A&A 387, 258-270 (2002)
M. Steffen1 - H. Holweger2
1 - Astrophysikalisches Institut Potsdam, An der Sternwarte 16, 14482 Potsdam, Germany
2 - Institut für Theoretische Physik und Astrophysik, Universität Kiel, 24098 Kiel, Germany
Received 31 October 2001 / Accepted 1 March 2002
In an effort to estimate the largely unknown effects of photospheric temperature fluctuations on spectroscopic abundance determinations, we have studied the problem of LTE line formation in the inhomogeneous solar photosphere based on detailed 2-dimensional radiation hydrodynamics simulations of the convective surface layers of the Sun. By means of a strictly differential 1D/2D comparison of the emergent equivalent widths, we have derived "granulation abundance corrections'' for individual lines, which have to be applied to standard abundance determinations based on homogeneous 1D model atmospheres in order to correct for the influence of the photospheric temperature fluctuations. In general, we find a line strengthening in the presence of temperature inhomogeneities as a consequence of the non-linear temperature dependence of the line opacity. The resulting corrections are negligible for lines with an excitation potential around Ei=5 eV, regardless of element and ionization stage. Moderate granulation effects ( dex) are obtained for weak, high-excitation lines ( eV) of C I, N I, O I as well as Mg II and Si II. The largest corrections are found for ground state lines (Ei=0 eV) of neutral atoms with an ionization potential between 6 and 8 eV like Mg I, Ca I, Ti I, Fe I, amounting to dex in the case of Ti I. For many lines of practical relevance, the magnitude of the abundance correction may be estimated from interpolation in the tables and graphs provided with this paper. The application of abundance corrections may often be an acceptable alternative to a detailed fitting of individual line profiles based on hydrodynamical simulations. The present study should be helpful in providing upper bounds for possible errors of spectroscopic abundance analyses, and for identifying spectral lines which are least sensitive to the influence of photospheric temperature inhomogeneities.
Key words: hydrodynamics - radiative transfer - convection - line: formation - Sun: abundances - Sun: photosphere
In contrast, the treatment of convection is still rather crude. Usually, stellar atmospheres are described as 1-dimensional hydrostatic configurations where the convective energy transport is calculated from the so-called mixing-length theory (MLT, Vitense 1953; Böhm-Vitense 1958) or variants thereof (e.g. Canuto et al. 1996). However, modern spectroscopic observations have reached a level of precision that requires improved model atmospheres for a reliable interpretation, including a proper treatment of hydrodynamical phenomena.
From the point of view of standard model atmospheres, convection affects the temperature structure in a twofold way: it influences the mean vertical stratification and introduces horizontal inhomogeneities. MLT is designed to model only the mean structure and is not suitable to construct realistic multi-component model atmospheres. Although realistic hydrodynamical model atmospheres do exist for the Sun and a few solar-type stars due to the pioneering work by Nordlund (1982), Nordlund & Dravins (1990), Stein & Nordlund (1989, 1998), and Ludwig et al. (1999), to name just a few important contributions, such complex models are not usually applied for the analysis of stellar spectra.
Observationally, the solar granulation is the visible imprint of thermal convection extending far into the photospheric layers where spectral lines are formed. The effect of the associated photospheric temperature inhomogeneities on the line formation process and the consequences for spectroscopic abundance determinations are not yet well understood, but it is generally believed that the errors introduced by representing a dynamic, inhomogeneous atmosphere by a static, plane-parallel model are small. However, reliable quantitative estimates of this effect are difficult to find. Earlier studies suffer from uncertainties in the underlying empirical two-component models, which are rather ad hoc and lack a firm physical foundation (e.g. Hermsen 1982). Later investigations based on relatively crude numerical convection models (Atroshchenko & Gadun 1994; Gadun & Pavlenko 1997) led to questionable and partly ambiguous conclusions. Recent determinations of the solar photospheric Fe and Si abundance by Asplund et al. (2000b) and Asplund (2000a), respectively, rely on state-of-the-art 3D numerical convection simulations and take all hydrodynamical effects fully into account. However, comparison of these results with those based on the empirical 1D Holweger-Müller (1974) solar atmosphere does not allow to separate the influence of inhomogeneities from the effect of the mean stratification, and to study the 1D/3D abundance corrections as a function of the line parameters in a systematic way.
In this work we investigate the spectroscopic effects of horizontal temperature inhomogeneities in the solar atmosphere based on detailed 2D hydrodynamical models of solar surface convection (Freytag et al. 1996; Ludwig et al. 1999; Steffen 2000). We do not address the question whether the mean temperature structure supplied by MLT is an appropriate representation of the mean stratification obtained from hydrodynamical simulations (the problem of the "right'' choice of the mixing-length parameter for recovering the correct mean temperature stratification was investigated by Steffen et al. (1995) in the context of white dwarf atmospheres). Rather, our main question is: how large are the systematic errors of standard spectroscopic abundance determinations due to the fact that one replaces the "real'' line profile, which actually is the result of averaging the spatially resolved line intensities over the granulation pattern, by the line profile computed from a single plane-parallel temperature stratification? In Sect. 2, we introduce a simple analytical argument to illustrate the basic effect of line strengthening in the presence of temperature fluctuations. Then we summarize the main characteristics of the numerical simulations of solar surface convection in Sect. 3, outline our method to derive "granulation abundance corrections'' from a differential 1D/2D comparison of computed line equivalent widths in Sect. 4, before finally presenting the results for the Sun in Sect. 5. A summary of our findings is given in Sect. 6.
In order to give a physical explanation for the numerical results obtained in this work, we have carried out a detailed investigation the process of LTE line formation in an inhomogeneous stellar atmosphere based on the analysis of the transfer equation and the evaluation of line depression contribution functions. This study is presented in the second paper of this series (Paper II).
|Figure 1: Spatial variation of W for a sinusoidal temperature variation according to Eq. (6), for and Ei=10 eV. Averaging over one spatial wavelength gives (dotted horizontal line).|
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A similar result is obtained when assuming a totally different distribution
Looking at Table 1, it turns out that the simple model severely overestimates the abundance correction in the case of the oxygen lines mentioned as an example above. This is because, in reality, the line formation process is much more complicated, since the line strength depends not only on the line opacity but also on the continuum opacity and on the local temperature gradient of the atmosphere (see Paper II). Moreover, spectral lines usually form over a substantial depth range, and the sign of may depend on depth, leading to partial cancellation of the local effects, especially for stronger lines. Finally, saturation can play a significant role as well. Hence, realistic abundance corrections can only be derived from detailed line formation calculations based on multi-dimensional hydrodynamical model atmospheres.
|Figure 2: Snapshot from a 2-dimensional numerical simulation of solar surface convection after 41 940 s of simulated time. This model was computed on a Cartesian grid with mesh points (tick marks along upper and right side), with periodic lateral boundary conditions (L = 5250 km). "Open'' boundary conditions at the bottom and top of the computational domain are designed to minimize artificial distortions of the flow. The velocity field is represented by pseudo streamlines, indicating the displacement of a test particle over 20 s (maximum velocity is 10.1 km s-1 at this moment); the temperature structure is outlined by temperature contours in steps of 500 K. Geometrical height z=0 (scale at left) corresponds to ; scales at right refer to the horizontally averaged gas pressure [dyn cm-2] and Rosseland optical depth .|
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Stellar surface convection is governed by the conservation equations of hydrodynamics, coupled with the equations of radiative energy transfer. Our hydrodynamical models result from the numerical integration of this set of partial differential equations. This approach constitutes an increasingly powerful tool to study in detail the time-dependent hydrodynamical properties of the solar granulation.
Just like "classical'' stellar atmospheres, the hydrodynamical models are characterized by effective temperature, , surface gravity, g, and chemical composition of the stellar matter. But in contrast to the mixing-length models, they account for "overshoot'' and no longer have any free parameter to adjust the efficiency of the convective energy transport. Based on first principles, RHD simulations provide physically consistent ab initio models of stellar convection which can serve to address a variety of questions, including the problem of line formation in inhomogeneous stellar atmospheres.
The models used for the present investigation comprise a small section near the solar surface, extending over 7 pressure scale heights in the vertical direction, including the photosphere, the thermal boundary layer near optical depth , and parts of the subphotospheric layers. Only the uppermost layers of the deep solar convection zone can be included in the model, requiring an open lower boundary. The simulations are designed to resolve the solar granulation. The effect of the smaller scales, which cannot be resolved numerically, is modeled by means of a subgrid scale viscosity (so-called Large Eddy Simulation approach). Spatial scales larger than the computational box are unaccounted for. We employ a realistic equation of state, including ionization of H, HeI, HeII and H2 molecule formation. In order to avoid problematic simplifications like the diffusion or Eddington approximation, we solve the non-local radiative transfer problem along a large number ( ) of rays in 5 opacity bins, with realistic opacities accounting adequately for the influence of spectral lines (see Nordlund 1982; Ludwig 1992). For further details see Ludwig et al. (1994), Freytag et al. (1996), Ludwig et al. (1999), Steffen (2000).
|Figure 3: Temperature as a function of geometrical depth z (top) and as a function of individual optical depth (bottom) for the snapshot shown in Fig. 2. Each line corresponds to a different x position. Note that the and the average are hardly distinguishable in this representation.|
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Oscillations excited by the stochastic convective motions contribute to the photospheric velocity field as well. Interestingly, the oscillation periods lie in the range 150 to 500 s for the solar simulation, in close agreement with the observed 5 min oscillations. Controlled by the balance between dynamical cooling due to adiabatic expansion and radiative heating in the spectral lines, the mean photospheric temperature stratification is slightly cooler than in radiative equilibrium, and the resulting temperature structure is not at all plane-parallel.
According to Eq. (3), the line strengthening depends on the key quantity , where . We have evaluated as a function of optical depth for our 2D numerical simulation of solar convection (Fig. 4). For the problem under investigation, temperature differences have to be taken over levels of constant optical depth, rather than at constant geometrical depth. According to the simulations, typical values in the lower photosphere are in the range . The minimum near is related to the "inversion'' of the temperature fluctuations at this height.
It is worth mentioning that we have recently carried out fully 3-dimensional simulations with a new code developed by B. Freytag and M. Steffen (Freytag; Steffen et al., in preparation). We find that the depth dependence of is qualitatively similar in 2D and 3D. However, the amplitude of the temperature fluctuations is systematically smaller at all heights in the 3D simulation. The difference is most pronounced in the higher photosphere (see Fig. 4).
The latter problems apply to state-of-the-art 3D convection simulations as well. We note that a perfect line profile fitting does not necessarily imply a perfect model atmosphere. Another important diagnostic is the center-to-limb variation of the continuum intensity. Even the best 3D solar granulation models still fail to reproduce the observed continuum properties with high accuracy (Asplund et al. 1999), indicating that "the temperature structure close to the continuum forming layers could be somewhat too steep''.
In view of the above mentioned limitations, we prefer to avoid a direct application of our convection simulations. Instead, we favor the somewhat less powerful but more reliable differential approach described in the following section.
|Figure 4: Time-averaged amplitude of rms temperature fluctuations as a function of Rosseland optical depth, derived from the numerical simulation of solar granulation described in Sect. 3 (solid). was obtained by averaging over surfaces of constant Rosseland optical depth for each snapshot and subsequently taking a long-term time average. For comparison, we show corresponding results from a 3-dimensional numerical simulations (dashed) recently carried out with a new code by B. Freytag and M. Steffen.|
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We have adopted a strictly differential procedure to quantify these errors. The basic idea is to identify our 2D hydrodynamical convection simulations with the real inhomogeneous atmosphere, and the mean structure of the simulations with the "best possible'' 1D representation. Note that it is not important for the hydrodynamical convection models to reproduce the mean temperature structure of the real solar atmosphere with great precision, because the related errors cancel to a first approximation in our differential comparison. Similarly, the details of the hydrodynamical velocity field are of secondary importance in the present context, since such 1D/2D differences are largely eliminated by a proper choice of the microturbulence parameter. In contrast, there is no free parameter to account for the presence photospheric temperature inhomogeneities. This is why we focus on the abundance corrections related to temperature fluctuations only. The most important quantity to be provided by the convection model is therefore the magnitude and height dependence of .
In the following, we outline our method to derive LTE "granulation abundance corrections'', , which must be added to the logarithmic elemental abundance derived from standard 1D model atmosphere analysis in order to compensate for the effects of temperature inhomogeneities. Note that our procedure ensures that the corrections for vanishing horizontal temperature and pressure fluctuations.
Next we compute, for given elemental abundance and atomic line
parameters, the equivalent width of any given spectral line from
(i) the set of 2D snapshots by horizontal averaging of the spatially
resolved synthetic line profile
For the line formation calculations, we employed the Kiel spectrum synthesis code LINFOR under the assumption of LTE. For the 1D atmospheres, we have always adopted a depth-independent microturbulence velocity of km s-1. For snapshots from the convection simulations we have computed synthetic spectra for two different assumptions concerning the non-thermal velocity field: in case (A) we used the depth-dependent hydrodynamical velocity field v(x,z) obtained from the simulations, in case (B) we replaced the hydrodynamical velocity field by a depth-independent microturbulence velocity km s-1.
As the final step, we calculate the mean equivalent widths
resulting from the set of 1D and 2D line profiles,
For truly weak lines, the (logarithmic) "granulation abundance correction''
would simply be
Case (A) and (B) give essentially the same corrections for weak lines, but the results may differ substantially for stronger lines, where (implying that the hydrodynamical velocity field provides an effective "microturbulence'' km s-1). For lines of arbitrary strength, we strongly favor the granulation corrections derived from case (B), because this strictly differential 1D/2D comparison is based on the same non-thermal velocity field, and thus cleanly separates the effect of the temperature inhomogeneities from that of the small-scale velocity field. In the following, therefore denotes the granulation abundance corrections derived from case (B).
For practical reasons, we have used only a small number of snapshots
(S=11) for our present investigation. The statistical uncertainty of
the derived abundance corrections is therefore appreciable. In order
to quantify this uncertainty, we have also tabulated the standard
deviation of the corrections
Atoms with high ionization potential like C I, N I, and O I, as well as singly ionized elements are representative of species in the main ionization stage. Neutral nitrogen (ionization potential eV) can be taken as the prototype of these species. Unlike C and O, N is not significantly affected by molecule formation or changes in the degree of ionization.
|Figure 5: Logarithmic granulation abundance corrections as a function of excitation energy Ei for C I, N I, O I, and a number of singly ionized elements, representative of species in the main ionization stage. The plot is based on a subset of weak fictitious lines with Å, mÅ, taken from Tables 1-4. The upper symbol at Ei=12 eV for N I refers to Å.|
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|Figure 6: Granulation abundance corrections, , for the first 8 fictitious nitrogen lines listed in Table 1, compared with test results obtained by increasing the line strength by factors of 5 and 15, respectively.|
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|Figure 7: Granulation abundance corrections, , for the first 8 fictitious nitrogen lines listed in Table 1, compared with test results obtained by moving the weak lines from Å to different spectral regions, centered at , 8000, 10 000, 12 000, and 16 000 Å, respectively, keeping the reduced equivalent width fixed ( mÅ).|
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The corrections as a function of excitation energy, Ei, are displayed in Fig. 5 for our tabulated sample of weak fictitious lines ( Å, mÅ) of these species. Obviously, the corrections for high-excitation lines of the "majority species'' are similar for all elements, and can amount to -0.1 dex for weak lines with eV. This conclusion is independent of the assumed velocity field which is irrelevant here: for lines as weak as 5 mÅ (see tables).
The dependence of the granulation abundance corrections on line strength was investigated by test calculations for two additional sets of stronger N I lines. The two set of lines were obtained from the set of N I lines ( Å) listed in Table 1 by increasing their equivalent widths, , by a factor of 5 and 15, respectively. Thus, the test lines in first set of have mÅ, those in the second set mÅ. According to the results shown in Fig. 6, the granulation abundance corrections are systematically more positive for the stronger, otherwise identical lines. For Ei=12 eV the difference in between the lines with mÅ and 5 mÅ amounts to +0.07 dex.
The results shown in Fig. 7 indicate that the granulation abundance corrections indeed depend on wavelength, in the sense that lines in the blue part of the spectrum show systematically more negative corrections than those in the red part. This general trend is evident over the whole wavelength range, apart from a slight reversal between and 10 000 Å which is related to the Paschen jump at Å. For Ei=12 eV the difference in between a weak line at and 10 000 Å amounts to -0.1 dex.
|Figure 8: Logarithmic granulation abundance corrections as a function of excitation energy Ei for neutral atoms of predominantly ionized elements, and for molecules CN and MgH. The plot is based on a subset of weak fictitious lines with Å, mÅ, taken from Tables 1-4.|
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|Figure 9: Granulation abundance corrections, , for the 7 fictitious neutral iron lines listed in Table 3, compared with test results obtained by increasing the line strength by factors of 5 and 10, respectively.|
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|Figure 10: Granulation abundance corrections, , for the 7 fictitious neutral iron lines listed in Table 3, compared with test results obtained by moving the weak lines from Å to different spectral regions, centered at , 8000, 10 000, 12 000, and 16 000 Å, respectively, keeping the reduced equivalent width fixed ( mÅ).|
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The corrections for a sample of weak fictitious lines ( Å, mÅ) of these "minority species'' are displayed in Fig. 8 as a function of excitation energy, Ei. We note that the largest granulation effects are found for low-excitation lines of neutral atoms with an ionization potential between 6 and 8 eV. The magnitude of the correction depends on the element and is as large as -0.3 dex for Ti I, Ei=0 eV. Again, we point out that this result does not depend on the assumed non-thermal velocity field.
It is interesting to note that Asplund (2000a), using 3D models, finds corrections in the opposite direction for the "real'' Si I lines listed in Table 2. We emphasize that this is not an indication of fundamentally different "granulation corrections'' in 2D and 3D. The corrections given by Asplund with respect to the Holweger-Müller (1974) model include not only the effect of temperature inhomogeneities but also the influence of the different mean stratifications. Obviously, the difference in the mean structure is the more important of the two opposing effects in this case. Indeed, we find from our own models a total correction of -0.05 dex with respect to the Holweger-Müller model, compared to a the tabulated "granulation correction'' of +0.02 dex, explaining the apparently inconsistent results.
As for N I, the dependence of the granulation abundance corrections on line strength was investigated by test calculations for two additional sets of stronger Fe I lines, obtained from the set of Fe I lines listed in Table 3 by increasing their equivalent widths, , by a factor of 5 and 10, respectively. The test lines in first set of have then mÅ, those in the second set mÅ. The results are shown in Fig. 9. Qualitatively, the granulation abundance corrections for Fe I show the same dependence on line strength as already seen for N I: they are systematically more positive for the stronger, otherwise identical lines.
The corrections for Ei=0 and 1 eV, mÅ, shown in Fig. 9 are certainly too small (too positive), because a significant part of the line absorption comes from layers located outside the model atmosphere on which the line formation calculations are based ( ). This situation results in a stronger underestimation of the equivalent width for the inhomogeneous case. Hence, should actually be more negative, and we can expect abundance corrections dex even for the stronger ground state lines.
We have performed test calculations for five additional sets of Fe I lines, obtained by shifting the standard Fe I lines listed in Table 3 from Å to , 8000, 10 000, 12 000, and 16 000 Å, respectively, enforcing the same reduced equivalent width, , i.e. mÅ.
The results shown in Fig. 10 indicate that the wavelength dependence of the granulation abundance corrections for Fe I has the opposite sign as in the case of N I. Lines in the blue part of the spectrum tend to show smaller (less negative) corrections than those at longer wavelengths. This general trend is clearly evident from to 8000 Å; at longer wavelengths the variation of with is much reduced and no longer strictly monotonic (for an explanation see Paper II).
We have to point out that the results for 16 000 Å are uncertain, because a significant part of the line absorption comes from layers above , so the line formation region is not entirely covered by our model atmosphere.
In an effort to estimate the - hitherto largely unknown - effects of photospheric temperature fluctuations on spectroscopic abundance determinations, we have carried out numerical simulations of the problem of LTE line formation in the inhomogeneous solar photosphere based on detailed 2-dimensional radiation hydrodynamics models of the convective surface layers of the Sun.
For a variety of spectral lines of different elements, we have computed synthetic line profiles from the inhomogeneous 2D hydrodynamical atmosphere and from the corresponding 1D plane-parallel model, respectively. By means of a strictly differential 1D/2D comparison of the emergent equivalent widths, we have derived so called "granulation abundance corrections'' for the individual lines, which have to be applied to standard abundance determinations based on homogeneous 1D model atmospheres in order to correct for the influence of the photospheric temperature fluctuations. The "classical'' problem of finding the most appropriate mean vertical temperature stratification is not addressed here.
Using the concept of "fictitious'' spectral lines, we were able to investigate systematically the dependence of the "granulation abundance corrections'' on the basic line parameters like excitation potential, line strength, wavelength, element and ionization stage. For many lines of practical relevance, it should be possible to estimate the magnitude of the abundance correction by interpolation in the graphs and tables provided in this paper. This approach may often be an acceptable alternative to a detailed fitting of individual line profiles based on hydrodynamical simulations.
In general, we find a line strengthening in the presence of temperature inhomogeneities, implying mostly negative "granulation abundance corrections'', i.e. standard analysis based on plane-parallel atmospheres tends to overestimate abundances. The physical reason for the line strengthening is primarily the non-linear temperature dependence of the line opacity due to thermal ionization and excitation, as demonstrated in Paper II.
One remarkable result is that all lines of our sample with an excitation potential around eV are practically insensitive to granulation effects, regardless of element and ionization stage. Moderate granulation corrections ( dex) are found for weak, high-excitation lines ( eV) of ions and atoms with high ionization potential like N I. The largest corrections are found for ground state lines (Ei=0 eV) of neutral atoms with an ionization potential between 6 and 8 eV like Mg I, Ca I, Ti I, Fe I, amounting to dex in the case of Ti I.
For given excitation potential, the granulation corrections are systematically more positive for stronger lines. The wavelength dependence of , however, depends on the type of line ("majority'' or "minority'' species).
We recall that the corrections account only for the influence of the temperature and pressure fluctuations relative to a given mean structure. We have suppressed possible effects related to 1D/2D differences in the details of the non-thermal velocity field. Such additional effects, which apply to partly saturated lines only, should be small, however, since they are largely eliminated by the proper choice of the microturbulence parameter .
We point out that the magnitude of the "granulation abundance corrections'' derived in this work should be considered as upper limits ( ), because (i) possible NLTE-effects, which were ignored in this study, tend to reduce the fluctuations of the line opacity (e.g. Kiselman 1997; Cayrel & Steffen 2000; Asplund 2000b), and (ii) the amplitude of the temperature fluctuations is systematically overestimated in 2D relative to 3D convection models (see Fig. 4). The large negative corrections found for the low-excitation lines of atoms with low ionization potential are the result of the strongly increasing amplitude of the temperature fluctuations towards the upper photosphere as predicted by our present 2D hydrodynamical simulations. In a more realistic 3D atmosphere, this feature will therefore be less pronounced.
Even though the results are still somewhat uncertain, the present study should be helpful in providing upper bounds for possible errors of spectroscopic abundance analyses, and for selecting spectral lines which are as insensitive as possible to the effects of photospheric temperature inhomogeneities.
For a physical explanation of the numerical results obtained in this work, a detailed investigation of the process of LTE line formation in an inhomogeneous stellar atmosphere based on the analysis of the transfer equation and the evaluation of line depression contribution functions is presented in the second paper of this series (Paper II). With the help of this formalism, we can actually tell how the different atmospheric layers contribute to the granulation correction for any particular spectral line.
In the near future, we plan to extend the computation of abundance corrections to full 3D simulations for the Sun and other types of stars where convection extends into higher photospheric layers and the spectroscopic granulation effects may well be even larger.
|C I||12 700.0||8.7||80.479||78.720||(77.264)||+0.019||(+0.035)||0.007|
|N I||10 000.0||12.0||5.016||5.826||(5.767)||-0.078||(-0.073)||0.015|
|O I||11 500.0||10.0||58.720||58.650||(57.281)||+0.001||(+0.021)||0.008|
|O I||11 302.4||10.74||14.074||14.947||(14.807)||-0.033||(-0.028)||0.008|
|O I||13 164.9||10.99||17.303||17.894||(17.705)||-0.017||(-0.013)||0.006|
The 2D numerical convection simulations and line formation calculations were carried out on the CRAY T94 and CRAY SV1, respectively, at the Rechenzentrum der Universität Kiel. We thank the referee, Paul Barklem, for constructive criticism which helped to significantly improve the contents and presentation of this work.