11institutetext: Institut d’Astrophysique de Paris
98bis, Boulevard Arago, 75014 Paris, France

# Spectral line fluorescence in moving envelopes of stars

I. Theoretical background, numerical code, and proof of concept

Claude Bertout Retired. Current e-mail: claude.m.bertout (at) gmail.com

(Received February 29, 2024; accepted May 28, 2024)

###### Abstract

Context. Fluorescence of the optical FeI lines is observed in active T Tauri
stars, and is considered a defining characteristic of this class of young
stellar objects.

Aims. The formation of optical fluorescent lines in moving media has not yet
been studied in detail. This work represents a first step in investigating the
fluorescence process in different types of macroscopic velocity fields: (a)
accelerated outflows, (b) accelerated infalls, and (c) non-monotonic velocity
fields (such as an accelerating outflow followed by a deceleration region or
an accretion shock front). We aim to develop a general computer code for
studying the fluorescent emission in any 2D macroscopic velocity field. As an
illustration, we investigate FeI T Tauri-like fluorescent emission in these
moving stellar envelopes.

Methods. We solved the radiative transfer equations for the lines involved in
the fluorescent process, assuming spherical symmetry and a simplified atomic
model. We used the framework of the generalized Sobolev theory to compute the
interacting, nonlocal source functions. The emergent line fluxes were then
integrated exactly.

Results. Because of Doppler shifts in the moving gaseous envelope, photons of
the three lines involved in T Tauri star FeI fluorescence (CaII H, FeI$\lambda
3969$, and Hϵ) interact with each other in a complex way, such that
fluorescent amplification of the line flux occurs not only for FeI$\lambda
3969$, but also for the other two lines, in all velocity fields that we
investigated. With the assumption of local thermodynamic equilibrium
populations, the line source functions of moderately optically thick lines are
not strongly affected by line interactions, while they are depressed in the
inner envelope for optically thick lines because of stellar photon absorption
in the interaction regions. Fluorescent amplification takes place mainly in
the observer’s reference frame during the flux integration. We define a
measure of fluorescence based on the line equivalent widths and perform a
parameter study for an accretion flow with a broad range of envelope
temperatures and densities while including approximate collisional de-
excitation rates in the source function computations. Significant fluorescence
occurs over the entire temperature range of the investigated flow, but only in
the higher density range, suggesting that relatively high mass accretion rates
are needed to trigger the fluorescence process.

Conclusions. Further comparison with observations will require solving the
rate equations for the atomic populations involved, along with the radiation
field computed with the method presented here. The main product of this
research is the open-source computer code SLIM2 (Spectral Line Interactions in
Moving Media), written in Python/NumPy, which numerically solves the
fluorescence problem for arbitrary 2D velocities.

###### Key Words.:

Line: formation – Radiative transfer – Methods: numerical – Stars: variables:
T Tauri, Herbig Ae/Be

##  1 Introduction

The physical process of fluorescence, also known as photoexcitation by
accidental resonance, is associated in stellar astronomy with the name Ira S.
Bowen, who showed that the spectra and composition of gaseous nebulae could be
explained by fluorescent emission of the ions OIII and NIII excited by
neighboring HeII transitions (Bowen, 1934, 1935). This solved a long-standing
mystery that had even led Higgins (1898) to postulate the existence of a
chemical element he called “nebuliu”. Fluorescent emission was also successful
in explaining features of the line spectra in long-period variables
(Thackeray, 1937; Willson, 1972) as well as in symbiotic stars (e.g., Kastner,
1991; Pereira et al., 1999; Eriksson et al., 2004; Selvelli et al., 2007;
Hyung et al., 2018).

A similar mechanism was proposed by Herbig (1945) to account for the anomalous
Fe I emission observed in active T Tauri stars (TTSs). Specifically, the
transitions $\lambda$4063 and $\lambda$4132 of Fe I appear in strong emission
in active TTSs, while other lines of the multiplet $a^{3}F-y^{3}F^{0}$ are in
very weak emission or are not detected (Joy, 1945). The two anomalous lines
originate from the same $y^{3}F^{0}_{3}$ upper level together with the
$\lambda$3969 line. Herbig conjectured that “the explanation lies in the near
coincidence of the $\lambda$3969 line of Fe I with the red component of the H
line of Ca II. Absorption of the proper frequencies by the Fe I atoms would
ensure a supply of atoms to $y^{3}F^{0}_{3}$, while the other sub-levels of
$J=2\ \rm{and}\ 4$ would lack such an active source.” The $\lambda$3969.3 line
is flanked by Ca II H at $\lambda$3968.5 on the blue side and by the H I
$H_{\epsilon}$ line at $\lambda$3970.1 on the red side. Both of these lines
are in strong emission in TTSs, and given the complexity of motions in the
surroundings of these stars, it is conceivable that either one could be
responsible for pumping atoms into the upper level of Fe I $\lambda$3969\.
Herbig did not consider the possible role of Hϵ in the Fe I fluorescence,
presumably because it was believed in 1945 that TTS activity was a scaled-up
version of solar-type, chromospheric activity accompanied by a strong stellar
wind. However, Willson (1974, 1975) concluded from a detailed investigation of
the statistical equilibrium of the lines involved that both H and Hϵ were
contributing to the fluorescence mechanism in the active TTS RW Aur, and he
suggested that rotational velocities were greater than ejection velocities in
the envelope of that star. While we know now that TTSs are not fast rotators,
and that TTS activity is fueled mainly by accretion onto the star of disk
material, it is certainly plausible that either H or Hϵ, or both, are at work
in the fluorescence process. And given the wealth of mid- and high-resolution
spectroscopic data that has become available with the advent of such
instruments as ESO’s ESPRESSO, UVES, and X-Shooter (see, e.g., Armeni et al.,
2023), it is obvious that a renewed quantitative study of fluorescence in
young stars is overdue. Our aim is to present the theoretical background as
well as a numerical code that enables such a study. The problem is relatively
complex, and we will therefore solve it in two steps.

The first one, presented in the present paper, focuses on the formation of the
$\lambda$3969 FeI line and its interactions with the CaII H and HI Hϵ lines,
using the equivalent two-level atom approach for each of the lines and the
assumption of local thermodynamic equilibrium (LTE) for computing the
populations of the atomic levels involved. Our aim is to create a code for
solving the radiative interactions between lines that occur in the moving
media that we consider and to gain some understanding of the physical
conditions of the envelope that lead to amplification of the $\lambda$3969
line flux.

This first investigation will not make a direct comparison between models and
observations possible, since we will not be solving the statistical
equilibrium equations for the level populations of FeI self-consistently with
the radiation field. In a forthcoming extension of this work, we will
therefore couple the code developed here with a computation of the non-LTE
rate equations for a multilevel Fe atom model, which will then allow us to
compare the anomalous $\ce{FeI}\lambda$4063 and $\lambda$4132 lines, as well
as the nonfluorescent lines of the same multiplet, to observations.

The physical processes taking place in the most active T Tauri envelopes
involve several components: circumstellar disk, magnetospheric accretion
columns between disk and star, bipolar disk wind, and stellar wind. Modeling
the line formation process in this dynamic environment is challenging, but
considerable advances have been achieved over the past 20 years (e.g.,
Tambovtseva et al., 2001, 2014; Kurosawa & Romanova, 2012). All of this work
has been done in the framework of the generalized Sobolev approximation, which
makes this problem tractable.

In a moment of remarkable insight, Sobolev (1947, English translation in 1960)
realized that the Doppler shift simplifies the formation of spectral lines in
moving media such as stellar envelopes and winds. The basic approximation made
in what is now called the Sobolev theory is that, because of the macroscopic
velocity gradient in the moving gas, the source function and optical depth of
a given transition are constant over the narrow region where emitted photons
are not Doppler-shifted with respect to atoms in the rest of the envelope, and
equal to zero everywhere else. This is true in particular in outwardly
accelerating, supersonic winds, which represent the simplest case for applying
the Sobolev theory; the line formation problem then reduces to finding regions
of constant radial velocity with respect to the observer, each of which
corresponds to a single frequency in the emergent line profile.

In more quantitative terms, Sobolev assumed in his original derivation that
the local absorption and emission profiles were constant over the width
$\Delta\nu_{ik}=2(u/c)\nu_{ik}$ and equal to zero everywhere else. Here,
$\nu_{ik}$ is the frequency of transition $ik$, $u$ is the mean thermal
velocity of atoms, and $c$ is the velocity of light. For a hydrogen line at
$10^{4}$K, the emission and absorption profile widths are on the order of 10
km/s, so $\Delta\nu_{ik}/\nu_{ik}$ is on the order of $10^{-4}$, or 0.5Å for a
line wavelength of 5000Å. This is to be compared to the overall line width,
which will extend, for example, over 16Å in a 500 km/s outflow. In other
words, whenever the Sobolev approximation holds, absorption and emission line
profiles can be seen as delta-like functions, which is consistent with
considering that the emitting regions at a given line frequency are radial
velocity surfaces rather than extended volumes.

The Sobolev theory has been used extensively for modeling line formation in
envelopes of stars and outflows in a variety of objects, ranging from early-
type stars (e.g., Castor, 1970; Lamers et al., 1987) to young stellar objects
(e.g., Dmitriev et al., 2019) and from interstellar clouds (Goldreich & Kwan,
1974) to bipolar outflows (e.g., Cabrit & Bertout, 1986, 1990, 1992). A review
of the rich literature applying Sobolev’s theory in various contexts is beyond
the scope of this paper, but it is fair to say that this approach has led to
huge progress in our understanding of line formation in many different
astrophysical situations, as Sobolev himself had foreseen in his original
treatise.

Another reason for this success – besides the simplification in line formation
that we have alluded to – stems from the fact that the Sobolev method was
shown, by extensive comparisons with much more computationally challenging
numerical solutions of the exact line transfer equations, to be sufficiently
accurate for a large class of problems, especially when the approximation is
used only for computing the line source function and the emergent line flux is
integrated exactly (see, e.g., Bastian et al., 1980; Hamann, 1981).

Originally used for the study of accelerating outflows, the Sobolev method has
been generalized and extended over the years to more involved velocity fields,
including decelerated outflows, accelerated inflows, rotation, and various
combinations thereof. The generalization of the original Sobolev theory for
treating velocity laws that lead to complex radial velocity surfaces was
worked out by Rybicki & Hummer (1978) in a seminal paper. The same theoretical
framework has also been useful for studying the interactions between lines
that occur in some astrophysical situations, for example close doublets (cf.
Olson, 1982) and masers (cf. Pavlakis & Kylafis, 1996). The latter authors
considered the case of rectangular local emission and absorption profiles, in
which case the line source function computation of the locally interacting
lines becomes analytically tractable, albeit tedious.

In static media, spectral lines interact when their absorption profiles
overlap locally. If macroscopic velocity gradients are present in the line-
emitting region, however, it can also happen that the line of interest is
Doppler-shifted onto another one that is too far in wavelength for a local
interaction to occur; following Pavlakis & Kylafis (1996), we call this
situation a nonlocal interaction.

Quantitatively, two lines, A and B, with rest wavelengths $\lambda_{A}$ and
$\lambda_{B}$ and absorption profile widths $\Delta\lambda_{A}$ and
$\Delta\lambda_{B}$ interact locally when
$(\lambda_{B}-\lambda_{A})/\lambda_{A}\lesssim\Delta\lambda_{A}+\Delta\lambda_{B}$.
They interact non-locally whenever line-A photons emitted by matter moving at
velocity $v_{A}$ cross a distant region where the radial velocity,
$v_{r}^{B}$, of the B-line-emitting matter is such that
$(\lambda_{B}-\lambda_{A})/\lambda_{A}=(v_{r}^{B}-v_{r}^{A})/c$.111For the
sake of simplicity, we neglected the line widths in this condition. In the
example discussed earlier, the line at 5000Å will interact to some extent with
neighboring lines located within 8Å of the transition rest wavelength.

In the present work, we calculate the line source functions and emergent line
profiles of radiatively overlapping lines in moving media. While the
generalized Sobolev theory à la Hummer and Rybicki remains the conceptual
framework underlying the computation, we make use of the Sobolev approximation
for computing the source function and, following Bertout (1984), perform an
exact integration of the emerging flux, assuming Gaussian shapes for the local
absorption profiles.

We make several simplifying assumptions in the subsequent discussion. The
moving medium under consideration is a spherically symmetric, extended stellar
envelope with maximum radius Renv surrounding a star of mass $M_{*}$, radius
$R_{c}$, and effective temperature T${}_{\\!e\\!f\\!f}$. The envelope velocity
field is, at first, assumed to be a monotonically accelerating outflow from
velocity $V_{c}$ at the stellar core to $V_{max}$ at Renv. We then consider
velocity fields that lead to more complex distant radiative coupling. We
neglect continuous opacity in the envelope, which has temperature
T${}_{env}(r)$. The gas number density, $n_{t}$, is computed from the
continuity equation. The emitting gas has a solar composition, and the atomic
level populations and free electron number density, $n_{e}(r)$, follow from
the assumption of LTE. In the framework of equivalent two-level atoms, $n_{l}$
and $n_{u}$, the occupation numbers of both line levels in Eq. 8, are then
given by the Saha-Boltzmann equation evaluated at the local temperature and
electron density.

Collisional de-excitation of atoms is an important radiation source for the
lines that we investigate, so we included approximate rates, $\epsilon_{k}$,
in the source function computation for line $k$ . However, since most of the
previous theoretical work done on line formation in moving media neglects the
collisional term and since we want to be able to compare our results to
previous work, we will assume $\epsilon=0$ in the theoretical sections of this
work and reserve the use of collisional terms to Sects. 3 and 4.

In the next section, we study the fluorescence process in a monotonically
accelerating outflow. The Sobolev approximation for the source functions of
interacting lines is discussed in Sect. 2.1, while Sect. 2.2 extends Bertout’s
1984 exact flux integration method to the case at hand. In Sect. 3 we
generalize the analysis to the more complex radial velocity surfaces found in
accreting envelopes (Sect. 3.1) and in non-monotonic velocity fields (Sect.
3.2). A discussion of the physical conditions that lead to fluorescence
follows in Sect 4. The FeI fluorescence observed in TTSs is used as an example
in the various illustrations of the paper.

Finally, we briefly present in Appendix A the open-source computer code
(available on GitHub under CC 4.0 license) written in Python/NumPy that solves
the nonlocal radiative coupling between interacting spectral lines formed in
spherical, moving envelopes of stars.

##  2 Formation of interacting lines in a monotonic, accelerated outflow

We considered an outwardly accelerating stellar outflow with properties
loosely inspired by the Alfvén-wave-driven stellar wind originally proposed
for TTSs by Hartmann et al. (1982). At first, we used the data they give in
their table for the velocity of Model 5, interpolating between the few
provided radial points, but this resulted in discontinuous derivatives of the
velocity at the grid points that in turn led to spurious variations in the
source function of the optically thick hydrogen line. We therefore looked for
a least-square fit to the data, assuming a parameterized velocity of the form

$v(r)=v_{\infty}\left[1-(1-v_{min})\frac{r}{R_{c}}\right]^{\alpha}.$ (1)

The resulting parameters are given in Table 1 together with the other
computation parameters, and Fig. 2 shows the velocity law together with the
adopted gas density law, which follows from the continuity equation

$n_{t}(r)=n_{t}(R_{c})\frac{v(R_{c})}{v(r)}\left(\frac{R_{c}}{r}\right)^{2}.$
(2)

Our choice of computation parameters for the envelope properties is informed
by spectroscopic data of the active TTS system SCrA (e.g., Bertout et al.,
1982), which display the FeI fluorescence phenomenon discussed here. In these
spectra, the intensities of the CaII H + Hϵ blend and the CaII K line, both in
strong emission, are of comparable strength. This is also the case, more
generally, for many TTSs (Valenti et al., 1993). For this to occur (within the
limitations of the LTE assumption), the temperature of the line formation
region cannot be higher than $6-7\cdot 10^{3}$K. At such low temperatures, we
must have a relatively high gas density in order to get significant emission
line flux. In the range of values chosen here, the three lines are partially
optically thick at the bottom of the envelope (Fig. 2). Clearly, near-
photospheric conditions are required for producing moderately strong emission
lines when using the assumptions of LTE populations and isothermal gas, and
when neglecting collisions. As a consequence, the mass-flow rate determined
using these assumptions and the additional condition of spherical symmetry is
much higher than usually assumed for TTSs. The situation improves when
collisions are taken into account, as we will see in Sect. 4. But our primary
goal in this section, as already mentioned, is to present the method for
solving the fluorescent line formation in the case $\epsilon=0$, not to
compare the results with observations.

Table 1: Computation parameters for the outflow of Sect. 2. Rc | Renv | M∗ | T${}_{\\!e\\!f\\!f}$ | Tenv | $n_{t}(R_{c})$ | $v_{\infty}$ | $v_{min}$ | $\alpha$
---|---|---|---|---|---|---|---|---
2 R☉ | 10 Rc | 1 M☉ | 4.5 $\cdot 10^{3}$K |  $6\cdot 10^{3}$ K |  $2\cdot 10^{16}$ cm-3 | 250 km/s | 0.02 | 0.95

Figure 1: Velocity (Eq. 1) and density law (Eq. 2) of the stellar outflow.

Figure 2: Line absorption coefficient for the three lines considered.

###  2.1 Sobolev source functions

We consider three spectral transitions: $B,$ $C,$ and $R$ with rest
wavelengths $\lambda{}_{B}$ < $\lambda{}_{C}$ <$\lambda{}_{R}$ in a wavelength
interval

$(\lambda_{R}-\lambda_{B})=\lambda_{R}\cdot v(R_{env})/c.$ (3)

For convenience, the lines are denoted $B$ for blue, $C$ for center,and $R$
for red; these lines may stem from different chemical elements and have
respective thermal widths $\Delta\nu_{B}$, $\Delta\nu_{C}$, and
$\Delta\nu_{R}$.

The generalized Sobolev source functions of interacting lines can be computed
by modifying the formalism developed by Rybicki & Hummer (1978) for the study
of single lines emitted in velocity fields that lead to nonlocal radiative
coupling. We assume in the following that readers are familiar with their
work.

####  2.1.1 $B$ line

Figure 3: Selected CP surfaces for the spherically symmetric stellar wind,
with the properties given in the text. Each surface is associated with the
radial point of the same color. Note the variety of geometries for the
surfaces, which give rise to discontinuities in the source function

We first note that in any spherical, monotonically accelerating large-scale
flow, photons emitted by the bluest line (the $B$ line as defined above)
appear redshifted to atoms responsible for the redder $C$ and $R$ transitions
and thus escape the medium freely. The line source of $B$ is therefore
unaffected by $C$ and/or $R$, and its mathematical expression is the same as
the one derived for a single line by Rybicki and Hummer. For an equivalent
two-level atom, we then have

$S_{B}(r)=\frac{(1-\varepsilon_{B})\beta_{B}^{c}I_{B}^{c}+\varepsilon_{B}B_{B}}{\varepsilon_{B}+(1-\varepsilon_{B})\beta_{B}},$
(4)

where $\beta_{B}$ is the local escape probability, $\beta_{B}^{c}$ the
probability that an escaping photon will strike the stellar core, $I_{B}^{c}$
is the background continuous radiation emitted by the stellar core at line
frequency, $\varepsilon_{B}$ the ratio of collisional to total de-excitation
rates from the upper level, and $B_{B}$ is the Planck function at rest line
frequency and local electron temperature. All these quantities are defined
locally at radius $r$. With $\mu=\cos\theta$ defining the direction cosine of
a ray with respect to the outward radial direction, the local escape
probability is given by

$\beta_{B}=\frac{1}{2}\intop_{-1}^{+1}\frac{1-e^{-\tau_{B}}}{\tau_{B}}d\mu,$
(5)

while the probability for an escaping photon to reach the stellar core is

$\beta_{B}^{c}=\frac{1}{2}\intop_{\mu_{c}}^{+1}\frac{1-e^{-\tau_{B}}}{\tau_{B}}d\mu,$
(6)

where $\mu_{c}$ is the cosine of the half-angle subtended by the stellar core.
The quantity $\tau_{B}$ in the last two equations is the $B$ line-center
optical depth in direction $\mu$. Supposing that the transition occurs between
electronic levels $l$ and $u$, $\tau_{B}$ is given by (see, e.g., Hubeny &
Mihalas, 2014, Chapter 19, p.746ff)

$\tau_{B}(r,\mu)=\chi_{lu}(r)/Q(r,\mu),$ (7)

with the absorption coefficient given by

$\chi_{lu}(r)=(\pi
e^{2}/mc)f_{lu}[n_{l}(r)-(g_{l}/g_{u})n_{u}(r)]/\Delta\nu_{D}$ (8)

and

$Q(r,\mu)=|dv_{s}/ds|,$ (9)

where $v_{s}$ is the local fluid velocity projected in direction $s$ with
cosine angle $\mu$, $\Delta\nu_{D}=\nu_{0}v_{th}/c$ the local thermal line
width, and where the other symbols have their usual meaning. The absorption
coefficients of the three considered lines, as expressed by Eq. 8, are shown
in Fig. 2 for the parameters of Table 1.

####  2.1.2 $C$ line

We next consider photons of transition $C$ emitted in the outer parts of the
envelope. They will interact with $B$ photons emerging from deeper parts of
the envelope, where the emitting matter is moving slower, whenever the line-
of-sight velocity difference is equal to the rest velocity difference between
the two lines. In other words, interaction between $B$ and $C$ photons on any
given line of sight will take place whenever

$v_{rad}(r_{B})-v_{rad}(r_{C})=\Delta\lambda_{BC}/c,$ (10)

where $\Delta\lambda_{BC}$ is the wavelength difference between lines $B$ and
$C$. Solving Eq. 10 over the entire space defines a locus of distant points
$r^{{}^{\prime}}$ where emitted $B$ photons can interact with $C$ photons
emitted at $r$. This locus is called a common-point (CP) surface by Olson
(1982), using the nomenclature introduced by Rybicki & Hummer (1978) to
describe the line self-coupling that occurs in those velocity laws that lead
to distant radiative coupling. We use the same denomination and denote the CP
surface described above by $r^{{}^{\prime}}_{B}$ in the following equations.

Figure 3 shows some CP surfaces for the lines of interest in the stellar wind
with properties shown in Fig. 2 and parameters given in Table 1. In that
figure, the CP surfaces corresponding to a given radial point (identified by a
colored dot in the figure) are represented by dashed and dotted lines sharing
the same color as the radial point they are associated with. As mentioned
earlier, the bluest line $\lambda$3968 is formed locally and has no associated
CP surfaces. The dashed lines show the CP surfaces caused by interactions of
the two redder lines $C$ and $R$ with their closest neighboring line (i.e.,
the interaction of $\lambda$3969 with $\lambda$3968 photons in the left panel,
and interaction of $\lambda$3970 with $\lambda$3969 photons in the right panel
of Fig. 3). Finally, the dotted lines denote the CP surface caused by
interaction with the farthest spectral line (i.e., by the interaction of
$\lambda$3970 with $\lambda$3968 photons; right panel). In other words, the
left panel of Fig. 3, with only one set of CP surfaces, represents the
situation for the $C$ line discussed here, while the right panel represents
the situation for the $R$ line discussed in the next section. 222Please note
that we keep the same graphical conventions for the representation of CP
surfaces in subsequent figures.

As mentioned by Olson (1982) and also seen in our figures, the topology of CP
surfaces is quite complex and varies with radial distance from the stellar
core. Close to the star, the CP surface is an arc of circle surrounding the
reference point and extending from the abscissa to the angle at which the line
of sight intercepts the core. At larger distances it becomes a closed surface
entirely surrounding the reference point, which then opens farther out as the
envelope limit is reached. We see below that these topological changes are
reflected in the source functions of optically thick lines.

The source function for transition $C$ takes into account the presence of
resonant CP surfaces coupling it to the $B$ transition. It is written as (see
also Olson 1982)

$S_{C}(r)=\frac{(1-\varepsilon_{C})\beta_{C}^{c}I_{C}^{c}+\varepsilon_{C}B_{C}+(1-\varepsilon_{C})S_{nl}^{C}(r)}{\varepsilon_{C}+(1-\varepsilon_{C})\beta_{C}},$
(11)

where the nonlocal contribution $S_{nl}^{C}$ to the source is

$S^{nl}_{C}=\frac{1}{2}\intop_{-1}^{+1}\frac{1-e^{-\tau_{C}}}{\tau_{C}}S(r^{{}^{\prime}}_{B})(1-e^{-\tau(r^{{}^{\prime}}_{B})})d\mu$
(12)

and the probability that a photon strikes the core is modified to take into
account the intervening $B$ layer:

$\beta_{C}^{c}=\frac{1}{2}\intop_{\mu_{c}}^{+1}\frac{1-e^{-\tau_{C}}}{\tau_{C}}e^{-\tau(r^{{}^{\prime}}_{B})}d\mu.$
(13)

####  2.1.3 $R$ line

Similarly, the third and most redward transition $R$ will be affected by
photons of both lines $C$ and $B$, and we must locate the two CP surfaces
$r^{{}^{\prime\prime}}_{B}$ and $r^{{}^{\prime\prime\prime}}_{C}$ where $B$
and $C$ have the same line-of-sight velocity as the gas at the considered $R$
emission point $r$, using Eq. 14:

$\left\\{\begin{array}[]{l}v_{rad}(r_{R})-v_{rad}(r_{B})=\Delta\lambda_{BR}/c\\\
v_{rad}(r_{R})-v_{rad}(r_{C})=\Delta\lambda_{RC}/c\end{array}\right.\ .$ (14)

The generalization of Eqs. 10 to 13 to line $R$ is obvious and leads to the
following expression for $S_{R}(r)$:

$S_{R}(r)=\frac{(1-\varepsilon_{R})\beta_{R}^{c}I_{R}^{c}+\varepsilon_{R}B_{R}+(1-\varepsilon_{R})S_{nl}^{R}(r)}{\varepsilon_{R}+(1-\varepsilon_{R})\beta_{R}},$
(15)

with the nonlocal integral $S_{R}^{nl}$ given by

$S^{nl}_{R}=\frac{1}{2}\intop_{-1}^{+1}\frac{1-e^{-\tau_{R}}}{\tau_{R}}\left[S(r^{{}^{\prime\prime}}_{B})(1-e^{-\tau(r^{{}^{\prime\prime}}_{B})})+S(r^{{}^{\prime\prime\prime}}_{C})(1-e^{-\tau(r^{{}^{\prime\prime\prime}}_{C}}))e^{-\tau(r^{{}^{\prime\prime}}_{B})}\right]d\mu$
(16)

and

$\beta_{R}^{c}=\frac{1}{2}\intop_{\mu_{c}}^{+1}\frac{1-e^{-\tau_{R}}}{\tau_{R}}e^{-\tau(r^{{}^{\prime\prime}}_{B})-\tau(r^{{}^{\prime\prime\prime}}_{C})}d\mu.$
(17)

The geometry of CP surfaces, where the various quantities entering Eqs. 12 to
17 are computed, depends only on the gas velocity law and the transitions
wavelengths. It is computed numerically by integrating over (a) the entire
space when computing the nonlocal parts of the source functions and (b) the
angle subtended by the stellar core when computing the stellar continuum
contribution to the source. The computation of $S_{B}$ is straightforward
algebra in this approximation since all quantities are local, while computing
$S_{C}$ and $S_{R}$ requires an iterative process to solve for the nonlocal
parts $S^{nl}$ consistently.

Figure 4: Source functions for the three lines when they are computed
independently.

Figure 5: Source functions for the three lines when radiative interactions are
taken into account.

The above discussion suggests constructing an interaction matrix $I$ to keep
track of each possible line interaction, a concept that proved useful in
coding the problem at hand. The matrix elements are 0 for a line that can be
considered single or 1 when an interaction is present. For the current case of
three lines in an accelerating outflow, with line ordering $B,C,R$, the
interaction matrix is

$I=\left(\begin{array}[]{ccc}0&0&0\\\ 1&0&0\\\ 1&1&0\end{array}\right),$ (18)

meaning that the $B$ line (first row) has no interaction with itself, $C$, or
$R$, while the $C$ line (second row) interacts with $B$, and the $R$ line
(third row) interacts with both $B$ and $C$.

Figure 5 displays the computed source functions for the three lines under
consideration when they are treated as independent, that is to say, when local
equations similar to Eq. 4 are used for computing all of them. Figure 5 shows
the source functions computed when considering the radiative interactions
(i.e., when using Eq. 11 for the $C$ line and Eq. 15 for the $R$ line). While
the source function of the $B$ line is evidently the same in both cases,
discontinuities caused by the different topologies of CP surfaces are seen;
but since the lines are only moderately optically thick, the induced changes
remain small and do not affect the resulting source functions very much. The
assumption of LTE atomic populations, which results in constant radial optical
depths as the source functions are iterated, also plays a role here. Allowing
for an iterative estimate of populations is thus likely to result in very
different source functions. While the two-level atom assumption is clearly a
limitation of the current code, it is a first step that allows us to
understand the radiative couplings of the lines occurring in both the comoving
reference frame and the observer’s frame, and the next section deals with the
latter.

###  2.2 Exact line profile integration

Figure 6: Regions and frequencies contributing to the line profiles of the
$B$, $C$, and $R$ lines (respectively for each column) in the wind model of
Sect. 2. Shown here is a meridional cut in the half spherical stellar envelope
of radius $R_{max}=10R_{c}$ surrounding a stellar core of radius $R_{c}$.
Displayed are the constant radial velocity surfaces (CRVSs) seen by an outside
observer located at infinity on the right side of the figure for the three
lines considered (left-right diagonal), as well as the displaced CRVSs of
neighboring lines. The scale of the color bars is in km/s, and blue (red)
indicates positive (negative) line frequencies.

We refer to Bertout (1984) for a detailed discussion of the line profile
integration method for single lines and focus here on the specific
modifications that are needed to introduce the possibility of line interaction
in the flux integration. The emergent flux $F_{x,l}$ of line $l$ at frequency
displacement $x=(\nu-\nu_{0,l})/\Delta\nu_{l}$ is

$F_{x,l}=2\pi\int_{0}^{R_{max}}I_{l}(p,x,z=\infty)pdp,$ (19)

where impact parameter $p$ and abscissa $z$ are related to the usual $r$ and
$\theta$ coordinates by the relationships $r^{2}=p^{2}+z^{2}$ and
$z=r\cos\theta$. The quantity $I_{l}(p,x,z=\infty)$ is the emergent intensity
at frequency $x$ on the ray $p$, and $\Delta\nu_{l}$ is the width of the local
absorption profile. The Doppler equation at $(p,z)$ is

$x(p,z)=\frac{v(r)}{\xi_{l}(r)}\frac{z}{r},$ (20)

where $\xi_{l}(r)$ is the width of the line absorption profile in the same
units as $v(r)$. The local absorption profile $\xi_{l}(r)$ results from both
thermal broadening and an additional turbulent widening $v_{turb}$, chosen to
be equal to 1km/s in the actual computations. The only constraint on the
choice of the overall absorption profile width in velocity space is that its
integral over velocity (or frequency) be the same for the three considered
lines (see below). To provide a frequency rest frame common to these lines, we
thus chose the width $\xi$ to be equal to or larger than the widest local line
width expressed in velocity units. In other words,

$\xi\equiv(\alpha+1){\rm max}[\xi_{l}(r)],$ (21)

where $\alpha$ is typically in the range $[0,1]$.

In an accelerated outflow, there is a unique correspondence between line
frequencies and positions on the line of sight. This is shown in the top left
panel of Fig. 6, which displays the envelope’s gas radial velocities seen by
an outside observer located at $z=\infty$. The color code of radial velocities
is given in the bar under the figure. Impact parameter $p=0$ contains the full
range of velocities (or, equivalently, frequency displacements) present in the
three emergent line profiles, ranging from $-V_{max}$ at $z=-R_{max}$ to
$+V_{max}$ at $z=R_{max.}$ We set up a uniform grid over this velocity range
with the understanding that the velocity step $dv$, which defines the
numerical resolution of the line profile computation, is small enough to allow
the narrowest line absorption profile, which has width $\xi_{l}$, to be
resolved with sufficient accuracy. If we consider an impact parameter $p_{j}$,
it will contain velocities in the range

$v_{rad}^{min}=V_{max}\frac{z_{min}}{R_{max}}\leq v_{rad}\leq
V_{max}\frac{z_{max}}{R_{max}}=v_{rad}^{max}$ (22)

with

$z_{min}=\Biggr{\\{}\begin{array}[]{cc}\sqrt{R_{c}^{2}-p_{j}^{2}},&p_{j}\leq
R_{c}\\\
-\sqrt{R_{max}^{2}-p_{j}^{2}},&p_{j}>R_{c}\end{array}{\rm{\rm\;and}\quad}z_{max}=\sqrt{R_{max}^{2}-p_{j}^{2}}.$

####  2.2.1 Line-$C$ flux

In the observer’s frame, there is no preferred flow direction, so all
transitions are treated in the same way. We now consider the formation of line
$C$ in some detail, since it is our primary candidate for fluorescence, with
the understanding that the two other lines are formed in a similar way. For
the following discussion, the reader is invited to refer to the central column
of Fig. 6, which illustrates the origins of line $C$.

When they reach the observer at $z=\infty$, line-$C$ photons with apparent
frequency $\tilde{x}$ traveling in the $p$ direction originate in atomic
transitions taking place in three different regions;

  1. (a)

Transition $C$ photons are emitted from atoms in the $z$-range corresponding
to frequencies around $\tilde{x}$ in the local line emission profile
$\Delta\nu_{D_{C}}$. This is the situation illustrated by the central figure
of Fig. 6, which displays the constant radial velocity surfaces (CRVSs) of
$C$-emitting atoms seen by the observer. As explained above, each of the CRVSs
corresponds to a frequency $\tilde{x}$ in the overall line profile.

  2. (b)

A second contribution to the $C$-line intensity on $p$ stems from transition
$B$ photons that are emitted by atoms moving slower by the velocity difference
between the two lines $c\Delta\nu_{BC}/\nu_{C}\pm\Delta\nu_{D_{C}}$ and are
therefore located deeper in the envelope (as seen by the observer) than atoms
emitting transition $C$ frequency $\tilde{x}$. The top panel of the central
column in Fig. 6 displays the frequency range of this contribution or,
equivalently, the envelope parts from which it originates.

  3. (c)

Finally, there is a third contribution from transition $R$ photons emitted by
atoms that are moving faster by $c\Delta\nu_{CR}/\nu_{C}\pm\Delta\nu_{D_{C}}$
and are therefore located in front of the atoms that emit transition $C$ at
frequency $\tilde{x}$. The bottom panel of the central column in Fig. 6
displays the space-frequency distribution of this contribution.

Similar considerations hold for the other two lines and the resulting space-
frequency contributions to these lines are illustrated by the first and third
columns of Fig. 6.

We next considered the contributions (a) to (c) above to the optical depth and
emergent $C$-line intensity at $\tilde{x}$ in a more quantitative way.

##### Local contribution to the C-line intensity.

The $z$-range corresponding to the local absorption profile frequency range is
$z(\tilde{x}-\xi/2)<\tilde{z}<z(\tilde{x}+\xi/2)$, and the optical depth along
$p$ is given by

$\tau_{C}(p,z,\tilde{x})=\frac{1}{\sqrt{\pi}}\int_{z(\tilde{x}-\xi/2)}^{z}\frac{\kappa_{C}(r^{\prime})}{\Delta\nu_{D_{C}}(r^{\prime})}\varphi(x^{\prime})dz^{\prime},$
(23)

where $\kappa_{C}$ is the line center opacity and $\varphi$ the line
absorption profile (see also Eqs. 7 to 8). Let us denote with $\tau_{C}^{*}$
the maximum value333We use a similar notation for the maximum values of the
other lines’ optical depths. of $\tau_{C}$ on the ray:

$\tau_{C}^{*}=\frac{1}{\sqrt{\pi}}\int_{z(\tilde{x}-\xi/2)}^{z(\tilde{x}+\xi/2)}\frac{\kappa_{C}(r)}{\Delta\nu_{D_{C}}(r)}\varphi(x)dz.$
(24)

Moving in the $-z$ direction along impact parameter $p$, and remembering that
the line frequency displacement decreases with $z$ until the line center is
reached at $z=0$, the line optical depth then varies as

$\tau_{C}(p,z,\tilde{x})=\left\\{\begin{aligned} 0,&\quad
z>z(\tilde{x}+\xi/2)\\\
\frac{1}{\sqrt{\pi}}\int_{z(\tilde{x}+\xi/2)}^{z}\frac{\kappa_{C}(r^{\prime})}{\Delta\nu_{D_{C}}(r^{\prime})}\varphi(x^{\prime})dz^{\prime},&\quad
z(\tilde{x}+\xi/2)\geq z>z(\tilde{x}-\xi/2)\\\ \tau_{C}^{*},&\quad
z<z(\tilde{x}-\xi/2).\\\ \end{aligned}\right.$ (25)

In other words, the optical depth varies from $0$ to $\tau_{C}^{*}$ over a
short distance and then remains constant over the ray. The line intensity at
$\tilde{x}$ is

$I_{C}^{c}(p,z,\tilde{x})=\int_{0}^{\tau(\tilde{x})}S_{C}(p^{\prime},z^{\prime},\tilde{x}^{\prime})\exp[-\tau_{C}(p^{\prime},z^{\prime},\tilde{x}^{\prime})]d\tau(p^{\prime},z^{\prime},\tilde{x}^{\prime}),$
(26)

where the $c$ subscript indicates that this intensity part is caused by
$C$-line photons.

Following Bertout (1984), we rewrite Eq. 23 as

$\tau_{C}(p,z,x_{j})=\sum_{ij}\Delta\tau_{ij}$ (27)

with

$\Delta\tau_{ij}=\frac{1}{\sqrt{\pi}}\biggl{\langle}\frac{\kappa_{l,i}}{\Delta\nu_{D_{C}}}\left(\dfrac{dx}{dz}\right)^{-1}_{i}\biggr{\rangle}\int_{x_{j-1/2}}^{x_{j+1/2}}\varphi(x^{\prime})dx^{\prime}.$
(28)

In Eq. 27, the summation over $i$ runs over the points $z_{i}$ of the $z$ grid
defined on impact parameter $p,$ while the summation over j runs over
frequency displacements at which a contribution is expected at $z_{i}$ (i.e.,
for $x_{j}$ in the range $x_{i}-\xi/2<x_{j}<x_{i}+\xi/2$, where
$x_{i}=x(z_{i})$). The averaged quantity in Eq. 28 is evaluated over the
frequency step $\delta x$, and the integral is equal to
$[{\rm{erf}(x_{j+1/2})-{\rm{erf}(x_{j-1/2})]/2}}$ if the local absorption
profile is Gaussian.

Equation 26 can now be integrated to obtain the emergent intensity of line $C$
if it were a single line without interactions with its neighbors:

$I_{C}^{c}(p,z=\infty,\tilde{x})=I_{C}^{*}(\tilde{x})e^{-\tau_{C}}+S_{C}(1-e^{-\tau_{C}}),$
(29)

where $I_{C}^{*}$ is the stellar continuum intensity at $\lambda_{C}$ and
$\tau_{C}$ the optical depth value at $\tilde{x}$ on $p$, given by Eqs. 27 and
28.

While the similarity with the expression derived in the framework of the
Sobolev theory is evident, we did not factor out the profile function, but
integrated it explicitly over the region where the optical depth goes from
zero to its maximum value, $\tau_{C}^{*}$, on the ray. In other words, we did
not make use of the Sobolev approximation, so that Eq. 29 is equivalent to Eq.
25 in Rybicki & Hummer (1978) rather than to Eq. 26 of the same paper.

We next consider the additional contributions to the intensity on $p$ caused
by the neighboring lines.

Figure 7: Line profiles of the three lines considered and their blend for the
wind model of Sect. 2 when the lines are computed independently.

Figure 8: Line profiles of the three lines considered and their blend for the
wind model of Sect. 2 when radiative interactions between lines are taken into
account.

##### Contribution from B-line photons to line C.

As noted earlier, line $B$ is displaced by $\delta v$ from line $C$ such that
line-$B$ photons traveling in the $z$ direction on a given impact parameter
will be able to interact with atoms emitting transition $C$ that are located
at apparent velocities

$v_{rad}^{min}\leq v_{rad}\leq v_{rad}^{max}-\delta v$ (30)

since these are Doppler-shifted to the rest frequency of line $C$. As
illustrated by the top central panel of Fig. 6, line $C$ will get a
contribution from $B$ photons that are emitted deeper in the envelope and are
moving slower by an amount $\delta v$. This contribution is taken into account
by additional terms in Eqs. 23 and 26. Naming $\delta x$ the frequency
displacement corresponding to $\delta v$, the optical depth of line $C$ along
the line of sight is augmented by the quantity

$\tau_{B}(p,z,\tilde{x})=\frac{1}{\sqrt{\pi}}\int_{z(\tilde{x}-\xi/2-\delta
x)}^{z}\frac{\kappa_{B}(r^{\prime})}{\Delta\nu_{D_{B}}(r^{\prime})}\varphi(x^{\prime})dz^{\prime},$
(31)

while the line intensity becomes

$I_{C}^{bc}(p,z,\tilde{x})=\int_{0}^{\tau(\tilde{x})}\\{S_{C}(p^{\prime},z^{\prime},\tilde{x}^{\prime})\exp[-\tau_{C}(p^{\prime},z^{\prime},\tilde{x}^{\prime})]+S_{B}(p^{\prime},z^{\prime},\tilde{x}^{\prime})\exp[-\tau_{B}(p^{\prime},z^{\prime},\tilde{x}^{\prime})-\tau_{C}(p^{\prime},z^{\prime},\tilde{x}^{\prime})]\\}d\tau(p^{\prime},z^{\prime},\tilde{x}^{\prime})$
(32)

since line $B$ forms deeper than $C$ (as seen by an outside observer). The
subscript $bc$ indicates that the expression for the $C$ intensity now takes
contributions of both $B$ and $C$ into account. The optical depth along impact
parameter $p$ for frequency $\tilde{x}$, from $z=z_{max}(p)$ inward, varies
as444This assumes that $p$ is close to the stellar core and $\tilde{x}$ is
both positive and lies in the vicinity of line center, so that the entire
frequency range of interest is present on the impact parameter.

$\tau_{C}^{bc}(p,z,\tilde{x})=\left\\{\begin{aligned} 0,&\quad
z>z(\tilde{x}+\xi/2)\\\
\frac{1}{\sqrt{\pi}}\int_{z(\tilde{x}+\xi/2)}^{z}\frac{\kappa_{C}(r^{\prime})}{\Delta\nu_{D_{C}}(r^{\prime})}\varphi(x^{\prime})dz^{\prime},&\quad
z(\tilde{x}+\xi/2\geq z>z(\tilde{x}-\xi/2)\\\ \tau_{C}^{*},&\quad
z(\tilde{x}-\xi/2)\geq z>z(\tilde{x}+\xi/2-\delta x)\\\
\tau_{C}^{*}+\frac{1}{\sqrt{\pi}}\int_{z(\tilde{x}+\xi/2-\delta
x)}^{z}\frac{\kappa_{B}(r^{\prime})}{\Delta\nu_{D_{B}}(r^{\prime})}\varphi(x^{\prime})dz^{\prime},&\quad
z(\tilde{x}+\xi/2-\delta x)>z\geq z(\tilde{x}-\xi/2-\delta x)\\\
\tau_{C}^{*}+\tau_{B}^{*},&\quad z<z(\tilde{x}-\xi/2-\delta
x).\end{aligned}\right.$ (33)

We then integrate Eq. 32 to obtain the emergent intensity at $\tilde{x}$:

$I_{C}^{bc}(p,z=\infty,\tilde{x})=I_{C}^{*}(\tilde{x})e^{-\tau_{B}-\tau_{C}}+S_{C}(1-e^{-\tau_{C}})+S_{B}(1-e^{-\tau_{B}})e^{-\tau_{C}},$
(34)

where the notations are as defined in the previous paragraphs.

##### Contribution from R-line photons to line C.

We next consider the contribution of the reddest of these three lines to $C$,
which is displaced from $C$ by a velocity amount $-\delta v^{\dagger}$ (or the
corresponding frequency displacement $-\delta x^{\dagger}$), and interactions
will occur with atoms moving at velocities

$v_{rad}^{min}\leq v_{rad}\leq v_{rad}^{max}+\delta v^{\dagger}.$ (35)

While photons of line $B$ contributing to $C$ emerge from deeper layers than
those of line $C$, we now have the opposite situation: line-$R$ photons
contributing to $C$ originate from layers above $C$ (as seen by an observer at
$z=\infty$). The optical depth along the line of sight is increased by an
amount

$\tau_{R}(p,z,\tilde{x})=\frac{1}{\sqrt{\pi}}\int_{z(\tilde{x}-\xi/2-\delta
x^{\dagger})}^{z}\frac{\kappa_{R}(r^{\prime})}{\Delta\nu_{D_{R}}(r^{\prime})}\varphi(x^{\prime})dz^{\prime},$
(36)

while the line intensity becomes

$\begin{split}I_{C}^{bcr}(p,z,\tilde{x})=&\int_{0}^{\tau(\tilde{x})}\\{S_{B}(p^{\prime},z^{\prime},\tilde{x}^{\prime})\exp[-\tau_{B}(p^{\prime},z^{\prime},\tilde{x}^{\prime})-\tau_{C}(p^{\prime},z^{\prime},\tilde{x}^{\prime})-\tau_{R}(p^{\prime},z^{\prime},\tilde{x}^{\prime})]\\\
&+S_{C}(p^{\prime},z^{\prime},\tilde{x}^{\prime})\exp[-\tau_{C}(p^{\prime},z^{\prime},\tilde{x}^{\prime})-\tau_{R}(p^{\prime},z^{\prime},\tilde{x}^{\prime})]+S_{R}(p^{\prime},z^{\prime},\tilde{x}^{\prime})\exp[-\tau_{R}(p^{\prime},z^{\prime},\tilde{x}^{\prime})]\\}d\tau(p^{\prime},z^{\prime},\tilde{x}^{\prime}),\end{split}$
(37)

where the subscript $bcr$ indicates that contributions of the three
interacting lines to $C$ are now taken in account. The optical depth variation
along the ray is written, with the same caveat as before, as

$\tau_{C}^{bcr}(p,z,\tilde{x})=\left\\{\begin{aligned} 0,&\quad
z>z(\tilde{x}+\xi/2+\delta x^{\dagger})\\\
\frac{1}{\sqrt{\pi}}\int_{z(\tilde{x}+\xi/2+\delta
x^{\dagger})}^{z}\frac{\kappa_{R}(r^{\prime})}{\Delta\nu_{D_{R}}(r^{\prime})}\varphi(x^{\prime})dz^{\prime},&\quad
z(\tilde{x}+\xi/2-\delta x^{\dagger})\geq z>z(\tilde{x}-\xi/2+\delta
x^{\dagger})\\\ \tau_{R}^{*},&\quad z(\tilde{x}-\xi/2+\delta x^{\dagger})\geq
z>z(\tilde{x}+\xi/2)\\\
\tau_{R}^{*}+\frac{1}{\sqrt{\pi}}\int_{z(\tilde{x}+\xi/2)}^{z}\frac{\kappa_{C}(r^{\prime})}{\Delta\nu_{D_{C}}(r^{\prime})}\varphi(x^{\prime})dz^{\prime},&\quad
z(\tilde{x}+\xi/2)>z\geq z(\tilde{x}-\xi/2)\\\
\tau_{R}^{*}+\tau_{C}^{*},&\quad z(\tilde{x}+\xi/2)\geq
z>z(\tilde{x}+\xi/2-\delta x)\\\
\tau_{R}^{*}+\tau_{C}^{*}+\frac{1}{\sqrt{\pi}}\int_{z(\tilde{x}+\xi/2-\delta
x)}^{z}\frac{\kappa_{B}(r^{\prime})}{\Delta\nu_{D_{B}}(r^{\prime})}\varphi(x^{\prime})dz^{\prime},&\quad
z(\tilde{x}+\xi/2-\delta x)>z\geq z(\tilde{x}-\xi/2-\delta x)\\\
\tau_{R}^{*}+\tau_{C}^{*}+\tau_{B}^{*},&\quad z<z(\tilde{x}-\xi/2-\delta
x).\end{aligned}\right.$ (38)

Integrating Eq 37, we obtain the final emergent intensity of line $C$, which
now takes all interactions into account:

$I_{C}^{bcr}(p,z=\infty,\tilde{x})=I_{C}^{*}(\tilde{x})e^{-\tau_{B}-\tau_{C}-\tau_{R}}+S_{C}(1-e^{-\tau_{C}})e^{-\tau_{R}}+S_{B}(1-e^{-\tau_{B}})e^{-\tau_{C}-\tau_{R}}+S_{R}(1-e^{-\tau_{R}}).$
(39)

The line flux then follows from an integration of Eq. 39 over all impact
parameters.

####  2.2.2 Lines $B$ and $R$

As mentioned earlier, a reasoning similar to what has been done above for line
$C$ shows that formation of the $B$ and $R$ lines proceeds in exactly the same
way, and the resulting expressions for the line intensities are very much the
same as Eq. 39 above. The reason for this is the intrinsic symmetry of the
problem, which is made very clear by Fig. 6. We can thus generalize Eq. 39 to

$I_{B,C,R}^{bcr}(p,z=\infty,\tilde{x})=I_{B,C,R}^{*}(\tilde{x})e^{-\tau_{B}-\tau_{C}-\tau_{R}}+S_{B}(1-e^{-\tau_{B}})e^{-\tau_{C}-\tau_{R}}+S_{C}(1-e^{-\tau_{C}})e^{-\tau_{R}}+S_{R}(1-e^{-\tau_{R}}).$
(40)

Even though emergent intensities are basically the same for the three lines,
their contributions to the emergent profiles are in fact quite different. This
is because the frequencies to which each of the resonant regions contributes
vary from line to line, as can also be seen in Fig. 6.

####  2.2.3 Line profiles

In order to show how the interactions affect the line profiles, we display
them in two situations: in Fig. 8 the lines are computed independently, that
is, without taking the interactions discussed above into account555All line
profiles shown in the paper are computed with a numerical resolution of
1km/s.. Unsurprisingly, the individual profiles display the characteristic
features of lines formed in accelerating outflows, with absorption features
reaching almost to the rest wavelength of the line and the sharp drop in
intensity near zero velocity on the blue line side (see, e.g., Lamers et al.,
1987). In Fig. 8, we show the emergent profiles when interactions are
considered. Line $C$, which has the lowest optical depth of the three lines,
is weak in the noninteracting case, but goes into strong emission when
interactions are considered. Lines $R$ and $B$ have comparable optical depths,
and are strongly enhanced by the interactions. The blends shown in these
figures result from co-adding the individual line fluxes.

What can we conclude from this first example of line fluorescence? Clearly,
line $C$ is enhanced by photons of the two other lines in an outwardly
accelerating outflow. So are $B$ and $R$, and the resulting blend is a wide,
triple peaked profile. The blue-displaced absorption characteristic of
outflows is essentially that of the $B$ line, somewhat filled up by emission.
Given that the interacting line source functions are qualitatively quite
similar to what they would be if the lines were independent (compare Figs. 5
and 5), the gist of line amplification occurs during flux integration.
Actually, test computations demonstrated that line profiles obtained with
source functions computed simply with the local Sobolev approximation are
often qualitatively indistinguishable from those obtained with the interacting
source functions used above. Again, this is likely an artifact introduced by
the assumption of LTE atomic populations. But it is quite interesting to note
that fluorescence caused by line amplification in the observer’s frame occurs
despite this limiting condition. This is of course a direct consequence of the
additional emission in the resonance zones caused by Doppler-shifted
neighboring lines in macroscopic velocity fields.

We next considered a more complex case, one with a different topology for the
CRVSs: that of an accretion flow.

##  3 Extension to more complex velocity fields

Figure 9: Velocity and density laws of the accretion flow discussed in Sect.
3.

Figure 10: Line absorption coefficient for the three lines considered.

###  3.1 Accretion flow

Free-falling accretion flows, as well as outwardly decelerating ballistic
outflows or rotational flows, all lead to situations where distant parts of
the emitting region can interact radiatively even when considering the case of
single spectral lines; this is because the constant radial velocity surfaces
discussed above are multi-valued along some rays. As we mentioned earlier, a
generalization of the Sobolev theory to handle this case was worked out by
Rybicki & Hummer (1978) for single lines. We recall their main results in the
next paragraph before extending the same formalism to interacting lines.

####  3.1.1 Sobolev source functions

We consider an accelerated gravitational accretion flow with velocity law

$v(r)=-V_{c}\cdot(R_{c}/r)^{\alpha},$ (41)

where $V_{c}$ is the free-fall velocity at $R_{c}$. The computation parameters
are summarized in Table 2. Since the maximum envelope gas density is 20 times
lower than in the case discussed in the previous section, the line absorption
coefficients of Fig. 10 are correspondingly smaller, but sufficient for
producing sizable line emission in this velocity field, as seen below.

Table 2: Computation parameters for the accretion flow of Sect. 3. Rc | Renv | M∗ | T${}_{\\!e\\!f\\!f}$ | Tenv | $n_{t}(R_{c})$ | $V_{c}$ | $\alpha$
---|---|---|---|---|---|---|---
2 R☉ | 10 Rc | 1 M☉ | 4.5 $\cdot 10^{3}$K |  $6\cdot 10^{3}$ K |  $1\cdot 10^{15}$ cm-3 | 436.76 km/s | 0.5

Since the envelope velocity is varying monotonically, the photons of the
reddest considered line $R$ are seen redshifted by atoms emitting bluer line
photons, so that the source function of line $R$ remains unaffected by the
other transitions considered. We therefore start our discussion with
transition $R$, to which the Rybicki & Hummer (1978) equations directly apply.

#####  $R$ transition.

The generalized Sobolev source function for the case of an equivalent two-
level atom in an inwardly accelerating radial flow is given by

$S_{R}(r)=\frac{(1-\varepsilon_{R})\bar{\beta}_{R}^{c}I_{R}^{c}+\varepsilon_{R}B_{R}+(1-\varepsilon_{R})S_{R}^{nl}}{\varepsilon_{R}+(1-\varepsilon_{R})\beta_{R}},$
(42)

with

$S^{nl}_{R}(r)=\frac{1}{2}\int_{-1}^{+1}\frac{1-e^{-\tau_{R}}}{\tau_{R}}(1-e^{-\tau(r^{*}_{R})})S(r^{*}_{R})d\mu,$
(43)

where $\tau(r^{*}_{R})$ and $S(r^{*}_{R})$ are evaluated at the CP surface
$r^{*}_{R}$ associated with $O(r)$. The third term of Eq. 42 represents the
main nonlocal contribution to the source function that arises in these flows.
A second nonlocal contribution appears in the term
$\bar{\beta}_{R}^{c}I_{R}^{c},$ which represents the contribution of the
stellar continuum flux at line wavelength, attenuated by the intervening CP
layers between core and $O(r)$:

$\bar{\beta}_{R}^{c}I_{R}^{c}=\frac{I_{R}^{c}}{2}\intop_{\mu_{c}}^{+1}\frac{1-e^{-\tau_{R}}}{\tau_{R}}e^{-\tau(r^{*}_{R})}d\mu.$
(44)

The left panel of Fig. 11 shows the CP surfaces associated with line $R$ (in
this case the H_$\epsilon$ line), which can be compared to the CP surfaces
found by Rybicki & Hummer (1978). These are self-CP surfaces in the sense that
transition $R$ interacts radiatively with distant photons at the same rest
wavelength.

Figure 11: Selected CP surfaces for the spherically symmetric accretion flow,
with properties given in the text. Each surface is associated with the radial
point of the same color.

Figure 12: Source functions for the three lines when they are computed
independently.

Figure 13: Source functions for the three lines when radiative interactions
are taken into account.

The analogy of these equations with the case of interacting lines $B$ and $C$
(Eqs. 11 to 13) of the previous section is obvious. In the previous case of
transition $C$, the integration ran over a resonance surface corresponding to
the locus of $B$ photons that were Doppler-shifted by the right amount,
providing an additional source term that represents a net gain of photons for
transition $C$. For a single line formed in an accretion flow, on the other
hand, the local source term for $R$ is supplemented by nonlocal photons of the
same transition $R$, originating from the CP surface associated with the
considered emission point in the envelope.

#####  $C$ and $B$ transitions.

The expressions for $S_{C}(r)$ and $S_{B}(r)$ are similar to Eq. 42 but with
the following changes in the nonlocal parts of the source function, reflecting
the fact that the self-CP surface $r^{*}$ of the line is not the only region
that must be taken into account in the nonlocal integral. Indeed, atoms
emitting line $C$ will be able to absorb some blue-displaced line-$R$ photons,
while atoms emitting transition $B$ will interact with blue-displaced photons
from both $C$ and $R$. Figure 11, in its center and right panels, show the set
of CP surfaces that are contributing to the nonlocal parts of the respective
source functions. The graphical convention is the following: solid lines
indicate line self-interactions, dashed lines show the CP surfaces
corresponding to the nearest transition, and dotted lines indicate the CP
surfaces corresponding to the third, farthest transition.

As previously, the condition of interaction for the $C$ line is given by

$v_{rad}(r_{R})-v_{rad}(r_{C})=\Delta\lambda_{RC}/c,$ (45)

which defines a CP surface denoted by $r^{{}^{\prime}}_{R}$ in the following
expressions. With $\tau(r^{{}^{\prime}}_{R})=\tau^{{}^{\prime}}_{R}$ and
$\tau(r^{*}_{C})=\tau^{*}_{C}$, we have

$S_{C}(r)=\frac{(1-\varepsilon_{C})\bar{\beta}_{C}^{c}I_{C}^{c}+\varepsilon_{C}B_{C}+(1-\varepsilon_{C})S_{C}^{nl}}{\varepsilon_{C}+(1-\varepsilon_{C})\beta_{C}},$
(46)

with

$S^{nl}_{C}(r)=\frac{1}{2}\int_{-1}^{+1}\frac{1-e^{-\tau_{C}}}{\tau_{C}}\left[(1-e^{-\tau^{*}_{C}})S(r^{*}_{C})+(1-e^{-\tau^{{}^{\prime}}_{R}})S(r^{{}^{\prime}}_{R})e^{-\tau^{*}_{C}}\right]d\mu$
(47)

and

$\bar{\beta}_{C}^{c}=\frac{1}{2}\intop_{\mu_{c}}^{+1}\frac{1-e^{-\tau_{C}}}{\tau_{C}}e^{-(\tau^{*}_{C}{+\tau^{{}^{\prime}}_{R}})}d\mu.$
(48)

For line $B$, two resonance conditions on the radial velocity lead to line
interactions, namely

$\left\\{\begin{array}[]{l}v_{rad}(r_{B})-v_{rad}(r_{C})=\Delta\lambda_{BC}/c\\\
v_{rad}(r_{B})-v_{rad}(r_{R})=\Delta\lambda_{BR}/c\end{array}\right.\ .$ (49)

Figure 14: Regions contributing to the formation of the considered line
profiles in the accretion flow of Sect. 4. See the caption of Fig. 6 for
details.

Figure 15: Line profiles of the three lines considered and their blend for the
accretion flow discussed here when the lines are computed independently.

Figure 16: Line profiles of the three lines considered and their blend for the
accretion flow discussed here when radiative interactions between lines are
taken into account.

We must therefore locate the resonance regions corresponding to these
conditions, the loci of which are noted $r^{{}^{\prime\prime}}_{C}$ and
$r^{{}^{\prime\prime\prime}}_{R}$ in the following equations. In a direct
extension of Eq. 15, we then have

$S_{B}(r)=\frac{(1-\varepsilon_{B})\bar{\beta}_{B}^{c}I_{B}^{c}+\varepsilon_{B}B_{B}+(1-\varepsilon_{B})S_{B}^{nl}}{\varepsilon_{B}+(1-\varepsilon_{B})\beta_{B}},$
(50)

with

$S^{nl}_{B}(r)=\frac{1}{2}\int_{-1}^{+1}\frac{1-e^{-\tau_{B}}}{\tau_{B}}\left[(1-e^{-\tau^{*}_{B}})S(r^{*}_{B})+(1-e^{-\tau^{{}^{\prime\prime}}_{C}})S(r^{{}^{\prime\prime}}_{C})e^{-\tau^{*}_{B}}+(1-e^{-\tau^{{}^{\prime\prime\prime}}_{R}})S(r^{{}^{\prime\prime\prime}}_{R})e^{-(\tau^{*}_{B}+\tau^{{}^{\prime\prime}}_{C})}\right]d\mu$
(51)

and

$\bar{\beta}_{B}^{c}=\frac{1}{2}\intop_{\mu_{c}}^{+1}\frac{1-e^{-\tau_{B}}}{\tau_{B}}e^{-(\tau^{*}_{B}{+\tau^{{}^{\prime\prime}}_{C}}+{\tau^{{}^{\prime\prime\prime}}_{R})}}d\mu.$
(52)

The interaction matrix for this case must include the self-interaction of each
line, so that it becomes, with line ordering $R,C,B$,

$I=\left(\begin{array}[]{ccc}1&0&0\\\ 1&1&0\\\ 1&1&1\end{array}\right).$ (53)

Figure 13 displays the computed source functions for the three lines in
consideration when they are treated as independent, that is, when Eq. 4 is
used for computing all of them. Figure 13 shows the source functions computed
when taking the radiative interactions into consideration, that is, when using
Eq. 46 for the $C$ line and Eq. 50 for the $B$ line. The changes caused by
core photon absorption at the CP surfaces are small for the optically thin Fe
I line but lead to a strong depression of the optically thick Ca II line
source function below $R\sim 4$. We recall that the H_ϵ line source function
is the same in both cases.

####  3.1.2 Flux integration

In an accelerating infall or decelerating outflow, the unique correspondence
between line frequencies and positions on the line of sight found in
accelerating outflows is lost, as can be seen in the top left panel of Fig.
14, which displays the envelope’s gas radial velocities seen by an outside
observer located at $z=\infty$. Depending on the impact parameter and envelope
extent, the CRVSs are either open or closed, so that there are one or two
locations on the line of sight that have the same frequency displacement. As
explained in Bertout (1984), the flux integration method that we use handles
this situation without modification. It finds all line frequency displacements
present on a given impact parameter and orders them for the line intensity
computation.

The line flux of $R$ is made up of several contributions shown in the first
column of Fig. 14. They are first the line-$R$ photons originating in the
CRVSs shown in the top panel, then the displaced $C$-line photons originating
in deeper parts of the envelope (middle panel), and also the displaced
$B$-line photons originating even deeper in the envelope (bottom panel). The
final contribution to the red side of line $R$ comes from $C$ and $B$ photons
emitted by fast-moving atoms located in front of the star. As illustrated by
the second column of Fig. 14, line $C$ gets a contribution from $R$ photons
that are emitted in the outside part of the envelope (as seen by the observer)
where atoms are moving slower by an amount $\delta v$, so the flux is enhanced
by emission from the front part of the envelope. Line $C$ gets an additional
contribution from $B$ photons that are coming from the deeper parts of the
envelope and displaced by the right amount. We again note the additional
contribution to the $C$-line red wing from $B$ photons emitted by fast-moving
atoms located in front of the star. Similar considerations hold for line $B$,
and the various origins of its flux are shown in the right column of Fig. 14.

Figures 16 and 16 display the computed line profiles for two situations. In
Fig. 16 the lines are computed as if they were single lines, without taking
all interactions discussed above into account, while in Fig. 16 we show the
profiles that result when interactions are considered. Profiles of Fig. 16 are
typical accretion flow profiles with intermediate optical depths. Line $C$ has
the lowest optical depth of the three lines; it is indistinguishable from the
continuum when the lines are computed independently but goes in moderate
emission when interactions are taken into account. Line $R$ has a somewhat
lower optical depth than $B$ but is enhanced by contributions of photons from
the other transitions. The flux of these last two lines are similar, so the
blend is in emission, with a single peak approximately centered on the
strongest line $R$; it displays a moderately deep but wide absorption
component that results from a combination of the three individual absorption
features.

###  3.2 Non-monotonic velocity fields

The method developed here readily extends to non-monotonic velocities, as
found, for example, in an accreting envelope with an accretion shock close to
the stellar surface. To illustrate this, we considered a qualitative TTS
accretion shock toy model, the velocity and density fields of which are
displayed in Fig. 18. The gas free-fall velocity increases toward the star
with the usual $v(r)\propto r^{-1/2}$ radial dependence until it reaches the
shock at $R_{s}$, assumed to be located at 1.2 $R_{c}$. The shock is assumed
to be isothermal, so the gas velocity decreases by a factor of 4 at $R_{s}$.
This jump is followed by an (assumed) linear deceleration and reaches the
stellar radius $R_{c}$ at 10 km/s. The density follows from the continuity
equation, and the gas density at the shock is $n_{t}(R_{s})=5\cdot 10^{15}$
cm-3. We did not consider the complex physics involved in the shock, but
simply solved the formation of our three lines with these values of velocity,
temperature, and density fields.

Figure 17: Velocity and density laws of the non-monotonic accretion flow
discussed in this section.

Figure 18: Line absorption coefficient and rates of collisional de-excitation
for the three lines considered.

Figure 19: Selected CP surfaces for the spherically symmetric non-monotonic
accretion flow, with properties given in the text. Each surface is associated
with the radial point of the same color.

For more realistic source functions, we introduced collisional de-excitation
rates in an approximate way by using formulas from Allen’s Astrophysical
Quantities (Cox, 2000), which actually go back to van Regemorter (1962).
Clearly, more modern data should be used for a detailed comparison with
observations, but at this preliminary stage a crude estimate of the collision
effects is sufficient. The collisional de-excitation rates (dashed lines) and
the absorption coefficients (solid lines) of the three lines are shown in Fig.
18. The collisional de-excitation rate of the calcium ion line is an order of
magnitude higher than the rates of neutral hydrogen and iron lines, which are
indistinguishable in this graph. The three lines are moderately optically
thick in the post-shock region, where much of the fluorescent amplification
occurs.

The CP surfaces for the three lines considered are shown in Fig. 19. As in
previous panels showing CPs, the solid lines display the self-CP surfaces of
each line, while the dashed lines show the interaction surfaces with the first
neighboring line and the dotted lines display the interaction surfaces with
the second neighboring line. We see that, unlike the monotonic cases, all
lines are, as expected, involved in the interactions, and the CP surfaces are
a mix of those found in the previous cases of monotonic flows. They introduce
discontinuities in the source functions, particularly in the shock region, as
can be seen in Fig. 21.

Figure 20: Source functions of the three lines considered and their blend for
the non-monotonic accretion flow discussed here.

Figure 21: Line profiles of the three lines considered and their blend for the
non-monotonic accretion flow discussed here.

The resulting line profiles are displayed in Fig. 21. They do not resemble
those found for the accretion flow of the previous section, partly because
most of the flux originates in the post-shock deceleration region close to the
star, and partly because of the collision-term effect. Despite the fact that
the optical depths of the H and CaII lines are very similar, the emergent
profiles are quite different, with CaII H showing a steep central peak caused
by the collision term that enhances its source function in the innermost
envelope region (see Fig. 21). We note that the three lines participate in the
fluorescence process, with the iron line getting contributions from both
neighboring lines and thus being amplified most.

##  4 Discussion and conclusion

Figure 22: Parameter space study for the net EWs of FeI$\lambda 3969$ line and
its blend with H and Hϵ, in the case of an accretion flow with $\epsilon\neq
0$.

We have investigated a rather small parameter space in the previous sections
of this paper, mainly because we required that CaII H and K have comparable
strengths, as is observed in many TTSs. As discussed in Sect. 2, this requires
– when LTE and isothermal envelope are assumed – that gas temperatures be
restricted to a limited range around 6000K, which in turn implies, when
ignoring collisional excitation, high gas densities at the envelope’s bottom
and resulting mass flow rates that appear unrealistically high for TTSs.

Table 3: Computation parameters for the parameter space study discussed here Rc | Renv | M∗ | T${}_{\\!e\\!f\\!f}$ | Tenv | $n_{t}(R_{c})$ | $V_{c}$ | $\alpha$
---|---|---|---|---|---|---|---
2 R☉ | 10 Rc | 0.5 M☉ | 3.0 $\cdot 10^{3}$K |  $10^{12}-10^{14}$cm-3 K |  $6\cdot 10^{3}-9\cdot 10^{3}$K | 436.76 km/s | 0.5

Figure 23: Source functions of the three lines considered for an accretion
flow with $n_{t}(R_{c})=6\cdot 10^{12}$cm-3 and T${}_{env}=8.5\cdot 10^{3}$K
when collisional de-excitation rates are included.

Figure 24: Line profiles of the three lines considered for an accretion flow
with the properties given in the previous figure caption.

However, fluorescence is not limited to this narrow range of parameters, but
can also take place at higher temperatures and at lower densities when we
include the collisional de-excitation terms in the source functions. This is
because the collision term enhances the source function, leading to sizable
flux for low line optical depths. Figure 22 shows the result of a parameter
space exploration that includes collisional de-excitation rates in the source
function computation. There, we display the net equivalent widths (EWs) of
both FeI$\lambda$3969 and the blend of the three lines, as computed in a more
extended parameter space investigation. We vary $n_{t}(R_{c})$ and Tenv in the
ranges indicated in Table 3, which also gives the other computation
parameters. The collisional terms are included in the approximation discussed
earlier. The net EW666By convention, emission line components have positive
EWs here.  of a line is defined as its computed EW when the computation
includes interactions with the other two lines, from which we subtracted a
baseline EW computed under the assumption that the three lines are
independent. The net EW is thus a measure of the fluorescent flux.

In Fig. 22 we plot the logarithm of the net EWs, which ranges from -2 to +1.5
in the parameter space investigated. We find that a significant
FeI$\lambda$3969 fluorescent flux (EW${}_{net}\gtrapprox 1\AA$) is produced
over a wide range of temperatures for $n_{t}(R_{c})\gtrapprox 4\cdot
10^{12}$cm-3. The interacting lines at these relatively low densities are all
optically thin, with pure, almost featureless emission profiles driven by the
collision terms; the sources and profiles of a representative example are
shown in Figs. 24 and 24. The minimum density above corresponds to a spherical
mass flow rate of $\approx 10^{-6}$M☉/yr. This mass accretion rate value
remains quite high for TTSs in general, but it may represent a plausible value
for the most active members of the class (Hartigan et al., 1995). One should
also recall that TTSs are highly variable, which indicates that the accretion
process is not steady (cf. Bouvier et al., 2007). Fluorescence might therefore
occur in periods of increased accretion but disappear during more quiescent
phases, as observed recently in VW Cha (A. Armeni, priv. comm.).

As an absolute calibration of TTS mass accretion rates is notoriously
difficult (see the discussion in Gullbring et al., 1998), it seems worthy to
improve the atomic model and try computing the precise threshold mass
accretion rate for fluorescence to set in. This will provide a useful new
comparison tool for other mass flow rate determination methods.

With this work, we are now in a position to compute the formation of
fluorescent lines in spherically moving media. However, as already mentioned,
we are still far from being able to meaningfully compare theoretical line
profiles to observed TTS ones. The reasons are threefold: (a) velocity fields
in TTS envelopes are not spherically symmetric and so require a 3D analysis;
(b) the atomic line level populations, which are dominated by non-LTE effects,
must be computed iteratively with the radiation field; and (c) stellar
envelopes are not isothermal. Point (a) can be addressed relatively easily by
extending the present formalism to axisymmetric flow geometries. Point (b)
requires setting up an iterative procedure that couples levels equations and
the radiation field; several convergence acceleration methods have been
proposed (cf. Auer, 2009) and can be used to solve this problem. Point (c) is
easily remedied; actually, a version of our code already makes provisions for
varying envelope temperatures. Another aspect requiring improvement for
further progress is the approximation used for the collisional de-excitation
rates, which should be replaced by more recent data. This will be a direct
consequence of setting up realistic atom models for addressing point (b).

###### Acknowledgements.

It is a pleasure to thank Sylvie Cabrit for useful discussions and Joli Adams
for improving the English. I am grateful to an anonymous referee for useful
comments and suggestions that improved the paper’s presentation.

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##  Appendix A Numerical implementation

SLIM2 (Spectral Line Interactions in Moving Media) is a publicly
available777under Creative Commons 4.0 BY-NC-SA license. The code is available
at https://github.com/claude-bertout/SLIM2 Python/NumPy code for solving the
line formation problem in 2D moving media, for multiple lines stemming from
multiple elements, in the special case of interacting lines leading to
fluorescence effects. In the framework of the well-tested hybrid approach to
the problem of line formation described above, it first computes the
generalized Sobolev source functions and then integrates the line flux exactly
.

The code makes extensive use of the NumPy ndarray((dim1, dim2), object) 3D
construct, which is particularly useful for ray-tracing procedures, although
it is frowned upon by some pythonistas because if complicates array
broadcasting.

Version 1.0 presented here is stable and optimized for solving the problem of
forming the three-line blend around 3969Å in TTS spectra. The atomic level
populations are LTE and remain constant during the iterative procedure used to
compute the source functions. Plotting routines are restricted to this
specific three-line case. They are included in the project so that interested
users will be able to reproduce easily this publication’s figures, thus
gaining experience with the code before starting on their own developments.

A more general version of the code, which is able to compute many interacting
and noninteracting lines simultaneously and allows for variable temperature in
the envelope, will be made available at a later date.

The following modules are included in the GitHub TTS Fluorescence package:

  * •

Fluorescence _ Main.py contains a single function main _ module that calls all
other modules in the following order to perform the computation and to store
computational parameters, source functions, and line profiles in a
subdirectory of results directory marked with a date-time stamp of the start
of computation.

  * •

import _ parameters.py imports all input parameters from computation _
parameters.conf. For this, an ancillary module parameter _ parser.py is
needed. Several different options are possible for the computation (local or
nonlocal Sobolev source, noninteracting or interacting lines, Sobolev or exact
flux mode, single model or grid of models, various velocity fields, several
levels of logging, etc.) so that functional parameters are numerous. There are
also a few physical parameters, such as the maximum density, envelope
temperature, and stellar properties, that must be defined. The meaning of each
of these parameters is explained in the .conf file and default values are
indicated.

  * •

setup _ grids.py sets up the spatial, velocity, and density grids and analyzes
the velocity field structure.

  * •

import _ line _ data.py reads the line data from import _ atomic _ data.py and
computes and stores as arrays the line optical depths using LTE _ module.py.
It also computes the absorption line profiles, defines the line interaction
matrices, and sets up the overall frequency grid for the flux integration.
Databases for partition functions and abundances are stored in directory input
_ data.

  * •

source _ module.py first finds and orders all the CP surfaces discussed above
on high-resolution angular grids. This step is a major CPU-time consumer, and
we store the CP-surface geometries for repeated use when a string of models
with the same geometrical properties are calculated. All integrals over solid
angles that appear in the computation of the source function are solved by
Gaussian quadratures. The local source functions are computed and iterated
with the nonlocal contributions until convergence occurs. Although there are
often more than two resonant surfaces on each line of sight, convergence
occurs relatively rapidly for the velocity fields considered above and for the
many others that we tested.

  * •

flux _ integration.py integrates the line intensities over the impact
parameter grid. A spatial grid is defined on each impact parameter and is
transformed by interpolation into a grid of line frequencies present on the
given ray. Contributing frequencies to each line are then found by taking
their velocity displacements into account, and resonant regions are deduced
from there. Incremental line intensities are then computed, and the flux
follows from integrating the emergent intensities.

  * •

function _ library.py contains ancillary functions for source _ module.py and
flux _ integration.py.

  * •

in _ out _ library.py contains functions for storing the results in a
subdirectory of results and reading them for future use.

  * •

Finally, plot _ library.py reads stored results and plots various quantities,
storing the plots for future reference in the same subdirectory of results
(see above).

All these modules are heavily commented to make it easier for the user to
follow the procedures used for solving the problem.

The computing time for three interacting lines with high-resolution grids (100
radial points, 2000 angles, 100 impact parameters, and 2000 frequencies across
the line profiles) is typically on the order of 30 seconds on an Apple MacBook
Pro with M1 Pro silicon running PyCharm.