nu}(\vec{\Omega})}}\,.$
(15)

This assumption is also consistent with the low–temperature assumption, which
is also compatible with $n_{1}\gg n_{2}$.

Furthermore, the respective (and potentially also depth-dependent) emission
and absorption profiles are defined as:

$\varphi_{\nu}(\vec{\Omega})=\int{\alpha_{12}(\xi)f_{1}(\vec{v})d^{3}\vec{v}}\,,$
(16)

and,

$\psi_{\nu}(\vec{\Omega})=\int{\eta_{21}(\xi,\vec{\Omega})f_{2}(\vec{v})d^{3}\vec{v}}\,.$
(17)

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 $\eta_{21}$ and $\alpha_{12}$ are _not_ identical.
Moreover, the emission profile $\eta_{21}$ will depend on the radiation field.
We shall return to this issue in our section 8.

##  V A two–distribution, two-level 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 though, by considering coherent
scattering i.e., $\alpha_{12}(\xi)=\delta(\xi-\nu_{0})$, where $\nu_{0}$ is
the central wavelength of the spectral line associated with our two-level
atom.

Let us therefore start with the additional simplifications of, a stationnary,
force-free and no advection (or streaming) case for which:

$\left({{\delta F_{2}}\over{\delta t}}\right)_{\rm rad.}+\left({{\delta
F_{2}}\over{\delta t}}\right)_{\rm inel.}+\left({{\delta F_{2}}\over{\delta
t}}\right)_{\rm el.}=0\,.$ (18)

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: $C_{ij}\leftarrow n_{e}C_{ij}$ – see
also Eq. (9.51) of Hubeny & Mihalas (2014) for instance, we have:

$n_{2}(A_{21}+C_{21})f_{2}(\vec{v})+n_{2}Q_{2}[f_{2}(\vec{v})-f^{M}(\vec{v})]=n_{1}(B_{12}J_{12}(\vec{v})+C_{12})f_{1}(\vec{v})\,.$
(19)

Should we assume that $f_{1,2}$ are both Maxwellian, which is a common
assumption even for non–LTE 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 two-level atom case,
neglecting stimulated emission:

$n_{2}(A_{21}+C_{21})=n_{1}(B_{12}\int_{\vec{v}}{J_{12}(\vec{v})f^{M}(\vec{v})d^{3}\vec{v}}+C_{12})\,.$
(20)

We shall come back soon to the integral term after Eq. (30). Note here that,
because this particular contribution vanishes after integration over
velocities, the potential effects of velocity-changing elastic collisions can
be considered explicitly and reasonably discussed though, _only_ by using such
a kinetic description.

Now from Eq. (19), we are able to derive an _explicit_ expression for the
_non_ –equilibrium VDF $f_{2}$, assuming that all atoms in their fundamental
state of energy follow a Maxwellian VDF i.e., $f_{1}\equiv f^{M}$. 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:

${\cal{B}}_{W}=\left(\frac{2h\nu_{0}^{3}}{c^{2}}\right)e^{-h\nu_{0}/kT}\,.$
(21)

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:

$\bar{J}_{12}={J}_{12}/{\cal{B}}_{W}\,,$ (22)

as well as use the following relationships, first between Einstein
coefficients:

$\frac{A_{21}}{B_{21}}=\left(\frac{2h\nu_{0}^{3}}{c^{2}}\right)\,,$ (23)

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:

$n_{1}^{*}C_{12}=n_{2}^{*}C_{21}\,,$ (24)

where $n_{1,2}^{*}$ are the LTE values of the respective densities of
population, satisfying the Boltzmann law:

$\left(\frac{n_{2}^{*}}{n_{1}^{*}}\right)=\left(\frac{g_{2}}{g_{1}}\right)e^{-h\nu_{0}/kT}\,.$
(25)

We shall therefore now introduce the _normalized_ population, which will allow
some measurement of deviations from LTE: $\bar{n}_{2}=n_{2}/{n_{2}^{*}}$.

Let us now introduce:

$\varepsilon=\frac{C_{21}}{A_{21}+C_{21}}\,,$ (26)

which is another parameter otherwise traducing the amount of “departure from
LTE” in a given atmosphere, and the less common:

$\zeta=\frac{Q_{2}}{A_{21}+C_{21}}\,,$ (27)

which characterizes the amount of elastic collisions. Assuming that
${n_{1}}={n_{1}^{*}}$ and $f_{1}\equiv f^{M}$, we may now rewrite the
following:

$\bar{n}_{2}f_{2}(\vec{v})-\bar{n}_{2}\zeta[f^{M}(\vec{v})-f_{2}(\vec{v})]=\left[\varepsilon+(1-\varepsilon)\bar{J}_{12}(\vec{v})\right]f^{M}(\vec{v})\,.$
(28)

This expression is indeed identical to Eq. (2.44) of Borsenberger et al.
(1986) when one neglects streaming of particles (his $\eta=0$).

Now, integrating Eq. (28) _over all velocities_ , we find that:

$\bar{n}_{2}=\varepsilon+(1-\varepsilon){\cal{J}}_{12}\,,$ (29)

where we defined:

${\cal{J}}_{12}=\int_{\vec{v}}{\bar{J}_{12}(\vec{v})f^{M}(\vec{v})d^{3}\vec{v}}\,,$
(30)

which is equivalent to the common $\bar{J}$ scattering integral used in the
standard approach. We finally recover the following:

$f_{2}(\vec{v})=\left[\frac{\zeta}{1+\zeta}+\left(\frac{1}{1+\zeta}\right)\frac{\varepsilon+(1-\varepsilon)\bar{J}_{12}(\vec{v})}{\varepsilon+(1-\varepsilon){\cal{J}}_{12}}\right]f^{M}(\vec{v})\,.$
(31)

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 $I$.

##  VI 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: $S=n_{2}A_{21}/n_{1}B_{12}$ i.e.,
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 i.e.,
$\alpha_{12}\equiv\eta_{21}$, CRD also assumes implicitly the identity:

$f_{2}\equiv f_{1}(\equiv f^{M})\,,$ (32)

whose effects therefore result from expressions given in Eqs. (16) and (17).
In that frame, we also have that the normalized source function
$\bar{S}=S/B_{W}$ is such that:

$\bar{S}=\bar{n}_{2}=\varepsilon+(1-\varepsilon){\cal{J}}_{12}\,.$ (33)

This expression is indeed consistent with the classical (CRD) “two-level atom”
case described in any radiation transfer textbook, since Avrett & Hummer
(1965).

However, 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 $\zeta=0$ _and_ $\varepsilon=1$ i.e. assuming
LTE! Therefore non–LTE 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:

$\gamma=\frac{A_{21}+C_{21}}{A_{21}+C_{21}+Q_{E}}$ (34)

is introduced (see Omont et al. 1972, Heinzel & Hubený 1982, Heinzel et al.
1987), to form a linear combination of $R_{\rm II}$ and $R_{\rm III}$
redistribution such as:

$R(x^{\prime},x)=\gamma R_{\rm II}(x^{\prime},x)+(1-\gamma)R_{\rm
III}(x^{\prime},x)\ ,$ (35)

following the nomenclature of Hummer (1962), and where $x^{\prime}$ and $x$
are, respectively, the usual incoming and outgoing reduced frequencies,
usually defined as:

$x=\frac{\nu-\nu_{0}}{\Delta\nu_{D}}\,,$ (36)

with the Doppler width $\Delta\nu_{D}=(\nu_{0}/c){v}_{th.}$, and the most
probable velocity ${v}_{th.}=\sqrt{2kT/M}$ as usual.

The elastic collision rate $Q_{E}$ which appears in the branching ratio
$\gamma$ 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 _velocity-changing_ elastic collisions i.e., 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 §13.2) estimate that $Q_{2}$ should
be very small in solar–like atmospheres, so that the so-called 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 velocity-changing 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 though, the elastic
collision term introduced in Eq. (7) just _vanishes_.

##  VII Quantifying departures from equilibrium

The stage is now set for a 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:

$\alpha_{12}(\xi)=\eta_{21}(\xi)=\delta(\xi-\nu_{0})\,,$ (37)

where both atomic profiles are also assumed to be _isotropic_. Should we
moreover assume _n_ o elastic collisions so far i.e. $\zeta=0$, then after Eq.
(31) we end-up with:

$f_{2}(\vec{u})=\left[\frac{\varepsilon+(1-\varepsilon)\bar{J}_{12}(\vec{u})}{\varepsilon+(1-\varepsilon){\cal{J}}_{12}}\right]f^{M}(\vec{u})\,,$
(38)

where we also introduced the _normalized_ velocity $u=v/v_{th.}$.

Figure 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.

Non–LTE implies that $\varepsilon<1$, 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:

$\bar{J}_{12}(u)=\frac{1}{2u}\int_{-u}^{+u}{I(x)dx}\,.$ (39)

Then, since $f_{1}\equiv f^{M}$, using this classical expression for the
(normalized to unity) macroscopic absorption profile:

$\varphi_{x}=\frac{1}{\sqrt{\pi}}e^{-x^{2}}\,,$ (40)

we can also establish that:

${\cal{J}}_{12}=\frac{1}{\sqrt{\pi}}\int_{-\infty}^{+\infty}{I(x)e^{-x^{2}}dx}\,,$
(41)

which finally allows to evaluate departures of $f_{2}$ from the equilibrium
distribution depending on the thius far given (and normalized to unity)
radiation field $I(x)$.

Figure 2: Ratios $f_{2}/f^{M}$ as a function of the normalized velocity $u$,
for different radiation field profiles, and for $\varepsilon=0$ and $\zeta=0$.
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 $I_{\rm max.}/I(0)$.

According to these expressions, one can notice that, with coherent scattering
in the atom’s frame, it remains however a _ultimate_ option for “saving” the
CRD case. Indeed, assuming isotropic “white” light illumination i.e., a
frequency independent radiation field such as $I(x)=I_{0}$ is the only
condition left to us in order to keep $f_{2}\equiv f^{M}$, and therefore
$\psi_{x}\equiv\varphi_{x}$.

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 i.e., $I_{\rm max.}/I(0)$, as
well as peaks located at different, and increasing positions away from line
center. These may mimic some observed and well-resolved resonance line
profiles, similar to one of the good candidates for future modeling such as
Ly$\alpha$ of H i.

Then, we computed ratios of $f_{2}/f^{M}$ according to Eq. (38), and adopting
further that $\varepsilon=0$. 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).

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 $I_{\rm max}$ and its central value $I(0)$.

Finally, it is also possible to compute the ratio between the macroscopic
emission profile $\psi_{x}$ and the (Gaussian) absorption profile
$\varphi_{x}$. Indeed, after Eqs. (38) and (17), and with $\varepsilon=0$
though, one can derive (see also Appendix A) that:

$\psi_{x}=\frac{2}{\sqrt{\pi}{\cal{J}}_{12}}\int_{\lvert
x\rvert}^{\infty}{\bar{J}_{12}(u)ue^{-u^{2}}du}\,,$ (42)

for the case of coherent scattering in the atom’s frame.

Figure 3: Ratios $\psi_{x}/\varphi_{x}$ 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.

Note also that, at this stage, with the knowledge of the _radiation dependent_
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 profiles ratios as a function of $x$, for the same
set of incident radiation profiles as before.

We observe for these ratios, and the special case of coherent scattering in
the atom’s frame, 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.

Figure 4: Ratio $f_{2}/f^{M}$ vs. the normalized velocity $u$ for
$\varepsilon=10^{-7}$ ($\zeta=0$), and a realistic H i Ly$\alpha$ illumination
profile as observed by SoHO/Sumer. The peak of the line profile is located
$x\sim 5$, and the central self–reversal is about 0.7 for this resonance line
(see Fig. 5 of Gunár et al. 2020).

##  VIII 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 Ly$\alpha$ 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 $x\sim 30$. Then we
fixed $\zeta=0$ and $\varepsilon=10^{-7}$ i.e., 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 $x\sim 5$ in our case, and
in the line wings.

But most important will be to address further the self–consistent computation
of the two distributions $f_{2}$ and $I$. This should be possible within an
iterative cycle involving: an initial guess for the source function $S$
assuming for instance both LTE and CRD, then a _formal solution_ of the RTE
allowing for the computation of $\bar{J}_{12}$ and ${\cal{J}}_{12}$. Then
$f_{2}$ will be estimated, as well as $\psi_{x}$, making it possible to update
further the source function, start another iteration and so forth, until
reaching convergence for _both_ $f_{2}$ and $I$.

We believe that state-of-the-art 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 more
detailed theoretical frame, beyond what could be achieved at that time by
Borsenberger et al. (1986, 1987) and Atanackovi$\check{\rm c}$ et al. (1987).

In parallel with this numerical set-up, 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 explicitely on the radiation field. The
expression for this emission profile $\eta_{21}$ is discussed in Appendix B of
Oxenius (1986); see also Hubený et al. (1983; their §4.3). The derivation of
the latter expressions remains however questionable, and it more likely leads
to _erroneous_ coefficients (V. Bommier, private communication). Therefore,
the modified emissivity consisting in the sum of a “classical” (order-2) term
with an extra “order-4” contribution proposed by Bommier (1997) will be
preferred. In that frame, based on sound physical grounds, it appears that
_only_ the order-2 term emissivity would be associated with the iterative
modifications of the VDF of the excited atoms. Indeed, the so-called order-4
coefficient introduced by Bommier (1997) involves the _lower level_ atomic
state – see her Eq. (92) for the unpolarized case. However, as well as the
order-2 contribution (via the statistical equilibrium equation), the order-4
contribution222This very contribution is also reminiscent of $(R_{\rm
II}-R_{\rm III})$, which clearly appears looking at Eq. (102) of Bommier
(1997), for the special case of an infinitely sharp lower atomic level. to
emissivity depends on the radiation field, so that it will _also_ need to be
“runned to convergence” together with the distributions $I$ and $f_{2}$. The
associated numerical burden will mainly amount to recomputing Voigt-like
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 velocity-changing _elastic_
collisions, which are “by construction” though not taken into account in the
standard non–LTE approach could be properly discussed for different
astrophysical situations, using the present formalism.

##  IX Conclusion

We have revisited and rewritten, using more conventional notations basic
elements of Oxenius’ kinetic approach for non–LTE spectral line formation. We
also brought clarification on some technical parts which were not always
clearly explained in his textbook and some other subsequent, but still
remaining “little aknowledged” studies. 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 §5.3). Therefore, in a next step, we shall implement
an iterative numerical scheme, based on pre-existing 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.

##  Appendix A Derivation of the macroscopic emission profile $\psi_{x}$

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). This is why we propose the following steps.

We shall first assume that:

$\oint{\delta(x-\vec{u}\cdot\vec{\Omega})d\Omega=\frac{2\pi}{u}H(u-|x|)}\,,$
(43)

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 $\varepsilon=0$
writes as:

$\psi_{x}=\frac{1}{{\cal{J}}_{12}}\oint{\left(\int{\bar{J}_{12}(\vec{u})\delta(x-\vec{u}\cdot\vec{\Omega})f^{M}(\vec{u})d^{3}\vec{u}}\right)\frac{d\Omega}{4\pi}}\,.$
(44)

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:

$d^{3}\vec{u}=u^{2}\sin(\theta)dud\theta d\varphi\,,$ (45)

where $\theta$ and $\varphi$ are the usual, respectively, polar angle and
azimuth, one gets:

$\psi_{x}=\frac{2\pi}{\pi^{3/2}{\cal{J}}_{12}}\int{\bar{J}_{12}(u)ue^{-u^{2}}H(u-|x|)du}\,,$
(46)

which finally leads to Eq. (42) using the property of the $H$ function which
is then different from 0 for $u\geq\lvert x\rvert$.

###### Acknowledgements.

We are grateful to Dr Stanislav Gunár (Astronomical Institute of the Czech
Academy of Sciences) who very kindly made his data available to us, and to Dr
Véronique Bommier (Cnrs and _Observatoire de Paris_ , France) for fruitful
discussions and her 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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