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Appendix A: Analysis of the abundance corrections for lines of neutral atoms

The temperature sensitivity of spectral lines originating from the neutral atoms of partially ionized species (e.g. Mg i, Ca i, Fe i) is governed by the Saha and Boltzmann equations, i.e. by changes of the degree of ionization and of the excitation of the line’s lower level. In the following, we develop a simplified model of the 3D– ⟨ 3D ⟩ abundance corrections, which result from horizontal fluctuations of the thermodynamical quantities.

Fig. A.1 Abundance correction Δ3D − ⟨ 3D ⟩ , computed according to Eqs. (A.1)–(A.4), versus the difference between ionization and excitation potential, Eion − χ. Each curve corresponds to a different value of Eion. The thermodynamic variables θ(x,y) and pe(x,y) were taken from the 3D model at monochromatic optical depth log τ850 = −0.48, where the mean temperature is 3360 ± 140 K. The temperature dependence of the partition functions U0 and U1 has been neglected.

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The ratio of the total number of neutral atoms n₀ (with ionization energy E_ion and partition function U₀) to the total number of singly ionized atoms n₁ (with partition function U₁) at temperature T = 5040/θ and electron pressure p_e is given by Saha’s equation (see e.g. Gray 2005, Eq. (1.20)) (A.1)The opacity of a line transition with lower level excitation potential χ is proportional to the number of absorbing atoms (per unit mass) in this state of excitation, (A.2)In the presence of horizontal fluctuations of θ and p_e, the average line opacity is amplified with respect to the ⟨3D⟩ case by a factor (A.3)where ⟨ . ⟩ _x,y denotes horizontal averaging at constant continuum optical depth τ_c. Assuming that fluctuations of the continuum opacity and the source function can be neglected, the abundance correction for weak lines can be estimated as (A.4)Figure A.1 shows the result of computing Δ_{3D − ⟨ 3D ⟩}  according to Eqs. (A.1)–(A.4) for different combinations of χ and E_ion. The figure illustrates how the curves Δ_{3D − ⟨ 3D ⟩}  versus E_ion − χ change systematically as the parameter E_ion increases from 0 to 9 eV. For E_ion ≲ 6 eV, the neutral atoms are a minority species, f ≪ 1, and we see from Eq. (A.2) that in this case (A.5)depends only on the difference (E_ion − χ), such that all curves with E_ion ≲ 6 eV fall on top of each other. As E_ion increases further, the vertex of the curves moves from (E_ion − χ)_max = 0 eV to ≈ 4 eV to 9 eV as the ionization balance shifts from ⟨ f ⟩ _x,y ≈ 0 (E_ion = 0 eV) to ≈ 1 (E_ion = 7 eV) to ≳ 1000 (E_ion = 9 eV). In fact, it can be shown analytically that (A.6)Admittedly, the description of the abundance corrections developed above is severely simplified. It ignores the fact that the line formation region is extended and that the location of its center of gravity depends sensitively on E_ion and χ (cf. Figs. A.2, B.1, C.2). Also, fluctuations of the continuum opacity and the source function were neglected. Nevertheless, the systematics seen in Fig. A.1 provides a basic explanation of the detailed numerical results presented in Sect. 3.2, Fig. 3 (especially middle left panel).

Fig. A.2 Disk-center (μ = 1) equivalent width contribution functions, ℬ(log τc), of a weak (artificial) Fe i line with excitation potential χ = 0 eV (top) and χ = 5 eV (bottom), at wavelengths λ 850 nm, evaluated according to the weak line approximation (Eqs. (B.5), (B.7)), for a single snapshot of the 3D model, the corresponding ⟨ 3D ⟩ average model, and the associated 1D LHD model used in this work. The contribution functions, originally defined on the monochromatic optical depth scale, have been transformed to the Rosseland optical depth scale.

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Appendix B: Analysis of the abundance corrections for high-excitation lines of ions

In the following, we analyze in some detail the abundance corrections derived for the high-excitation Fe ii lines, which are representative of the ionized atoms and show the largest 3D corrections (see Fig. 4). Evaluating the equivalent width contribution functions of this line in the 3D and the 1D models, we determine the physical cause of the abundance corrections. In particular, we can understand the sign of the 3D– ⟨ 3D ⟩ and ⟨ 3D ⟩ –1D corrections and explain why these corrections are so much smaller at λ 1600 nm than at λ 850 nm.

For simplicity, we consider only vertical rays (disk-center intensity) in a single snapshot from the 3D simulation, noting that the qualitative behavior of the abundance corrections is similar for intensity and flux, and does not vary much in time, i.e. the 3D– ⟨ 3D ⟩ correction is always strongly negative, while the ⟨ 3D ⟩ –1D correction is slightly positive at λ 850 nm.

B.1. Formalism

Following Magain (1986), the line depression contribution function (for vertical rays, LTE), , is defined in Linfor3D as (B.1)where τ_c is the continuum optical depth, η(Δλ,τ_c) = κ_ℓ(Δλ,τ_c)/κ_c(τ_c) is the ratio of line opacity to continuum opacity, is the difference between outgoing continuum intensity and source function⁴, and (τ_c + τ_ℓ(Δλ)) is the total optical depth in the line; angle brackets ⟨.⟩_x,y indicate horizontal averaging at constant continuum optical depth. Note that η (and τ_ℓ) vary with wavelength position in the line profile, Δλ, whereas τ_c and u_c can be considered as constant across the line profile. Then the absolute line depression at any wavelength in the line profile is (B.2)Defining further the equivalent width contribution function as (B.3)the equivalent width of the line is finally computed as (B.4)where ⟨ I_c ⟩ is the horizontally averaged emergent continuum intensity, and ℬ is defined as (B.5)In the limit of weak lines, we can assume that τ_ℓ ≪ τ_c over the whole line formation region, and Eqs. (B.1) and (B.3) simplify to (B.6)and (B.7)where (B.8)In this weak line limit, , and hence are strictly proportional to the line opacity, and the equivalent width scales linearly with the gf-value of the line (or the respective chemical abundance). Then the 3D abundance corrections can simply be obtained from the equivalent widths as (B.9)

B.2. Analysis of mixed contribution functions

Fig. B.1 Disk-center (μ = 1) equivalent width contribution functions, ℬ(log τc), of a weak (artificial) Fe ii line with excitation potential χ = 10 eV, at wavelengths λ 850 nm (top) and λ 1600 nm (bottom), evaluated according to the weak line approximation (Eqs. (B.5), (B.7)), for a single snapshot of the 3D model, the corresponding ⟨ 3D ⟩ average model, and the associated 1D LHD model used in this work. The contribution functions, originally defined on the monochromatic optical depth scale, have been transformed to the Rosseland optical depth scale as a common reference.

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Figure B.1 compares the equivalent width contribution functions, ℬ, of a weak (artificial) Fe ii line with excitation potential χ = 10 eV at two wavelengths, λ 850 nm (top) and λ 1600 nm (bottom), for the 3D model, the corresponding ⟨ 3D ⟩ average model, and the associated 1D LHD model used in this work. For each of the different models the area below the corresponding curve is proportional to the equivalent width of the emerging line profile. At λ 850 nm, the equivalent width produced by the 3D model is significantly larger than that of the 1D LHD model, which in turn is significantly lager than that of the ⟨ 3D ⟩ model, assuming the same iron abundance in all cases. The abundance corrections derived with Eq. (B.9) are Δ_3D − 1D = −0.33, Δ_{3D − ⟨ 3D ⟩}  = −0.46, and Δ _{⟨ 3D ⟩ − 1D} = + 0.13 dex. These numbers are fully consistent with the results shown in Fig. 4. At λ 1600 nm, on the other hand, the equivalent widths obtained from the three different model atmospheres are obviously very similar; all abundance corrections are much smaller than those at λ 850 nm, again in basic agreement with the results shown in Fig. 4.

Figure B.1 also shows that the formation region of this high-excitation Fe ii line is well confined to a narrow region in the deep photosphere, mainly below τ_Ross = 1, where the degree of ionization of iron is changing rapidly with depth. As expected, the line originates from somewhat deeper layers at λ 1600 nm (minimum of H⁻ opacity) than at λ 850 nm (maximum of H⁻ opacity). According to Fig. 1, both the amplitude of the horizontal temperature fluctuations and the 1D– ⟨ 3D ⟩ temperature difference increase with depth in the range − 1 < log τ_Ross < + 1, such that they are larger in the line formation region at λ 1600 nm. Naively, one would thus expect the amplitude of the abundance corrections Δ_{3D − ⟨ 3D ⟩}  and Δ _{⟨ 3D ⟩ − 1D} to be larger at λ 1600 nm than at λ 850 nm. However, this reasoning obviously fails. As we have seen before, the abundance corrections are found to be strikingly smaller at λ 1600 nm. Additional analysis is necessary to resolve this apparent contradiction.

To understand the origin of the abundance corrections, we need to understand the role of the different factors that make up the contribution function ℬ, essentially u_c and η₀ = κ_ℓ/κ_c. The different behavior of these factors in the different types of models determines the sign and amplitude of the abundance corrections. In the following, subscripts 1, 2, and 3 refer to the 1D LHD model, the ⟨ 3D ⟩ model, and the 3D model, respectively. With this notation in mind, we define the mixed contribution functions (B.10)where i = 1...3, j = 1...3, k = 1...3. The three mixed contribution functions with three identical subscripts i = j = k are thus the normal contribution functions for the 1D, ⟨ 3D ⟩ , and 3D model, respectively. From each of the ℬ_i,j,k we can compute an equivalent width according to Eq. (B.4), which we denote as W_i,j,k. The equivalent widths can then be used to derive abundance corrections via (B.11)The numerical evaluation of the relevant abundance corrections is compiled in Table B.1.

B.2.1.  ⟨ 3D ⟩ –1D abundance corrections

With the help of Table B.1, the physical interpretation of the Δ _{⟨ 3D ⟩ − 1D} abundance correction is straightforward. Columns (6) and (7) show the effect of the different factors that contribute to the ⟨ 3D ⟩ –1D abundance correction. Owing to the different thermal structure of the two model atmospheres, all three factors, u_c, κ_ℓ, and κ_c change simultaneously, and the full correction is Δ _{⟨ 3D ⟩ − 1D} = Δ_2,2,2,1, listed in the last row of the table as case (7). The other cases (1)–(6) refer to “experiments” where only one or two of the factors are allowed to change while the remaining factors are fixed to expose the abundance corrections due to the individual factors. Case (1), for example, shows the correction that would result for fixed opacities, κ_ℓ(τ_c), κ_c(τ_c), accounting only for the differences in the source function gradient u_c. Case (6) shows the complementary experiment where u_c is fixed and both opacities are changing in accordance with the different thermodynamical conditions.

At λ 850 nm, the continuum opacity is dominated by H⁻ bound-free absorption, which shows its maximum at this wavelength. The high-excitation Fe ii line forms around log τ₈₅₀ ≈ + 0.7 (log τ_Ross ≈ + 0.6), i.e. significantly below continuum optical depth unity. At this depth, both the temperature and the temperature gradient are slightly lower in the ⟨ 3D ⟩ model than in the 1D model. As a consequence, both u_c and κ_c decrease toward the ⟨ 3D ⟩ model, approximately by the same factor (see cases 1 and 4), and hence their effects cancel out. The highly temperature-dependent line opacity (∂log κ_ℓ/∂θ ≈ − 10; θ = 5040/T) is thus the dominating factor and determines the total Δ _{⟨ 3D ⟩ − 1D} abundance correction (compare cases 2 and 7).

At λ 1600 nm, the situation is different. Here the continuum opacity is mainly due to H⁻ free-free absorption. The important difference is that the temperature sensitivity of the H⁻ free-free opacity is significantly higher than that of the H⁻ bound-free absorption (see Fig. B.2). In the line formation region around log τ₁₆₀₀ ≈ + 0.15 (log τ_Ross ≈ + 0.85), the temperature sensitivity of κ_ℓ and κ_c is now comparable, such that the ratio of both opacities is nearly the same in the two models. The corrections due to κ_ℓ and κ_c are almost equal and of opposite sign (cases 2 and 4), and hence cancel out. At the same time, the source function gradient is very similar in both models, and thus the correction due to u_c is small (case 1). The total Δ _{⟨ 3D ⟩ − 1D} abundance correction is therefore significantly smaller than at λ 850 nm (case 7).

Fig. B.2 Continuous opacity due to H at λ 850 nm (solid line, cloud of blue dots), and due to H at λ 1600 nm (dashed line, cloud of red dots), as a function of temperature in the 3D model (dots) and in the ⟨ 3D ⟩ model (lines), respectively. The opacities have been computed according to Eqs. (8.12) and (8.13) by Gray (2005); they are given in units of cm2 per neutral hydrogen atom, and have been normalized to 1 at T = 6000 K. Both opacities are proportional to the electron pressure, and their ratio is a simple monotonic function of temperature. Note that κ(H is almost temperature insensitive between T = 4000 and 5000 K.

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B.2.2. 3D– ⟨ 3D ⟩ abundance corrections

The physical interpretation of the Δ_{3D − ⟨ 3D ⟩}  abundance correction proceeds along similar lines. Columns (3) and (4) of Table B.1 show the influence of the different factors that contribute to the 3D– ⟨ 3D ⟩ “granulation correction”. The three factors, u_c, κ_ℓ, and κ_c differ between the 3D and the ⟨ 3D ⟩ model due to the presence of horizontal fluctuations of the thermodynamical conditions at constant optical depth τ_c, which then lead to more or less nonlinear fluctuations of the factors that make up the contribution function. The full correction, Δ_{3D − ⟨ 3D ⟩}  = Δ_3,3,3,2, allows for fluctuations in all three factors and is listed in the last row of the table as case (7). The other cases (1)–(6) refer to “experiments” where the fluctuations are artificially suppressed for one or two of the factors to study the impact of the fluctuations of the individual factors on the resulting the abundance correction. For example, case (5) shows the correction that would result if the fluctuations of the line opacity, κ_ℓ(τ_c), were suppressed. case (2) shows the complementary experiment where only κ_ℓ(τ_c) is allowed to fluctuate, while u_c and κ_c are fixed.

At λ 850 nm, our weak Fe ii line is strongly enhanced in the 3D model due to the nonlinear fluctuations of the line opacity. The fluctuations lead to a line enhancement, and hence to negative 3D– ⟨ 3D ⟩ abundance corrections, whenever ⟨ κ_ℓ(T) ⟩ _x,y > κ_ℓ( ⟨ T ⟩ _x,y), which happens to be the case as κ_ℓ ∝ exp { − E/kT }  (roughly speaking because ∂²κ_ℓ/∂T² > 0). As can be deduced from the comparison of cases (2) and (7), suppression of the fluctuations of both u_c and κ_c does not change the resulting 3D abundance correction. We can furthermore see that the fluctuations of u_c enhance the nonlinearity of the fluctuations of κ_ℓ (case 3) and that the fluctuations of κ_c diminish the nonlinearity of the fluctuations of κ_ℓ (case 6). We conclude that the fluctuations of u_c and κ_c must be substantial, but essentially linear, such that they do not produce any significant abundance corrections on their own (cases 1, 4, and 5).

At λ 1600 nm, the continuum opacity κ_c is lower than at λ 850 nm, and our weak Fe ii line forms at somewhat deeper layers where the temperature is higher. Equally important, the temperature sensitivity of κ_c is distinctly higher at λ 1600 nm than at λ 850 nm, as is demonstrated in Fig. B.2. This fact is the key to understanding the drastically smaller abundance corrections found at λ 1600 nm.

Comparing the effect of the line opacity fluctuations for the two wavelengths (case 2), we see that the corresponding abundance correction is significantly smaller at λ 1600 nm. This result is unexpected, because according to Fig. 1 the temperature fluctuations, δT_rms, ought to be larger in the deeper layers where the near-IR line forms, which in turn should lead to more nonlinear fluctuations of the line opacity and hence larger abundance corrections at λ 1600 nm compared to λ 850 nm.

Further investigations revealed that the opposite is true. The point is that we have to distinguish between fluctuations at constant Rosseland optical depth, τ_Ross, and fluctuations at constant monochromatic optical depth, τ_c, which are relevant in the present context. In fact, the higher temperature sensitivity of the continuum opacity at λ 1600 nm reduces the amplitude of the temperature fluctuations at constant continuum optical depth τ₁₆₀₀ with respect to the fluctuations at constant τ₈₅₀, as illustrated in Fig. B.3 (top panel). The degree of nonlinearity of the line opacity fluctuations, as measured by the ratio of average line opacity to line opacity at mean temperature, , is shown in the bottom panel of Fig. B.3. Over the whole depth range, the nonlinearity of the κ_ℓ fluctuations is higher at constant τ₈₅₀ than at constant τ₁₆₀₀. Remarkably,  increases toward lower temperatures, even though the amplitude of the temperature fluctuations decreases with height. This is because the temperature sensitivity of κ_ℓ increases strongly as Fe ii becomes a minority species at lower T (cf. Fig. 2). The fact that is significantly higher for the red line at λ 850 nm than for the near-IR line at λ 1600 nm explains the wavelength dependence of the abundance corrections found for case (2), Cols. (3) and (4).

Fig. B.3 Amplitude of the (relative) temperature fluctuations, δTrms/ ⟨ T ⟩  (top), and ratio of average line opacity to line opacity at mean temperature, of a weak (artificial) Fe ii line with excitation potential χ = 10 eV (bottom) as a function of the mean temperature ⟨ T ⟩ , where angle brackets indicate horizontal averaging at constant monochromatic optical depth. Solid and dashed curves show the results of averaging on surfaces of constant τ850 and constant τ1600, respectively. Filled dots indicate the location of the center of the line formation regions at λ 8500 and 1600 nm.

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Comparing cases (2) and (6) for the near-IR line, we see that the abundance correction essentially vanishes when combining the fluctuations of the line opacity with the fluctuations of the continuum opacity. We note that the fluctuations of κ_c are significantly larger at log τ₁₆₀₀ = 0.15, where the near-IR line forms, than at log τ₈₅₀ = 0.7, where the red line forms, even though the temperature fluctuations are lower (see Fig. B.3). This is again a consequence of the enhanced temperature sensitivity of the continuum opacity at λ 1600 nm (Fig. B.2). It thus happens that the abundance corrections due to the fluctuations of the continuum opacity and the line opacity, respectively, are comparable (compare cases 2 and 4). The net result is a cancelation of the two effects. The total Δ_{3D − ⟨ 3D ⟩}  abundance correction λ 1600 nm is therefore small.

B.3. Saturation effects

So far we have considered the abundance corrections for the limiting case of weak, unsaturated lines. In this limit, the abundance corrections are independent of the equivalent width of the line and of the microturbulence parameter ξ_mic chosen for the spectrum synthesis with the 1D models. Figure B.4 shows how the results change if saturation effects are fully taken into account, again for the example of the high-excitation Fe ii line.

Fig. B.4 Total 3D abundance correction Δ3D−1D for the artificial Fe ii line with excitation potential χ = 10 eV at λ 850 nm (top) and λ 1600 nm (bottom) as a function of the equivalent width obtained from the 3D model. The fainter (green) curves and the thicker (black) curves refer to the intensity (μ = 1) and flux spectrum, respectively. The abundance corrections have been computed for three different values of the microturbulence parameter used with the 1D model, ξmic = 0.0 (dotted), 1.0 (solid), and 2.0 km s-1 (dashed lines). The weak line limit coincides with the horizontal part of the curves at low log W3D.

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Fig. B.5 Disk-center (μ = 1) equivalent width contribution functions, ℬ(log τc), of a weak (top) and strong (bottom) Fe ii line with excitation potential χ = 10 eV, at λ 850 nm. The contribution functions have been computed for a single snapshot of the 3D model (solid), the corresponding ⟨ 3D ⟩ average model (dashed), and the associated 1D LHD model (dotted) used in this work. They have been transformed from the monochromatic to the Rosseland optical depth scale. In all cases, the saturation factor exp { − τℓ } is properly taken into account (see Eq. (B.1)).

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Obviously, the total 3D abundance correction, Δ_3D − 1D, depends strongly on both the assumed value of ξ_mic and on the line strength, W_3D. We notice that this holds even for very weak lines, and conclude that even the weakest lines used for this study are already partly saturated. Plotting log W_3D versus log gf reveals that the curve-of-growth is linear, and thus saturation effects can be safely ignored as long as the equivalent width of the line is below  pm at λ 850 nm ( pm at λ 1600 nm). As soon as this line becomes detectable, it is no longer on the linear part of the curve-of-growth. This extreme behavior is of course related to the extreme temperature sensitivity of this high-excitation line, which changes the line-to-continuum opacity ratio from η ≪ 1 to η ≫ 1 within the line formation region. This is especially true at λ 850 nm, where the continuum opacity is less temperature dependent (see above). The partial saturation of weak lines is not a particular property of the 3D model, but is seen in 1D models, too.

The top panel of Fig. B.5 shows the equivalent width contribution functions ℬ(τ_Ross) of the same weak Fe ii line (χ = 10 eV, λ 850 nm) as in Fig. B.1 (top), but now including the saturation factor exp { − τ_ℓ } (see Eq. (B.1)). Comparison with Fig. B.1 (top) demonstrates that including the saturation factor reduces the equivalent width from W_3D ≈ 0.61 pm to W_3D ≈ 0.17 pm). Moreover, all contribution functions are shifted to slightly higher layers because of the presence of saturation effects. The upward shift is more pronounced for the 3D contribution function, because of the strongly nonlinear fluctuations of the saturation factor exp { − τ_ℓ } . As a result, ℬ(3D) now becomes smaller than ℬ( ⟨ 3D ⟩ ) and ℬ(1D) in the deepest part of the line-forming region. Therefore, the ratio of 3D to 1D equivalent width becomes smaller than in the weak line limit, where ℬ(3D) > ℬ(1D) over the whole optical depth range. Hence, the total 3D abundance correction, Δ_3D − 1D, becomes less negative if saturation is taken into account.

If the line strength is increased even more, the contribution functions become wider and extend to higher atmospheric layers, as shown in the bottom panel of Fig. B.5. Recalling that the equivalent width contribution function is a superposition of the line depression contribution functions for the individual wavelength positions in the line profile, it seems evident that the double peak structure is related to the contributions of the line core (left peak) and of the extended line wings (right peak). Test calculations confirm this interpretation. Proceeding form the top to the bottom of the line formation region, the difference ℬ(3D) − ℬ(1D) changes sign from positive to negative to positive to negative. Because the contributions of the different layers to the abundance correction cancel partially, a straightforward interpretation of the resulting abundance correction becomes difficult. In principle, a detailed analysis of the line depression contribution functions at individual wavelengths might lead to further insights. Noting, however, that the situation becomes even more complicated when considering flux spectra (involving inclined rays), we have some doubts that such an investigation is worthwhile.

Appendix C: Molecule concentrations, line opacities, and height of formation

The equilibrium number density of diatomic molecules with constituents A and B, N_AB, is given by the Saha-like relation (C.1)where N_A and N_B are the number densities (per unit volume) of free neutral atoms (in the ground state) of elements A and B, with partition functions U_A and U_B, respectively; the molecule is characterized by its mass, m_AB, its partition function, Q_AB, and its dissociation energy D₀ (cf. Cox 2000). Defining the number densities per unit mass as X_i = N_i/ρ, where ρ is the mass density, we obtain (C.2)Figure C.1 shows the number densities of our selection of diatomic molecules (normalized to the total number of carbon nuclei, N_AB/ ∑ N_C = X_AB/ ∑ X_C) as a function of Rosseland optical depth in the 1D LHD model used in this work. In the photosphere (log τ_Ross < 0), essentially all carbon is locked up in CO. The decrease of X_OH, X_NH, X_CN toward lower optical depths is a consequence of the density factor ρ in Eq. (C.2). The destruction of all molecules beyond τ_Ross ≈ 1 is due to the Boltzmann factor exp { D₀/kT } ; a higher dissociation energy corresponds to a steeper drop of the molecule concentration with T.

	Fig. C.1 Number density of different molecules, normalized to the total number density of carbon (sum over all molecules and ionization states) as a function of the Rosseland optical depth in the 1D LHD model.
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	Fig. C.2 Same as Fig. A.2, but for two weak (artificial) molecular lines: a CO line with excitation potential χ = 0 eV (top) and a C2 line with χ = 4 eV (bottom), both at wavelength λ 850 nm.
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The opacity of a line with lower transition level energy χ, is proportional to X_AB/Q_QB exp { − χ/kT } , and thus the line opacity per unit mass can be written as (C.3)where θ = 5040/T, and the temperature dependence of the partition functions U_A and U_B has been ignored; the molecular partition function Q_AB cancels out.

If both atoms A and B are majority species (e.g. H, N, O for the conditions in our red giant atmosphere), then X_A and X_B are constant, and the temperature dependence of the line opacity is given by In general, the temperature dependence of the molecular line opacity is more complicated, because X_A and/or X_B are more or less strongly temperature dependent due to ionization and/or formation of different molecules. In our red giant atmosphere, for example, the concentration of carbon atoms is controlled by the formation of CO molecules. This leads to a strong increase of κ_ℓ with temperature (∂log κ_ℓ/∂θ < 0) for CH and C₂ at τ_Ross < 1, such that these molecules can only form in a narrow region centered around log  τ_Ross ≈ 0 (see Fig. C.2).

Finally, we point out that the molecular lines form in the same height range as the lines of neutral atoms and ions.

Figure C.2 displays the contribution functions for the most extreme examples. The ground state CO line (top panel) shows the most extended formation region, centered around log τ_Ross ≈ − 1. The contribution function of this line is almost identical to that of the ground state Fe i line shown in Fig. A.2. The high-excitation C₂ line (bottom panel) originates from a very narrow formation region located in the deep photosphere around log τ_Ross ≈ 0. The contribution functions of the other molecular lines considered in this study lie somewhere in between these two extremes; the entire formation region of the molecular lines is always inside the height range covered by our model atmospheres.

© ESO, 2012