Issue 
A&A
Volume 690, October 2024



Article Number  A213  
Number of page(s)  11  
Section  Stellar atmospheres  
DOI  https://doi.org/10.1051/00046361/202449722  
Published online  09 October 2024 
Full nonLTE spectral line formation
III. The case of a twolevel atom with broadened upper level
^{1}
Indian Institute of Astrophysics,
Koramangala,
Bengaluru
560034,
India
^{2}
Université de Toulouse, UPSObservatoire MidiPyrénées, Cnrs, Cnes, Irap,
Toulouse,
France
^{3}
Cnrs, Institut de Recherche en Astrophysique et Planétologie,
14 av. E. Belin,
31400
Toulouse,
France
^{4}
LESIA, Observatoire de Paris, Université PSL, Sorbonne Université, Université Paris Cité, CNRS,
Meudon,
France
^{★} Corresponding author; sampoorna@iiap.res.in
Received:
24
February
2024
Accepted:
10
August
2024
In the present paper we consider the full nonlocal thermodynamic equilibrium (nonLTE) radiation transfer problem. This formalism allows us to account for deviation from equilibrium distribution of both the radiation field and the massive particles. In the present study, twolevel atoms with broadened upper level represent the massive particles. In the absence of velocitychanging collisions, we demonstrate the analytic equivalence of the full nonLTE source function with the corresponding standard nonLTE partial frequency redistribution (PFR) model. We present an iterative method based on operator splitting techniques that can be used to numerically solve the problem at hand. We benchmark it against the standard nonLTE transfer problem for a twolevel atom with PFR. We illustrate the deviation of the velocity distribution function of excited atoms from the equilibrium distribution. We also discuss the dependence of the emission profile and the velocity distribution function on elastic collisions and velocitychanging collisions.
Key words: line: formation / line: profiles / radiation mechanisms: general / radiative transfer / methods: numerical / stars: atmospheres
© The Authors 2024
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1 Introduction
In a series of two papers (Paletou & Peymirat 2021; Paletou et al. 2023), we revisited the problem of socalled “full” nonlocal thermodynamic equilibrium (nonLTE) radiation transfer originally formulated by Oxenius (1986). This formalism accounts not only for deviation of the radiation field from the Planckian equilibrium distribution, but also for the deviation of the velocity distribution of massive particles from the Maxwellian equilibrium distribution. While Paletou & Peymirat (2021) focused on reformulating the basic elements of full nonLTE formalism using standard notations, Paletou et al. (2023) considered its numerical solution for the case of coherent scattering (CS) in the frame of the atom, which corresponds to scattering on a two level atom with (putative) infinitely sharp energy levels. In the present paper, we further extend these works to scattering on a twolevel atom with an infinitely sharp lower level and a more realistic broadened upper level (which may already be suitable for the modeling of strong resonance lines).
The full nonLTE radiation transfer formalism is based on the kinetic theory of particles and photons (Oxenius 1986). In particular, it is founded on a semiclassical description of light scattering in spectral lines. As described in Hubeny & Mihalas (2014), the semiclassical picture combines concepts from classical theory and the more exact quantum mechanical description of the problem at hand, thereby providing a very intuitive and compelling approach to the problem. Clearly, the semiclassical picture is not a selfconsistent theory and therefore contains a number of not so well defined concepts. However, it has been very successful in describing several of the linescattering mechanisms in astrophysical conditions (see the abovecited books for details). For the problem considered in this paper, the poorly defined concepts from the physical point of view are those related to the rate of velocitychanging collisions and a clear distinction between these and elastic collisions. However, despite this, we introduce separate rates for the elastic and velocitychanging collisions. Although it is not completely clear how they would be evaluated for actual cases, and indeed a proper quantummechanical definition of these quantities is uncertain, their introduction and usage in the present paper is fully in line with the semiclassical picture that we adopt here. Indeed this semiclassical theory provides a way to treat the problem, namely including a selfconsistent determination of the velocity distribution of atoms in the upper level of the transition. For a more detailed outline of the semiclassical picture, we refer the reader to Hubeny & Mihalas (2014, see their Chapter 10, specifically pp. 291–294).
In the full nonLTE formalism, the kinetic equation for the velocity distribution of the massive particles (namely the atoms or ions and free electrons) and that for the photons (namely the radiative transfer equation for the intensity of the radiation field) have to be formulated and solved simultaneously and selfconsistently. As the velocity distribution functions (VDFs) of the atomic levels are not known a priori, the absorption and emission profiles that enter the radiative transfer equation need to be obtained by convolving the corresponding atomic quantities with the VDFs, wherein the velocity of the massive particle is measured in the observer’s frame. An evaluation of the need to use this formalism has so far remained unexplored because of the numerical complexity involved in its implementation. The aim of the present series of papers is to clarify this question through detailed numerical calculations. For this purpose, we have embarked upon developing suitable numerical techniques to implement this formalism. As a first step, Paletou et al. (2023) considered the twodistribution problem, namely the intensity of the radiation field and the VDF of the excited atoms are the only two distributions that need to be determined simultaneously and selfconsistently. In other words, twolevel atoms represent the massive particles, with their lower (ground) level exhibiting the equilibrium Maxwellian distribution (Oxenius 1986; Paletou & Peymirat 2021). Furthermore, stimulated emission was neglected, and the free electrons that are responsible for inelastic collisions between the two levels of the atom were also assumed to obey the equilibrium Maxwellian distribution. In the present paper, we continue to consider this twodistribution problem, with an important difference being that the lower level of the atom continues to be infinitely sharp, while the upper level is broadened. In this case, the atomic absorption profile is a Lorentzian and the atomic emission profile already depends on the radiation field. This introduces some difficulties in the numerical solution of the corresponding full nonLTE problem, namely we need to accurately compute Voigtlike function, which involves a Lorentzian function in its integrand (Paletou et al. 2020). Following a method developed previously by Bommier (1997a,b), in the present paper we present a simple and efficient technique to compute such an integral involving Lorentzian function.
The basic equations of the twodistribution problem are detailed in Paletou et al. (2023, see also Paletou & Peymirat 2021). Hence, we do not repeat them here, and only the equations relevant to the twolevel atom with broadened upper level are discussed. The outline of the present paper is as follows. In Section 2, we discuss the explicit forms of the absorption and emission profiles for a twolevel atom with a broadened upper level. The full nonLTE source function is presented in Section 3, wherein we also demonstrate the analytic equivalence with the corresponding standard nonLTE source function in the absence of velocitychanging collisions. In Section 4, we describe and clarify the three different types of collisions (namely the inelastic, elastic, and velocitychanging collisions) considered in this paper. In Section 5, we present the numerical method to solve the full nonLTE problem considered here. Our numerical results are illustrated and discussed in Section 6. Conclusions are presented in Section 7.
2 The absorption and emission profiles
As in the standard nonLTE formalism (see e.g., Hubeny & Mihalas 2014, who also adopt the semiclassical picture), the absorption and emission profiles (and also the frequency redistribution functions) are first determined in the atomic rest frame, and then transformed to the observer’s frame to account for the Doppler motion of the atoms in a stellar atmosphere. In the standard nonLTE formalism with complete frequency redistribution (CFR), the VDF of all the atomic levels is assumed to be the equilibrium Maxwellian distribution, while when partial frequency redistribution (PFR) is included, this assumption is limited to only the lower level. However, the standard nonLTE formalism with PFR does not provide access to the VDF of the upper level. This is provided by the full nonLTE formalism. Therefore, in this section we discuss the absorption and emission profiles first in the atomic frame and then in the observer’s frame.
For the case of a twolevel atom with a broadened upper level, the atomic absorption profile is given by (see Appendix B.2 of Oxenius 1986, see also Paletou & Peymirat 2021) $${\alpha}_{12}(\xi )=\frac{{\delta}_{w}}{\pi}\frac{1}{{\left(\xi {\nu}_{0}\right)}^{2}+{\delta}_{w}^{2}},$$(1)
where ξ is the photon frequency in the atomic frame, ν_{0} is the linecenter frequency, and the damping width δ_{w} = (A_{21} + Q_{1} + Q_{E})/(4π), with A_{21} being the Einstein coefficient for spontaneous emission or radiative deexcitation rate, Q_{I} the inelastic collisional deexcitation rate (denoted as C_{21} in Paletou & Peymirat 2021), and Q_{E} the total elastic collision rate.
The absorption profile in the observer’s frame is given by $${\phi}_{\nu}={\displaystyle {\int}_{u}{\alpha}_{12}}\left(\nu \mathrm{\Delta}{\nu}_{D}u\cdot \Omega \right){f}_{1}(u){d}^{3}u,$$(2)
where u is the atomic velocity^{1} normalized to the thermal velocity (${\nu}_{\text{th}}=\sqrt{2kT/M}$, with k being the Boltzmann constant, T the temperature, and M the mass of the atom), Ω is the propagation direction of the ray, and Δν_{D} is the Doppler width. In this paper, we do not account for bulk velocities resulting from mass motion of the massive particles. In other words only the Doppler motion of atoms is taken into account, consequently the corresponding velocities are in the nonrelativistic regime, wherein only the photon frequency is subject to Lorentz transformation between the atomic rest frame and the observer’s frame, while the photon direction remains unchanged (i.e., aberration is neglected; see also Eqs. (2.4.3a)–(2.4.3e) in page 54 of Oxenius 1986). Therefore, in the above equation, we used the Fizeau–Doppler relationship (see Eq. (9) of Paletou & Peymirat 2021), which relates the frequencies in the atomic frame (ξ) and the observer’s frame (ν). Furthermore, f_{1} represents the VDF of the lower level of the atom. In the weak radiation field regime, f_{1} can be assumed to be the equilibrium distribution, namely a Maxwellian f^{M} (Oxenius 1986; Paletou & Peymirat 2021). With this assumption, it is straightforward to show that the resulting absorption profile in the observer’s frame is a normalized Voigt function φ(x) = H(a, x), where a = δ_{w}/Δν_{D} and x = (ν − ν_{0})/Δν_{D}, with ν_{0} being the linecenter frequency. In the following subsections, we discuss the emission profile first in the atomic frame (see Section 2.1) and then in the observer’s frame (see Section 2.2).
2.1 The atomic emission profile
The explicit form of the atomic emission profile in the absence of velocitychanging collisions is given in Eq. (B.2.26) of Oxenius (1986) and in Eq. (4.32) of Hubeny et al. (1983a). Although the notations used in these publications are somewhat different, it can be readily shown that both the mentioned expressions are identical. In the presence of velocitychanging collisions, the atomic emission profile is given in Hubeny et al. (1983b, see their Section 4.1). In their notation, this atomic emission profile is given by $${\eta}_{21}(\xi ,\tau )=\frac{{B}_{12}{I}_{12}^{*}{j}_{121}(\xi ,\tau )+\left[{S}_{12}+{\gamma}_{2}\left({n}_{2}/{n}_{1}\right)\right]{r}_{12}(\xi )}{{B}_{12}{I}_{12}^{*}+{S}_{12}+{\gamma}_{2}\left({n}_{2}/{n}_{1}\right)},$$(3)
where B_{12} is the Einstein coefficient for radiative absorption, S_{12} is the collisional excitation rate, γ_{2} is the velocitychanging collision rate, n_{1} and n_{2} are the number density of the atoms in lower and upper levels, respectively, and τ is the line center optical depth. In Eq. (3), the generalized redistribution function r_{12}(ξ) = α_{12}(ξ) (see Eq. (6.3.55) of Oxenius 1986), the quantity ${I}_{12}^{*}$ is given by $${I}_{12}^{*}={\displaystyle \int {r}_{12}}(\xi )I(\nu ,\Omega ,\tau )d\xi ,$$(4)
and $${j}_{121}(\xi ,\tau )=\frac{1}{{I}_{12}^{*}}{\displaystyle \int {r}_{121}}\left({\xi}^{\prime},\xi \right)I\left({\nu}^{\prime},{\Omega}^{\prime},\tau \right)\text{d}{\xi}^{\prime},$$(5)
with $\int \text{d}}\xi ={\displaystyle \oint {\displaystyle \int \text{d}}}\nu \text{\hspace{0.17em}d}\Omega /(4\pi )$. In the above equations, I(ν, Ω, τ) is the specific intensity, and r_{121}(ξ′,ξ) is the generalized atomic redistribution function (Hubeny et al. 1983a), which describes the joint probability of absorbing a photon of frequency ξ′ and spontaneously reemitting a photon of frequency ξ.
We note that the quantity ${I}_{12}^{*}$ introduced by Hubeny et al. (1983a,b) is the same as I_{12} introduced in Eq. (B.2.20) of Oxenius (1986). In the notations of Paletou & Peymirat (2021, see also Paletou et al. 2023), we readily identify ${I}_{12}^{*}={J}_{12}(\mathit{u},\tau ),{S}_{12}={C}_{12}$, and γ_{2} = Q_{V}^{2}. Therefore, we rewrite Eq. (3) in the present notations as follows: $${\eta}_{21}(\xi ,\tau )=\frac{{B}_{12}{J}_{12}(u,\tau ){j}_{121}(\xi ,\tau )+\left[{C}_{12}+{Q}_{V}\left({n}_{2}/{n}_{1}\right)\right]{\alpha}_{12}(\xi )}{{B}_{12}{J}_{12}(u,\tau )+{C}_{12}+{Q}_{V}\left({n}_{2}/{n}_{1}\right)},$$(6)
where J_{12}(u, τ) is defined as (see e.g., Eq. (8) of Paletou & Peymirat 2021) $${J}_{12}(u,\tau )={\displaystyle \oint \frac{\text{d}\mathrm{\Omega}}{4\pi}}{\displaystyle {\int}_{0}^{\infty}{\alpha}_{12}}\left(\nu \mathrm{\Delta}{\nu}_{D}u\cdot \Omega \right)I(\nu ,\Omega ,\tau )\text{d}\nu .$$(7)
Also, in our notations, the quantity j_{121}(ξ, τ) takes the form $${j}_{121}(\xi ,\tau )=\frac{1}{{J}_{12}(u,\tau )}{\displaystyle \oint \frac{\text{d}{\mathrm{\Omega}}^{\prime}}{4\pi}}{\displaystyle {\int}_{0}^{\infty}{r}_{121}}\left({\xi}^{\prime},\xi \right)I\left({\nu}^{\prime},{\Omega}^{\prime},\tau \right)\text{d}{\nu}^{\prime}.$$(8)
The atomic frequencies appearing in the above equation are related to their counterparts in the observer’s frame through the FizeauDoppler relationship, namely, ξ′ = ν′ − Δν_{D} u • Ω′ and ξ = ν − Δv_{D} u • Ω.
2.2 The emission profile in the observer’s frame
The emission profile in the observer’s frame is given by $${\psi}_{\nu}(\Omega ,\tau )={\displaystyle {\int}_{u}{\eta}_{21}}\left(\nu \mathrm{\Delta}{\nu}_{D}u\cdot \Omega ,\tau \right){f}_{2}(u,\tau ){d}^{3}u.$$(9)
The VDF of the upper level, including the velocitychanging collisions, is given in Eq. (3) of Paletou et al. (2023). However, for the purpose of deriving the emission profile in the observer’s frame, we use the VDF as given in Eq. (19) of Paletou & Peymirat (2021), namely $${f}_{2}(u,\tau )=\frac{{n}_{1}}{{n}_{2}}\frac{{B}_{12}{J}_{12}(u,\tau )+{C}_{12}+{Q}_{V}\left({n}_{2}/{n}_{1}\right)}{{A}_{21}+{C}_{21}+{Q}_{V}}{f}^{M}(u).$$(10)
Substituting Eqs. (6) and (10) into Eq. (9), the latter takes the form $$\begin{array}{ll}{\psi}_{\nu}(\Omega ,\tau )=& {\displaystyle {\int}_{u}\{\frac{{n}_{1}}{{n}_{2}}\left[\frac{{B}_{12}{J}_{12}(u,\tau ){j}_{121}(\xi ,\tau )+{C}_{12}{\alpha}_{12}(\xi )}{{A}_{21}+{C}_{21}+{Q}_{V}}\right]}\hfill \\ \text{\hspace{0.17em}}& +\frac{{Q}_{V}{\alpha}_{12}(\xi )}{{A}_{21}+{C}_{21}+{Q}_{V}}\}{f}^{M}(u){d}^{3}u.\hfill \end{array}$$(11)
Following Section 5 of Paletou & Peymirat (2021), we assume LTE values for n_{1} (namely, ${n}_{1}={n}_{1}^{*}$), and introduce the normalized populations ${\overline{n}}_{2}={n}_{2}/{n}_{2}^{*}$. Substituting for j_{121} (cf. Eq. (8)), the first term in the curly brackets of the above equation can be rewritten as $$\frac{1}{{\overline{n}}_{2}}\frac{1}{1+\zeta}\left[\epsilon {\alpha}_{12}(\xi )+(1\epsilon ){\displaystyle \oint \frac{\text{d}{\mathrm{\Omega}}^{\prime}}{4\pi}}{\displaystyle {\int}_{0}^{\infty}{r}_{121}}\left({\xi}^{\prime},\xi \right)I\left({\nu}^{\prime},{\Omega}^{\prime},\tau \right)\text{d}{\nu}^{\prime}\right],$$
where we make use of Eqs. (23)–(27) of Paletou & Peymirat (2021), and I(ν′, Ω′, τ) is now normalized to the Planck function in the Wien limit. In the above expression, $$\epsilon =\frac{{Q}_{I}}{{A}_{21}+{Q}_{I}}$$(12)
is the usual collisional destruction probability and the quantity $$\zeta =\frac{{Q}_{V}}{{A}_{21}+{Q}_{I}}$$(13)
represents the amount of velocitychanging collisions. Similarly, the last term in the curly brackets of Eq. (11) reduces to ζα_{12}(ξ)/(1 + ζ). Using Eq. (29) of Paletou & Peymirat (2021), the emission profile can be rewritten as $$\begin{array}{l}{\psi}_{\nu}(\Omega ,\tau )={\displaystyle {\int}_{u}{f}^{M}}(u){d}^{3}u\{\frac{\zeta}{1+\zeta}{\alpha}_{12}(\xi )+\frac{1}{1+\zeta}\frac{1}{\epsilon +(1\epsilon ){\mathcal{J}}_{12}(\tau )}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\times \left[\epsilon {\alpha}_{12}(\xi )+(1\epsilon ){\displaystyle \oint \frac{\text{d}{\mathrm{\Omega}}^{\prime}}{4\pi}}{\displaystyle {\int}_{0}^{\infty}{r}_{121}}\left({\xi}^{\prime},\xi \right)I\left({\nu}^{\prime},{\Omega}^{\prime},\tau \right)\text{d}{\nu}^{\prime}\right]\},\hfill \end{array}$$(14)
where $${\mathcal{J}}_{12}(\tau )={\displaystyle {\int}_{u}{\overline{J}}_{12}}(u,\tau ){f}^{M}(u){d}^{3}u.$$(15)
In the above equation, ${\overline{J}}_{12}(u,\tau )={J}_{12}(u,\tau )/{\mathcal{B}}_{W}$, with 𝓑_{W} denoting the Planck function in the Wien limit. Substituting for J_{12}(u,τ) from Eq. (7) and using Eq. (2), it can be easily shown that $${\mathcal{J}}_{12}(\tau )={\displaystyle \oint \frac{\text{d}\mathrm{\Omega}}{4\pi}}{\displaystyle {\int}_{0}^{\infty}{\phi}_{\nu}}I(\nu ,\Omega ,\tau )\text{d}\nu $$(16)
is simply the frequency integrated mean intensity (also referred to as the “CFR scattering integral”) appearing in the standard nonLTE problem. Furthermore, we may readily identify that ʃ r_{121}(ξ′,ξ) f^{M} (u)d^{3} u = R_{121}(ν′, Ω′,ν, Ω), namely the angledependent generalized redistribution function in the observer’s frame. Furthermore, using Eq. (2), we obtain the emission profile in the observer’s frame as $$\begin{array}{l}{\psi}_{\nu}(\Omega ,\tau )=\frac{\zeta}{1+\zeta}{\phi}_{\nu}+\frac{1}{1+\zeta}\frac{1}{\epsilon +(1\epsilon ){\mathcal{J}}_{12}(\tau )}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\times \left[\epsilon {\phi}_{\nu}+(1\epsilon ){\displaystyle \oint \frac{\text{d}{\mathrm{\Omega}}^{\prime}}{4\pi}}{\displaystyle {\int}_{0}^{\infty}{R}_{121}}\left({\nu}^{\prime},{\Omega}^{\prime},\nu ,\Omega \right)I\left({\nu}^{\prime},{\Omega}^{\prime},\tau \right)\text{d}{\nu}^{\prime}\right].\end{array}$$(17)
We note that the emission profile given above is the same as that originally derived in Hubeny et al. (1983b, see their Eq. (4.15)), although we present it here in a slightly different form and also use the notations adopted in this paper. Furthermore, Hubeny & Cooper (1986) demonstrated that the emission profile derived from the semiclassical picture of Hubeny et al. (1983b) fully agrees with that derived from the quantum mechanical approach of Cooper et al. (1982).
In the present paper, we consider the angleaveraged emission profile, namely $$\psi (x,\tau )={\displaystyle \oint \frac{\text{d}\mathrm{\Omega}}{4\pi}}\psi (x,\mathbf{\Omega},\tau ),$$(18)
wherein we have transformed the real frequency ν into the nondi mensional frequency x. The resulting angleaveraged emission profile is given by $$\begin{array}{ll}\psi (x,\tau )=& \frac{\zeta}{1+\zeta}\phi (x)+\frac{1}{1+\zeta}\frac{1}{\epsilon +(1\epsilon ){\mathcal{J}}_{12}(\tau )}\hfill \\ \text{\hspace{0.17em}}\hfill & \times \left[\epsilon \phi (x)+(1\epsilon ){\displaystyle \oint \frac{\text{d}{\mathrm{\Omega}}^{\prime}}{4\pi}}{\displaystyle {\int}_{\infty}^{+\infty}{R}_{121}}\left({x}^{\prime},x\right)I\left({x}^{\prime},{\mathbf{\Omega}}^{\prime},\tau \right)\text{d}{x}^{\prime}\right].\end{array}$$(19)
For a twolevel atom with a broadened upper level and in the absence of velocitychanging collisions, the generalized redistribution function R_{121} is given by the observer’s frame counterpart of the usual atomic frame PFR function derived by Omont et al. (1972, see Milkey & Mihalas 1973 and Mihalas 1978 for the corresponding observer’s frame expression). In the presence of velocitychanging collisions, the explicit form of the PFR function R_{121} has been derived in Hubeny & Cooper (1986, see their Eqs. (3.15) and (3.16)) starting from the quantum mechanical approach of Cooper et al. (1982). In the present paper, we adopt this PFR function, which in our notation takes the following form: $${R}_{121}\left({x}^{\prime},x\right)={\gamma}_{\text{coh,\hspace{0.17em}},\text{V}}\text{\hspace{0.17em}}{R}_{\text{II}\text{A}}\left({x}^{\prime},x\right)+\left(1{\gamma}_{\text{coh},\text{\hspace{0.17em}V}}\right){R}_{\text{III}\text{A}}\left({x}^{\prime},x\right),$$(20)
where R_{II–A} and R_{III–A} are the typeII and typeIII angleaveraged PFR functions of Hummer (1962), respectively, and the coherence fraction is given by $${\gamma}_{\text{coh},\text{\hspace{0.17em}V}}=\frac{{A}_{21}+{Q}_{I}+{Q}_{V}}{{A}_{21}+{Q}_{I}+{Q}_{E}}.$$(21)
Clearly, in the absence of velocitychanging collisions, the coherence fraction as well as the PFR function become identical to those derived by Omont et al. (1972).
To clearly bring out the departure of the emission profile from CFR, we may rewrite Eq. (19) as $$\begin{array}{l}\psi (x,\tau )=\phi (x)+\frac{1}{1+\zeta}\frac{(1\epsilon )}{\epsilon +(1\epsilon ){\mathcal{J}}_{12}(\tau )}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\times \left[{\displaystyle \oint \frac{\text{d}{\mathrm{\Omega}}^{\prime}}{4\pi}}{\displaystyle {\int}_{\infty}^{+\infty}{R}_{121}}\left({x}^{\prime},x\right)I\left({x}^{\prime},{\Omega}^{\prime},\tau \right)\text{d}{x}^{\prime}{\mathcal{J}}_{12}(\tau )\phi (x)\right].\end{array}$$(22)
The above equation can easily be deduced from Eq. (19) by simply adding unity to and subtracting unity from ζ as it appears in the numerator of the first term of that equation^{3}.
3 The source function
In the full nonLTE formalism, the source function for a two level atom with a broadened upper level is of the form (see Eq. (7) of Paletou et al. 2023) $$S(x,\tau )=\left[\epsilon +(1\epsilon ){\mathcal{J}}_{12}(\tau )\right]\left[\frac{\psi (x,\tau )}{\phi (x)}\right],$$(23)
where ψ(x, τ) is given by Eq. (19) or Eq. (22) and φ(x) is the normalized Voigt function.
In the absence of velocitychanging collisions (namely, ζ = 0), the source function (23) reduces to the corresponding expression for the standard nonLTE PFR model for a twolevel atom with a broadened upper level. To demonstrate this, we first note that the emission profile for ζ = 0 (cf. Eq. (19)) takes the following form: $$\begin{array}{l}\psi (x,\tau )=\frac{1}{\epsilon +(1\epsilon ){\mathcal{J}}_{12}(\tau )}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\times \left[\epsilon \phi (x)+(1\epsilon ){\displaystyle \oint \frac{\text{d}{\mathrm{\Omega}}^{\prime}}{4\pi}}{\displaystyle {\int}_{\infty}^{+\infty}{R}_{121}}\left({x}^{\prime},x\right)I\left({x}^{\prime},{\mathbf{\Omega}}^{\prime},\tau \right)\text{d}{x}^{\prime}\right].\end{array}$$(24)
Substituting Eq. (24) into Eq. (23), we readily obtain $$S(x,\tau )=\epsilon +(1\epsilon ){\displaystyle \oint \frac{\text{d}{\mathrm{\Omega}}^{\prime}}{4\pi}}{\displaystyle {\int}_{\infty}^{+\infty}\left[\frac{{R}_{121}\left({x}^{\prime},x\right)}{\phi (x)}\right]}I\left({x}^{\prime},{\Omega}^{\prime},\tau \right)\text{d}{x}^{\prime},$$(25)
thereby establishing the equivalence of the source function derived from the full and standard nonLTE models. This is an important result that justifies the use of numerically relatively simple standard nonLTE formalism in the absence of velocitychanging collisions.
In the presence of velocitychanging collisions, the source function is obtained by substituting Eq. (22) into Eq. (23), namely $$\begin{array}{l}S(x,\tau )=\epsilon +(1\epsilon ){\mathcal{J}}_{12}(\tau )+\frac{(1\epsilon )}{1+\zeta}\\ \text{\hspace{0.17em}}\times \left\{{\displaystyle \oint \frac{\text{d}{\mathrm{\Omega}}^{\prime}}{4\pi}}{\displaystyle {\int}_{\infty}^{+\infty}\left[\frac{{R}_{121}\left({x}^{\prime},x\right)}{\phi (x)}\right]}I\left({x}^{\prime},{\Omega}^{\prime},\tau \right)\text{d}{x}^{\prime}{\mathcal{J}}_{12}(\tau )\right\}.\end{array}$$(26)
We recall that, as in Paletou & Peymirat (2021) and Paletou et al. (2023), the source function and intensity are normalized to the Planck function in the Wien limit. When velocitychanging collisions are significant (namely, ζ >> 1), the third term in Eq. (26) is negligible and the source function tends to the CFR source function.
4 A clarification regarding collisions
In the present paper, we consider three types of collisions:
inelastic collisions,
elastic collisions, and
velocitychanging collisions.
For the massive particles (twolevel atoms in the present paper), we distinguish between
the “internal variables” (the energy and quantum numbers of the levels), and
the “external variables” (the position and velocity).
The inelastic collisions are those responsible for collisional transitions between the lower and upper levels; they enter the kinetic (or statistical equilibrium) equation for atoms as inducing transitions, which further lead to the ε factor in the radiative transfer equation. In an inelastic collision, the internal variables change, whereas the external variables may or may not change.
The elastic collisions do not modify the level populations. They are responsible for line broadening. In a strictly elastic collision, both the internal and external variables remain unchanged before and after the collision. In a weakly elastic collision, the internal variables, namely the energy value of the level and the quantum numbers, may change (e.g., collisions between fine structure levels, between hyperfine structure levels, or between Zeeman sublevels in the presence of a weak magnetic field), while the external variables remain unchanged. It is also important to note that elastic collisions between degenerate Zeeman sublevels also lead to changes in the internal variables (as they change the magnetic quantum number before and after collisions, although not the atomic energy). Indeed, in the present paper, we consider isolated spectral lines and do not consider magnetic fields. Thus, the Zeeman sublevels are degenerate. If an elastic collision changes the Zeeman sublevel (namely the magnetic quantum number M is changed to M′ , with M ≠ M′), then it will have a polarizing or, more frequently, depolarizing effect (we note that polarization results from unequal populations between the Zeeman sublevels). On the contrary, if an elastic collision does not change the Zeeman sublevel (namely, M → M), then it only broadens the line, namely it is only a linebroadening collision. As clarified in SahalBréchot & Bommier (2019), line broadening and depolarizing collisions are different, and both contribute to the line broadening and hence to the elastic collision rate Q_{E} .
In addition to contributing to line broadening, the elastic collisions take part in the frequency redistribution of the scattered radiation in the line as shown by, for example, Omont et al. (1972, see also Bommier 1997a,b; SahalBréchot & Bommier 2019). In particular, they are responsible for CFR in the atomic frame. In the full nonLTE formalism, both these effects of elastic collisions are included following the works of Omont et al. (1972, see also Appendix B.2 of Oxenius 1986) and Hubeny & Cooper (1986) in the absence and presence of velocitychanging collisions, respectively. This is done by including Q_{E} in the total damping width of the absorption profile (see Eq. (1)) and using the appropriate PFR function of Omont et al. (1972) and Hubeny & Cooper (1986) for the generalized redistribution function (see Eq. (20)).
On the other hand, collisions where the internal variables remain unchanged while the external variables – such as atomic velocity (in particular its modulus) – are changed are called velocitychanging collisions. We believe that, as the internal variable is unchanged, Oxenius (1986) refers to velocitychanging collisions as elastic. However, as the atomic velocity is changed during the collision, we feel that it may not be appropriate to refer to this type of collision as elastic. Indeed Landi Degl’Innocenti & Landolfi (2004) do not refer to them as elastic. Also, most likely, the velocitychanging collisions represented by Q_{V} include both elastic and inelastic contributions. Furthermore, these types of collisions are close collisions or strong collisions with a small impact parameter. For the twodistribution problem considered in this paper, the velocitychanging collisions enter the kinetic equation for excited atoms as a relaxation term, as they are responsible for relaxing the VDF of the upper level to its equilibrium distribution function. This relaxation term is of the form given in Eq. (7) of Paletou & Peymirat (2021, see also Eq. (6.3.12) in page 167 of Oxenius 1986).
In general, a given collision can modify both the internal and external variables. Those collisions that modify only the internal variables are most likely longrange collisions (although shortrange collisions can also modify the internal variables), and those collisions that modify the external variable (namely velocity) are most likely strong shortrange collisions. In this respect, Q_{V} is a part of Q_{E} (see Sections 4.1.1 and 4.1.2 of Bommier 2016a, see also Section II(a) of Hubeny & Cooper 1986). Even if velocitychanging collisions are part of the linebroadening collisions, the method of calculation of Q_{V} is not the same as the method of calculation of Q_{E}, because Q_{E} addresses the atomic internal variables, while Q_{V} addresses the atomic external variables, although the colliding particles are the same. The velocitychanging collisions are atom–atom collisions, while the elastic and inelastic collisions may also be caused by electron–atom collisions (in addition to atom–atom collisions).
Hubeny & Cooper (1986) show that when lower state interaction is negligibly small (namely when the collisional scattering amplitude of the lower level is much smaller than that of the upper level), the total elastic collision rate Q_{E} associated with the upper level can be decomposed into two parts, one corresponding to only phase changes without change in velocity (denoted q_{E}) and the other corresponding to both phase and velocity changes (denoted ν by Hubeny & Cooper 1986). Such a decomposition is well suited for the resonance lines to which our present full nonLTE approach is applicable. Therefore, following Hubeny & Cooper (1986), we write Q_{E} = q_{E} + Q_{V} after identifying their ν as our Q_{V}. Furthermore, Hubeny & Cooper (1986) show that q_{E} = αQ_{E} and Q_{V} = (1 − α) Q_{E} with α in the range of 0 to 1. These authors also give an estimate of (1 − α). When m << M (with m denoting the mass of the perturber and M denoting the mass of the radiator), they show that (1 − α) = (m/M)^{2}, and when m ~ M, they estimate (1 − α) = 0.1.
It is known that the phasechanging elastic collision rate q_{E} can be obtained from Van der Waals approximation or using the more precise semiclassical theory developed in the 1990s by Anstee, Barklem, and O’Mara (the socalled ABO theory; see e.g., Barklem & O’Mara 1998; Barklem et al. 1998, and references cited therein). Regarding the velocitychanging collision rate Q_{V}, a precise calculation of the corresponding collision crosssection will depend on the atomic species under consideration. Landi Degl’Innocenti & Landolfi (2004) give only a rough order of magnitude for this crosssection (on the order of 10 to $100\text{\hspace{0.17em}}\pi {a}_{0}^{2}$, with a_{0} being the Bohr radius). Specific calculations would be necessary to achieve greater precision. A laboratory study was carried out by Brechignac et al. (1978), who measured the effects of the velocitychanging collisions between the excited Kr atoms and the He and/or Ar perturbers. Here the authors show that a “hardsphere” collision model is suitable for interpreting their experimental measurements. According to this model, the collisional crosssection is given by π(r_{A} + r_{B})^{2}, where r_{A} and r_{B} are the radii of the atoms participating in the collision. The atomic radii for any atomic species (including also the ions) are listed by Allen (1973, see page 45); the authors used these to compute the collisional crosssection for Kr*–He and Kr*–Ar collisions. In the present paper, all three collisional rates are assumed to be input parameters, and hence we do not compute them using the collisional dynamics. Consequently, we do not determine the velocity distribution of the colliders.
In order to evaluate the importance of velocitychanging collisions in a stellar atmosphere, Landi Degl’Innocenti & Landolfi (2004) provide a way to estimate the critical density of the perturbers or colliders (see their Eq. (13.6) in page 694). This is done by comparing an orderofmagnitude rate for velocitychanging collisions (given by nqv, where n is the number density of the perturbers, q is the cross section for velocitychanging collisions, and v is the average velocity of perturbers relative to the atom) with the rate for spontaneous emission (namely, A_{21}). The density for which these two rates are nearly the same gives the critical density: ${n}_{c}\approx 7.8\times {10}^{16}{A}_{21}/q/\sqrt{T}$ (in units of cm^{−3}). Here, A_{21} is in units of 10^{7} s^{−1}, temperature T is in units of 10^{4} K, and q is in units of $\pi {a}_{0}^{2}$. For densities greater than this critical density, velocitychanging collisions are significant. Landi Degl’Innocenti & Landolfi (2004) estimate q to be in a range^{4} that is rarely larger than 10 to 100. Using this, Bommier (2016a) provided an estimate of the critical density of colliders, which is in the order of 10^{20} cm^{−3}. We recalculated this estimate for the identical set of parameters used by this latter author (namely, A_{21} = 1). However, in Section 4.1.1 of Bommier (2016a), she does not mention the temperature value that she uses. Hence, we have chosen T = 0.5 here. We find the critical density n_{c} to be in the range of 1.103 × 10^{16} to 1.103 × 10^{15} cm^{−3}. This is about 4 to 5 orders of magnitude smaller than the density mentioned in Bommier (2016a). As the collisions with neutral hydrogen are expected to be the dominant source of both elastic and velocitychanging collisions in a stellar atmosphere, we compare n_{c} with the hydrogen density in the solar photosphere, which is on the order of 10^{17} cm^{−3} . Clearly, the velocitychanging collisions are important in the lower solar atmosphere, and hence have to be accounted for. Furthermore, the collisional destruction probability ε (which depends on the inelastic collision rate C_{21} or Q_{I}; see Eq. (12)) also becomes nonnegligible in the lower solar atmosphere (see e.g., Fig. 2c of Anusha et al. 2010). Finally, as discussed in Section 4.1.1 of Bommier (2016a), the elastic collision rate Q_{E} is also significant in the lower solar atmosphere. Thus, the present full nonLTE formalism is applicable in the lower solar atmosphere when velocitychanging collisions are important. In the upper solar atmosphere, where velocitychanging collisions are negligible, one may use the numerically relatively simple standard nonLTE PFR formalism.
5 The numerical method of solution
We solve the full nonLTE transfer problem for the case of a twolevel atom with broadened upper level using a modified version of the accelerated lambda iteration (ALI) method developed by Paletou & Auer (1995) for the corresponding standard PFR model. Here we present this ALI method in some detail, focusing on the changes brought about by the full nonLTE nature of the problem at hand. As in the standard PFR model, the source function given by Eq. (26) is iterated until convergence using the approximate lambda operator (ALO), which is chosen to be the diagonal of the full lambda operator (Olson et al. 1986).
In the onedimensional planar atmosphere considered here, the radiation field is axisymmetric. Thus, the specific intensity depends only on the inclination θ_{r} of the ray about the atmospheric normal. In other words I(x, Ω, τ) = I(x,µ,τ), where µ = cos θ_{r}. Thus, the formal solution of the radiative transfer equation can be stated as $${I}_{x\mu}={\mathrm{\Lambda}}_{x\mu}[{S}_{x}],$$(27)
where for notational convenience we have suppressed the dependence on optical depth, and the dependence on frequency and angular variables appear as subscript. Moreover, Λ_{xµ} denotes the frequency and angledependent integral operator. Given an estimate of the source function at the nth iteration, the iterative scheme will be given by $${S}_{x}^{(n+1)}={S}_{x}^{(n)}+\delta {S}_{x}^{(n)},$$(28)
where $\delta {S}_{x}^{(n)}$ is the iterative correction on the source function. Using the operator splitting technique (Cannon 1973), namely ${\mathrm{\Lambda}}_{x\mu}={\mathrm{\Lambda}}_{x\mu}^{*}+\left({\mathrm{\Lambda}}_{x\mu}{\mathrm{\Lambda}}_{x\mu}^{*}\right)\text{\hspace{0.17em}with\hspace{0.17em}}{\mathrm{\Lambda}}_{x\mu}^{*}$ being the ALO  chosen here to be the diagonal of the full lambda operator after Olson et al. (1986) –, and following a rather standard procedure (Paletou & Auer 1995, see also Sampoorna & Trujillo Bueno 2010), we arrive at the following expression for the iterative correction: $$\begin{array}{l}\delta {S}_{x}^{(n)}(1\epsilon ){\displaystyle {\int}_{0}^{\infty}{\phi}_{x}}{\mathrm{\Lambda}}_{x}^{*}\left[\delta {S}_{x}^{(n)}\right]\text{d}x\frac{(1\epsilon )}{(1+\zeta )}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\times \left\{{\displaystyle {\int}_{0}^{\infty}\left[\frac{{R}_{121}\left({x}^{\prime},x\right)}{{\phi}_{x}}\right]}{\mathrm{\Lambda}}_{{x}^{\prime}}^{*}\left[\delta {S}_{{x}^{\prime}}^{(n)}\right]\text{d}{x}^{\prime}{\displaystyle {\int}_{0}^{\infty}{\phi}_{x}}{\mathrm{\Lambda}}_{x}^{*}\left[\delta {S}_{x}^{(n)}\right]\text{d}x\right\}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}={r}_{x}^{(n)}.\end{array}$$(29)
In order to deduce the above equation, we used Eq. (16) and also the fact that, for a static atmosphere, the radiation field is symmetric about the linecenter, meaning that only half the profile can be considered. In the above equation, the frequencydependent ALO is given by $${\mathrm{\Lambda}}_{x}^{*}={\displaystyle {\int}_{1}^{+1}\frac{\text{\hspace{0.17em}d}\mu}{2}}{\mathrm{\Lambda}}_{x\mu}^{*},$$(30)
and the residual ${r}_{x}^{(n)}$ has the form $$\begin{array}{ll}{r}_{x}^{(n)}=& \epsilon +(1\epsilon ){\mathcal{J}}_{12}^{(n)}(\tau )+\frac{(1\epsilon )}{1+\zeta}\hfill \\ \text{\hspace{0.17em}}& \times \left\{{\displaystyle {\int}_{0}^{\infty}\left[\frac{{R}_{121}\left({x}^{\prime},x\right)}{{\phi}_{x}}\right]}{\mathrm{\Lambda}}_{{x}^{\prime}}\left[{S}_{{x}^{\prime}}^{(n)}\right]\text{d}{x}^{\prime}{\mathcal{J}}_{12}^{(n)}(\tau )\right\}{S}_{x}^{(n)},\hfill \end{array}$$(31)
where ${\mathcal{J}}_{12}^{(n)}(\tau )$ and the integral involving the angleaveraged PFR function are obtained from the formal solver using the nth iteration of the source function. To this end, we used the shortcharacteristic method of Olson & Kunasz (1987, see also Lambert et al. 2016). At each iteration, the system of linear equations (29) can be resolved using either a frequencyby frequency (FBF) method or a corewing method (Paletou & Auer 1995). In the following subsections, we briefly describe both these methods for the full nonLTE case considered here.
5.1 Frequencybyfrequency method
For a given depth point, the system of Equation (29) consists of N_{x} number of linear equations, with N_{x} representing the number of frequency points. In matrix form, this system of linear equations can be written as $$A\text{\hspace{0.17em}}\delta {S}^{(n)}={r}^{(n)},$$(32)
where at each depth point, δS^{(n)} and r^{(n)} are vectors of length N_{x} and A is a matrix of dimension N_{x} × N_{x}. Following Paletou & Auer (1995), we solve Eq. (32) using the LU decomposition scheme (see e.g., Press et al. 1986). As the FBF method involves matrix manipulations, such as inversion and multiplication, it is somewhat computationally expensive when compared to the corewing method presented in the following subsection.
5.2 Corewing method
Based on the behavior of the typeII PFR function of Hummer (1962), a corewing method was proposed by Paletou & Auer (1995) that allowed the computation of a system of linear equation (29) through simple algebraic manipulations, thereby considerably reducing the computational costs involved. In this method, the typeII PFR function is approximated by CFR in the line core and CS in the wings for the computation of the source function corrections. An extension of this method for the typeIII PFR function was given by Fluri et al. (2003), wherein this function is approximated by CFR in the line core and set to zero in the wings. We apply both the abovementioned corewing approximations to the R_{121} function appearing in Eq. (29). Furthermore, in Eq. (29) we make the approximation of computing the frequency integral involving the absorption profile φ_{x} only in the line core and set it to zero in the wings. This approximation is similar to the corewing approximation made for the typeIII PFR function. With these approximations, we can easily deduce the following corewing approximation for Eq. (29): $$\delta {S}_{x}^{(n)}=\frac{{r}_{x}^{(n)}+\left(1{\alpha}_{x}\right)\mathrm{\Delta}{T}^{\text{core\hspace{0.17em}}}}{1(1\epsilon ){\alpha}_{x}{\mathrm{\Lambda}}_{x}^{*}/(1+\zeta )},$$(33)
where α_{x} is the corewing separation coefficient given by $${\alpha}_{x}=\{\begin{array}{ll}0\hfill & \text{\hspace{0.17em}in\hspace{0.17em}the\hspace{0.17em}core\hspace{0.17em}}(x\u2a7d3.5),\hfill \\ {\gamma}_{\text{coh},\text{\hspace{0.17em}V}}{R}_{\text{II}\text{A}}(x,x)/{\phi}_{x}\hfill & \text{\hspace{0.17em}in\hspace{0.17em}the\hspace{0.17em}wings\hspace{0.17em}}(x>3.5)\hfill \end{array},$$(34)
and $$\mathrm{\Delta}{T}^{\text{core\hspace{0.17em}}}=(1\epsilon ){\displaystyle {\int}_{\text{core\hspace{0.17em}}}{\phi}_{x}}{\mathrm{\Lambda}}_{x}^{*}\left[\delta {S}_{x}^{(n)}\right]\text{d}x,$$(35)
which can be easily evaluated as described in Paletou & Auer (1995, see their Section 5.1). As in Sampoorna & Trujillo Bueno (2010), we find that when the elastic collision rate Q_{E} and/or velocitychanging collision rate Q_{V} are large, the approximation of setting the typeIII PFR function and the 𝒥_{12}(τ) integral to zero in the wings leads to convergence problems. In such cases, we use the CFR approximation throughout the line profile. This typically occurs for Q_{E} /A_{21} > 1 and/or Q_{V} /A_{21} > 1, when the medium is optically thick or semiinfinite. We verified that both the corewing and FBF methods give identical results. Therefore, all the solutions presented in this paper are computed with the corewing method.
5.3 Numerical computation of ${\overline{J}}_{12}(u,\tau )$
The full nonLTE formalism gives us access to the VDF f_{2} of the upper level, which depends on ${\overline{J}}_{12}(u,\tau )$ (see Eq. (3) of Paletou et al. 2023). In the case of a twolevel atom with broadened upper level, the computation of ${\overline{J}}_{12}(u,\tau )$ is numerically more complex than the CS case considered in Paletou et al. (2023). This is because, in the case of CS, the integrand in ${\overline{J}}_{12}(u,\tau )$ contained a delta function (see their Eq. (1)), while the integrand in the present case involves a Lorentzian function (see Eq. (7)). Integrals involving Lorentzian are known to be notoriously difficult to evaluate due to the sharp peaked nature of the Lorentzian function. Therefore, we need to devise a suitable method to evaluate such integrals accurately. Here we describe such a numerical method following Bommier (1997a,b).
In terms of the adimensional frequency x, the quantity ${\overline{J}}_{12}(u,\tau )$ is given by (cf. Eq. (7)) $${\overline{J}}_{12}(u,\tau )={\displaystyle \oint \frac{\text{d}\mathrm{\Omega}}{4\pi}}{\displaystyle {\int}_{\infty}^{+\infty}\frac{a}{\pi}}\frac{1}{{(xu\cdot \Omega )}^{2}+{a}^{2}}I(x,\mu ,\tau )\text{d}x.$$(36)
The dot product of the velocity vector u with the ray direction Ω is evaluated using Eq. (9) of Paletou et al. (2023), which requires us to construct the corresponding quadratures for polar angles θ_{u} and θ_{r} . As the radiation field is axisymmetric, it is sufficient to construct the quadrature directly for the azimuth difference (φ_{r} − φ_{u}). The dot product u • Ω can take both positive and negative values. Thus, the frequency integral in Eq. (36) has to include the entire range from −∞ to +∞. The intensity for the negative x values can be easily obtained from the corresponding positive values using the symmetry relation. We used 13 Gauss Legendre nodes for the direction cosines corresponding to both the ray and velocity vector in the [0,1] domain, and a quadrature made up of 20 equally spaced points for their azimuth difference (φ_{r} − φ_{u}) in the [0,2π] domain. For each pair of θ_{r}, θ_{u}, and (φ_{r} −φ_{u}), we first perform the frequency integral and then the angular integration.
Evaluating integrals involving Lorentzian function poses accuracy issues (see e.g., Fig. 1 of Paletou et al. 2020). In the present paper, we apply a method originally developed by Bommier (1997a,b) to compute the angledependent typeIII PFR function of Hummer (1962), which is known to involve an integration over the Lorentzian function (see e.g., Eq. (61) of Bommier 1997b). Following the method used by this latter author, we computed the frequency integral in Eq. (36) using the trapezoidal method with varying integration steps. The integration begins from the center of the Lorentzian (namely, at x = u • Ω), and proceeds symmetrically thereafter. The integration step is originally set as onetenth of the damping width a and is multiplied by 1.05 at each step of the integration (a geometric progression). As discussed above, the quantity u • Ω is evaluated as described in Paletou et al. (2023, see their Eq. (9)). Thus, the integration needs to be performed for each value of u and γ (which is the cosine of the angle between the velocity vector and the ray direction). As the frequency integration step size is varied as described above, the intensity computed on a standard frequency grid used for radiative transfer needs to be interpolated at every step of the frequency integration. We use the spline interpolation for this purpose.
However, the method described above to compute ${\overline{J}}_{12}(u,\tau )$ is somewhat slow. For a typical case of a semiinfinite atmosphere with 10 points per decade, 65 frequency points, and the angular quadrature mentioned above, it requires about 67 minutes of computing time on a Intel(R) Xeon(R) Gold 5122 processor with 3.6GHz, 10.4 GT/s clock speed. Clearly, computing this quantity and subsequently the VDF f_{2} of the upper level at every iteration would be computationally very expensive. However, as the ALI method described above does not require us to compute ${\overline{J}}_{12}(u,\tau )$ and f_{2} at every iteration (see Eq. (29)), we compute these quantities once the ALI solution has converged, which typically takes 15 seconds of computing time.
Fig. 1 Validation of our iterative method for the full nonLTE transfer problem (compare with Fig. 3c of Hummer 1969). The normalized source function is displayed as a function of frequency at different line center optical depths within the atmosphere: namely at τ = 0, 1, 10, 100, 10^{3}, and 10^{4}. For comparison, we also show the corresponding CFR source function (constant with frequency) as dashed lines. 
6 The numerical results
In this section, we first validate our iterative method by reproducing the benchmark result of Hummer (1969), and then illustrate the new quantities, namely the VDF of the upper level and the emission profiles together with source function with and without elastic and/ or velocitychanging collisions. We also illustrate a comparison of the normally emergent intensity profiles for the cases of a twolevel atom with infinitely sharp upper and lower levels considered in Paletou et al. (2023), and a two level atom with broadened upper level considered in this paper, along with the corresponding CFR standard nonLTE models. For the numerical studies presented here, we consider a onedimensional, isothermal, semiinfinite, planar atmosphere with a total optical thickness at line center of T = 10^{6} and є = 10^{−4}. The radiative width of the upper level parameterized as a_{R} = A_{21} /(4πΔν_{D}) is chosen to be 10^{−3}; it is related to the total damping parameter via a = a_{R}[1 + (Q_{I} + Q_{E})/A_{21}]. Unless otherwise mentioned, both the rates q_{E}/A_{21} and Q_{V}/A_{21} are set to zero.
6.1 Validation
For the atmospheric model described above, the standard non LTE source function for the R_{II–A} PFR model is illustrated in Fig. 3c of Hummer (1969). In order to validate our numerical method, we reproduced this benchmark result in our Figure 1, which displays the source function S (x, τ) for different optical depths as a function of frequency. These solutions are computed with the ALI method presented in Section 5 together with the corewing method (cf. Section 5.2) for calculating the source function corrections. A comparison of our Fig. 1 with Fig. 3c of Hummer (1969), clearly shows that our numerical method satisfactorily reproduces the benchmark solutions, thereby validating our iterative method. This is expected, as the source function derived from full nonLTE formalism is equivalent to the corresponding standard nonLTE PFR model when velocitychanging collisions are neglected (cf. Section 3).
Unlike the standard nonLTE PFR model considered by Hummer (1969), the full nonLTE model considered here gives access to the VDF of the upper level. The emission profile on the other hand can be obtained from both the abovementioned formalisms; however, it is rarely shown in the literature. Therefore, in this paper, we illustrate both the emission profile and the VDF of the upper level. Figures 2 and 3 exhibit respectively the ratios ψ(x,τ)/φ(x) and f_{2}(u, τ)/f^{M}(u) for different line center optical depths within the atmosphere. Figure 3 in the present paper is equivalent to Fig. 3 in Paletou et al. (2023), but for the case of scattering on a twolevel atom with naturally broadened upper level. For ease of comparison, in Figs. 2 and 3, we also show as dashed lines the corresponding quantities at τ = 1 for CS in the atomic frame (namely the case of a twolevel atom with infinitely sharp lower and upper levels) considered in Paletou et al. (2023).
For the standard nonLTE CFR model, the emission and the absorption profiles are identical (Hubeny & Mihalas 2014). Thus, to demonstrate the departure of the emission profile from CFR, we plot in Fig. 2 the ratio ψ(x,τ)/φ(x) at different line center optical depths within the atmosphere. Clearly, the emission profile departs from CFR for x > 1. As the optical depth increases, this departure from CFR decreases. Furthermore, the differences in ψ(x,τ)/φ(x) between the present and CS cases are significant (compare green solid and gray dashed lines in Fig. 2).
Because we consider an angleaveraged emission profile here, the VDF of the excited atom depending only on the modulus of velocity u is illustrated. As in the CS case, deviation from the Maxwellian distribution is significant for u > 2 and for optical depths close to the surface, which then decreases with increasing optical depth (compare the Fig. 3 here with Fig. 3 of Paletou et al. 2023). However, unlike the CS case, the overpopulation of the excited level for u > 2 is relatively small in the present case of a twolevel atom with a naturally broadened upper level (compare green solid and gray dashed lines in Fig. 3). We note here that a departure of the VDF of the upper level from the Maxwellian distribution was also obtained by Bommier (2016b, see her Section 5.3) through a selfconsistent solution of the statistical equilibrium equations for each velocity class of the velocitydependent atomic density matrix elements and the radiative transfer equation for the polarized radiation in the case of Na I D_{1} and D_{2} lines. This departure can be attributed to the radiative processes between the interacting atom and the incident radiation field, which is spectrally structured (i.e., nonflat) within the radiative width of the upper level.
Figure 4 displays a comparison of the normally emergent intensity for the CS and the present case of a twolevel atom with a radiatively broadened upper level. We also plot the corresponding CFR cases, namely for the damping parameter a = 0 and a = 10^{−3} . While the CS and the corresponding CFR (a = 0) cases nearly coincide (compare orange and green lines in Fig. 4), significant differences are seen in the wings for x > 3 between the CS and PFR cases (compare green and blue lines). Moreover, the PFR intensity differs significantly from the corresponding CFR intensity for x > 2 (compare blue and red lines in Fig. 4). In particular, we recover the lowering/dip of emergent intensity in the wings before finally reaching the continuum level, a wellknown effect of PFR (R_{II–A} ).
Fig. 2 Departure of emission profile ψ(x, τ) from CFR for the case of scattering on a twolevel atom with radiatively broadened upper level. Different lines correspond to ψ(x,τ)/φ(x) at different linecenter optical depths within the atmosphere (indicated in the figure legend). For comparison ψ(x, τ = 1)/φ(x), corresponding to scattering on a twolevel atom with infinitely sharp upper and lower levels (namely CS in the atomic frame), is shown as a dashed line. 
Fig. 3 Departure of the VDF of the naturally broadened upper level (f_{2}(u, τ)) of a twolevel atom from the Maxwellian equilibrium distribution f ^{M}(u) at different line center optical depths within the atmosphere (indicated in the figure legend). For comparison, the corresponding quantity at τ = 1 for the CS case is shown as a dashed line. There is clearly a greater overpopulation of f_{2} at large u in the CS case than in the present case of a twolevel atom with a naturally broadened upper level. 
6.2 Impact of velocitychanging collisions (Q_{V}/A_{21})
Unlike the standard nonLTE PFR formalism, the full nonLTE formalism of Oxenius (1986) takes into account the influence of velocitychanging collisions characterized here by Q_{V}/A_{21}. To highlight the impact of velocitychanging collisions, here we consider the extreme limit of a = 0, corresponding to the case of strong collisions (in the kinetic sense; see Hubeny & Cooper 1986). When a = 0, the total elastic collision rate Q_{E} is entirely provided by Q_{V}, which leads to simultaneous phase and velocity changes. In this respect, Q_{V} here actually represents the effective velocitychanging collision rate. Figures 5–7 display the influence of Q_{V}/A_{21} on the source function, emission profile, and the VDF of the upper level at optical depth τ = 1, respectively. The model parameters used are the same as those for Figures 1–3, but we now vary Q_{V}/A_{21} from 0 to 50 in much the same way as ζ is varied in Section 7 of Paletou et al. (2023) for the CS case. As we have chosen є = 10^{−4}, the ratio Q_{I}/A_{21} is relatively small, such that ζ (see Eq. (13)) is nearly the same as Q_{V}/A_{21}. Thus, our Fig. 5 is equivalent to Fig. 4 of Paletou et al. (2023), but for the case of the broadened upper level. However, unlike the CS case, the frequency grid is much more extended in the present case. This is to take into account the fact that the absorption profile is now a Voigt function with rather broad damping wings. As in the CS case, the source function at τ = 1 approaches the CFR limit with increasing values of Q_{V}/A_{21} or ζ (see Fig. 5). This is also in general the trend exhibited by the emission profile (see Fig. 6) and the VDF of the upper level (see Fig. 7). As discussed in Section 4, the velocitychanging collisions are nonnegligible in the lower solar atmosphere, wherein Q_{V}/A_{21} or ζ may take moderate values when full nonLTE formalism has to be adopted for an accurate determination of the source function (cf. Fig. 5) and the radiation field.
Fig. 4 Comparison of normally emergent intensity computed using the full nonLTE model for a twolevel atom with infinitely sharp levels (CS) and radiatively broadened upper level (PFR) and standard nonLTE model with CFR. We note that the intensity for CS and the corresponding CFR (a = 0) case nearly coincide. 
6.3 Impact of phasechanging elastic collisions (q_{E}/A_{21})
The phasechanging elastic collisions that are normally accounted for in spectral line formation theory through their effect on broadening the spectral line and leading to CFR in the atomic frame are characterized by q_{E} /A_{21}. By considering a = 1, here we present its impact on the source function, emission profile, and the VDF of the upper level at τ = 1. When a = 1, the total elastic collision rate Q_{E} is entirely contributed by the phasechanging collisions, which are weak collisions (Hubeny & Cooper 1986). The dependence of the source function on q_{E} /A_{21} is known from the standard nonLTE PFR formalism. With an increase in q_{E} /A_{21}, the source function approaches the CFR limit, which is indeed the case as seen from Fig. 8. This is also true for the emission profile, namely ψ(x, τ) → φ(x) with increasing values of the phasechanging elastic collision rate (see Fig. 9). Regarding the VDF of the upper level, its departure from the Maxwellian distribution initially increases until q_{E}/A_{21} = 0.5 and then decreases with further increase in q_{E}/A_{21} (see Fig. 10).
Fig. 5 Influence of velocitychanging collisions on the normalized source function at τ = 1. As expected, with increasing values of Q_{V}/A_{21}, the source function approaches the CFR limit, which is shown as a horizontal dashed line. 
Fig. 6 Dependence of the ratio of emission to absorption profile at τ = 1 (namely, ψ(x,τ = 1)/φ(x)) on velocitychanging collisions. As expected, the emission profile approaches the CFR limit, namely ψ(x, τ) → φ(x) with increasing values of Q_{V} /A_{21}. 
7 Conclusions
Full nonLTE radiative transfer, although formulated in the 1980s (Oxenius 1986), has remained largely unexplored because of the complexity involved in its numerical implementation (see however Borsenberger et al. 1986, 1987; Atanackovič et al. 1987, who considered the limiting case of a pure Doppler profile). More recently, Paletou & Peymirat (2021) reconsidered this problem, and expressed its basic elements in terms of the prevailing standard notations in this field of research. Paletou et al. (2023) then made a numerical implementation of this formalism for the case of CS in the atomic frame using the usual numerical iterative methods that are in use for the standard nonLTE transfer problem. In the present paper, we solve, for the first time, a full nonLTE radiative transfer problem considering the case of a twolevel atom with infinitely sharp lower level and broadened upper level. For this purpose, we applied, after suitable modifications, the wellknown operator perturbation methods developed for standard PFR models (Paletou & Auer 1995, see also Sampoorna & Trujillo Bueno 2010; Lambert et al. 2016). We validate our iterative method against the standard nonLTE transfer problem with an angleaveraged R_{II–A} PFR function (Hummer 1962, 1969). We illustrate the new quantities, namely the emission profile and the VDF of the upper level, and also make a comparison with the case of a twolevel atom with infinitely sharp lower and upper levels (namely the CS case considered in Paletou et al. 2023). We clearly demonstrate the influence of phasechanging elastic collisions (q_{E} , which lead to spectral line broadening and CFR in the atomic frame) and the velocitychanging collisions (Q_{V}) on the source function, emission profile, and the VDF of the upper level. In particular, we show that for moderate values of Q_{V}/A_{21} (or equivalently ζ; see Eq. (13)), one has to adopt the full nonLTE formalism presented here to accurately determine the source function (see Fig. 5) and the radiation field. Results presented in this paper may serve as benchmarks for future works on this topic (and will be made available upon request to the corresponding author).
In the present paper, we show that in the absence of velocitychanging collisions, the full nonLTE formalism is equivalent to the standard nonLTE PFR formalism, thereby validating the use of the numerically relatively simple standard nonLTE PFR formalism. However, unlike this latter, the full nonLTE formalism can also account for the velocitychanging collisions, which may become significant in the lower solar atmosphere (see Section 4). Given this, the accurate determination of velocitychanging collision rates for astrophysical applications becomes crucial. Until such calculations become available, the hardsphere collision model provides an excellent way to determine the crosssection for velocitychanging collisions.
For computational simplicity, in the present paper we consider the angleaveraged emission profile (cf. Eq. (18)), and thereby the angleaveraged redistribution functions (cf. Eq. (20)). A nearfuture goal would be to relax this assumption, which would allow us to explore the angular dependence of the VDF of the excited level.
The next crucial step will be to consider the full nonLTE transfer problem for multilevel atoms. In particular we intend to take this work forward by considering a threelevel atom, which would involve dealing with three distributions, one for the photon and two more for the excited atoms. Further, another important step would be to relax the usual assumption of Maxwellian velocity distribution for the free electrons.
Fig. 7 Dependence of the ratio of the VDF of the upper level to the Maxwellian distribution at τ = 1 (namely, f_{2}(u,τ = 1)/f^{M}(u)) on velocitychanging collisions. With increasing values of Q_{V}/A_{21}, the departure of the f_{2} from Maxwellian initially increases for Q_{V} /A_{21} = 0.25 in the regime of intermediate velocities and then decreases. 
Fig. 8 Influence of phasechanging elastic collision rate q_{E}/A_{21} on the normalized source function at τ = 1. As expected, the source function approaches the CFR limit (shown as horizontal dashed line) with increasing values of q_{E} /A_{21}. 
Fig. 9 Dependence of the ratio of the emission to absorption profile at τ = 1 (namely, ψ(x, τ = 1)/φ(x)) on the phasechanging elastic collision rate q_{E}/A_{21} (indicated in the figure legend). As expected, the emission profile approaches the CFR limit (namely, ψ(x, τ) → φ(x)) with increasing phasechanging elastic collision rate. 
Fig. 10 Dependence of the ratio of the VDF of the upperlevel to the Maxwellian distribution at τ = 1 (namely, f_{2}(u,τ = 1)/f^{M}(u)) on the phasechanging elastic collision rate q_{E}/A_{21} (indicated in the figure legend). With increasing values of q_{E}/A_{21}, the departure of the f_{2} from Maxwellian initially increases (until q_{E}/A_{21} = 0.5) and then decreases. 
Acknowledgements
M.S. acknowledges the support from the Science and Engineering Research Board (SERB), Department of Science and Technology, Government of India via a SERBWomen Excellence Award research grant WEA/2020/000012. We acknowledge the use of the highperformance computing facility (https://www.iiap.res.in/?q=facilities/computing/nova) at the Indian Institute of Astrophysics. M.S. would like to thank Prof. Helene Frisch of OCA, Nice, France for useful discussions. Authors thank an anonymous referee for constructive comments. Authors are also thankful to the referee Prof. Ivan Hubeny for carefully reviewing the work presented in this paper and for his valuable comments.
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In Paletou & Peymirat (2021) and Paletou et al. (2023), the velocitychanging collision rate Q_{V} is denoted Q_{2}. We adopt the Q_{V} notation here in order to avoid the possible confusion with the notation Q_{2} used in Hubeny & Mihalas (2014, see e.g., their Eq. (10.151) in page 327) for the elastic collision rate.
We remark that Eq. (22) can be more easily related to Eq. (4.15) of Hubeny et al. (1983b).
We verified that the range of q values suggested by Landi Degl’Innocenti & Landolfi (2004) approximately agrees with those determined from the hardsphere collision model. For example, using the atomic radii listed in Allen (1973, see page 45), we find q for H– H collisions to be 7, He–H collisions to be 12.9, and that for Cs–Cs collisions to be 137 (note that Cs has the largest atomic radius).
All Figures
Fig. 1 Validation of our iterative method for the full nonLTE transfer problem (compare with Fig. 3c of Hummer 1969). The normalized source function is displayed as a function of frequency at different line center optical depths within the atmosphere: namely at τ = 0, 1, 10, 100, 10^{3}, and 10^{4}. For comparison, we also show the corresponding CFR source function (constant with frequency) as dashed lines. 

In the text 
Fig. 2 Departure of emission profile ψ(x, τ) from CFR for the case of scattering on a twolevel atom with radiatively broadened upper level. Different lines correspond to ψ(x,τ)/φ(x) at different linecenter optical depths within the atmosphere (indicated in the figure legend). For comparison ψ(x, τ = 1)/φ(x), corresponding to scattering on a twolevel atom with infinitely sharp upper and lower levels (namely CS in the atomic frame), is shown as a dashed line. 

In the text 
Fig. 3 Departure of the VDF of the naturally broadened upper level (f_{2}(u, τ)) of a twolevel atom from the Maxwellian equilibrium distribution f ^{M}(u) at different line center optical depths within the atmosphere (indicated in the figure legend). For comparison, the corresponding quantity at τ = 1 for the CS case is shown as a dashed line. There is clearly a greater overpopulation of f_{2} at large u in the CS case than in the present case of a twolevel atom with a naturally broadened upper level. 

In the text 
Fig. 4 Comparison of normally emergent intensity computed using the full nonLTE model for a twolevel atom with infinitely sharp levels (CS) and radiatively broadened upper level (PFR) and standard nonLTE model with CFR. We note that the intensity for CS and the corresponding CFR (a = 0) case nearly coincide. 

In the text 
Fig. 5 Influence of velocitychanging collisions on the normalized source function at τ = 1. As expected, with increasing values of Q_{V}/A_{21}, the source function approaches the CFR limit, which is shown as a horizontal dashed line. 

In the text 
Fig. 6 Dependence of the ratio of emission to absorption profile at τ = 1 (namely, ψ(x,τ = 1)/φ(x)) on velocitychanging collisions. As expected, the emission profile approaches the CFR limit, namely ψ(x, τ) → φ(x) with increasing values of Q_{V} /A_{21}. 

In the text 
Fig. 7 Dependence of the ratio of the VDF of the upper level to the Maxwellian distribution at τ = 1 (namely, f_{2}(u,τ = 1)/f^{M}(u)) on velocitychanging collisions. With increasing values of Q_{V}/A_{21}, the departure of the f_{2} from Maxwellian initially increases for Q_{V} /A_{21} = 0.25 in the regime of intermediate velocities and then decreases. 

In the text 
Fig. 8 Influence of phasechanging elastic collision rate q_{E}/A_{21} on the normalized source function at τ = 1. As expected, the source function approaches the CFR limit (shown as horizontal dashed line) with increasing values of q_{E} /A_{21}. 

In the text 
Fig. 9 Dependence of the ratio of the emission to absorption profile at τ = 1 (namely, ψ(x, τ = 1)/φ(x)) on the phasechanging elastic collision rate q_{E}/A_{21} (indicated in the figure legend). As expected, the emission profile approaches the CFR limit (namely, ψ(x, τ) → φ(x)) with increasing phasechanging elastic collision rate. 

In the text 
Fig. 10 Dependence of the ratio of the VDF of the upperlevel to the Maxwellian distribution at τ = 1 (namely, f_{2}(u,τ = 1)/f^{M}(u)) on the phasechanging elastic collision rate q_{E}/A_{21} (indicated in the figure legend). With increasing values of q_{E}/A_{21}, the departure of the f_{2} from Maxwellian initially increases (until q_{E}/A_{21} = 0.5) and then decreases. 

In the text 
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