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Astron. Astrophys. 324, 281-288 (1997)

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5. Results

5.1. Iron

The abundance determination results ([FORMULA]) for our sample of Fe I and Fe II lines are given in Tables 3 and 4. The dependence of [FORMULA] and [FORMULA] on equivalent widths computed for OG and MS 2-D models is shown in Figs. 3 and 4.

5.1.1. Fe I and Fe II lines: [FORMULA] and [FORMULA] values obtained with 2-D models

From the modeling of Fe I   lines we found (Fig. 3):
- qualitatively, the behavior of [FORMULA] and [FORMULA] for OG 2-D models is the same as that obtained in Paper I for 3-D models: there are systematic differences between [FORMULA] and [FORMULA] (for the 10 weak lines given by Table 3 [FORMULA]  dex); [FORMULA] is strongly decreased for strong lines. As the main reason we may consider the deficit of a large-scale velocity field and NLTE effects (Paper I; Gadun 1996).
- The MS models have a statistically more extended spectrum of inhomogeneities. The absorption lines became less deep and wider. Note that [FORMULA] does not decrease even for strong lines. Moreover, the changes of [FORMULA] are weak too. As a result, the mean value of [FORMULA] strongly decreases down to [FORMULA] 0.01 dex for the 10 weak lines of Table 3.


Table 3. Iron abundance from Fe I lines


Table 4. Iron abundance from Fe II lines

[FIGURE] Fig. 3. Iron abundances obtained from equivalent widths (open symbols) and central depths (filled symbols) for Fe I lines as a function of equivalent width. a OG 2-D models: squares correspond to lines with an excitation potential of 0-1 eV, diamonds - 1-2 eV, triangles - 2-3 eV. b Data from MS 2-D models.
[FIGURE] Fig. 4. Iron abundances obtained from equivalent widths (open symbols) and central depths (filled symbols) for Fe II lines as a function of equivalent width. a OG 2-D models. b Data from MS 2-D models.

For Fe II lines we found (Fig. 4):
- the behavior of [FORMULA] and [FORMULA] for OG 2-D models is qualitatively the same one as for Fe I lines: for the 12 weak lines of Table 4 we obtained [FORMULA]  dex, [FORMULA] values drop with the increase of line strength.
- as mentioned above, MS models have a statistically more extended spectrum of inhomogeneities and another temperature structure. As a result, the [FORMULA] values computed for MS are systematically slightly increased in comparison with OG models (Table 4). The behaviour of [FORMULA] changes strongly. For MS models, [FORMULA] values are increased. This effect depends on the line strength. For strong lines with [FORMULA]  pm, [FORMULA] approaches to [FORMULA] - for the two strongest lines of our sample, the mean value of [FORMULA] is [FORMULA]  0.02 dex. But for the 12 weaker lines of Table 4 [FORMULA] remains large: [FORMULA] 0.1 dex.

Comparing these and other results obtained for 3-D models (Gadun 1996), we note that:
1) in the case of the 3-D self-consistent models the problem of the sharp increase of [FORMULA] (due to the decrease of [FORMULA]) has not been solved (Paper I). The problem is solved by extending the statistical spectrum of inhomogeneities in the frame of 2-D MS models.
2) As Solanki & Steenbock (1988) showed, NLTE effects for low-excitation lines of Fe I may be [FORMULA] 0.1 dex for cool HSRA-like models. Low values of [FORMULA], obtained for those lines with MS models in LTE may be a result of overestimation of the thermodynamic parameter and velocity field fluctuations in the upper and middle model photosphere.
3) On the contrary, using lines of Fe II, we have obtained for 2-D MS models too large [FORMULA] for weak lines in comparison with the 3-D case (Paper I). This cannot be explained by NLTE effects, because these effects for Fe II lines are weaker by a factor of 10 with respect to Fe I lines. Possibly, these differences are caused by the details of the structure of 2-D models (see Gadun 1996 for details).

5.1.2. Fe I lines: [FORMULA] for set of 1-D models

We found that:
- the averaged abundance of iron obtained for 1-D models using Fe I lines shows a strong dependence on the temperature structure of models: [FORMULA]. This result may be explained by the well known sensitivity of Fe I lines to the temperature structure of model atmospheres of solar-like stars.
- An impact of temperature inhomogeneities and velocity field on the iron abundance obtained for the Fe I lines is model-dependent. The [FORMULA] of Fe I lines formed in OG inhomogeneous model atmospheres are approximately equal to those corresponding to the averaged (OGA) ones (Table 3). But MS models show lower iron abundances in comparison with homogeneous MSA models. The difference increases up to 0.1 dex because MS models have a richer spectrum of inhomogeneities and a larger amplitude of their fluctuations (see Fig. 1) than OG models.
- The abundance of iron obtained for OGA models does not depend on the averaging procedure (over equal [FORMULA] or h level) of the models. This conclusion agrees well with that obtained previously for 3-D models (Paper I). However, in the case of MSA models we see a more pronounced difference between the results obtained for models averaged over [FORMULA] and h, because in the middle photosphere the differences in temperature structures of MSA models are more pronounced with respect to the OGA ones. We already mentioned above that in 2-D MS models there are convective flows of larger horizontal scales. They can penetrate into higher photospheric layers in comparison with modest overshooting flows in 2-D OG models.
- As mentioned above, our "cool" 3-D models were computed in the frame of a rough model of the radiative transfer without taking into account the effective warming by spectral lines. Besides, "hot models" were obtained with overestimation of the effect of line warming. As a result, "hot" and "cool" 3-D models give too large and too small iron abundances for Fe I lines, respectively. Note that our new iron abundances obtained for Fe I lines differ strongly from those of 3-D models (Paper I).

5.1.3. Fe II lines: [FORMULA] for set of 1-D models

Fe II lines are less sensitive to the temperature structure of the solar-like atmospheres in comparison with Fe I lines (see, for example, GK89). For our grid of 1-D model atmospheres we obtained a similar result. However, we point out that iron abundances derived for 1-D models: HOLMU, K93, and PK79, differ from averaged 2-D models: OGA and MSA. Averaged 2-D models show lower iron abundances due to the higher temperature gradient in the lower photosphere. In our work we used oscillator strengths of Fe II lines computed by GK89 for the HOLMU model atmosphere and A  = 7.64. Due to differences from the GK89 velocity field model and differences in computation procedure we have got a cloud of [FORMULA] values for our Fe II lines (Table 4).

The differences between 2-D and averaged models are less pronounced than for Fe I lines and are practically the same for both OG and MS model atmospheres. In the last case the differences do not exceed 0.05 dex.

The iron abundances obtained depend on the averaging procedure: for h -averaged models the abundances are higher. For OGA models this difference is not significant ([FORMULA] 0.02 dex), and the effect is more evident for MSA ([FORMULA] 0.05 dex).

Iron abundances obtained for 2-D models may be larger or smaller in comparison with values obtained for averaged models. The effect depends on the averaging procedure.

Iron abundance determined for 2-D models using Fe II lines agrees well with the value for 3-D models (Paper I), due to the low sensitivity of Fe II lines to temperature in solar-like atmospheres.

5.2. Lithium

We computed the strongest Li I resonance doublet line [FORMULA] 670.776 nm in the spectrum of the solar disk center. Observations provide [FORMULA]  = 0.18 pm (GK89).

5.2.1. 1-D models

In the frame of this work we studied the impact of the convective overshooting (Kurucz 1993) in 1-D model atmospheres on results of the lithium abundance determination. Two models with and without CoOv were computed with SAM92. We found that the lithium abundances [FORMULA] obtained for models with and without convective overshooting are 0.835 and 0.855, respectively. This means that the influence of CoOv on the Li abundance determinations in the solar atmosphere is rather weak.

5.2.2. 2-D models

Lithium abundances computed with 2-D and averaged models are given in Table 5. The value of [FORMULA] for 2-D inhomogeneous models is 0.05-0.07 dex, which lies between the values obtained for Fe I and Fe II lines.


Table 5. Lithium abundances

The impact of inhomogeneities on the lithium abundance, which may be estimated as a difference between the lithium abundances obtained from 2-D models and from averaged ones, is [FORMULA] 0.1-0.2 dex. We point out that these results are similar to those obtained for Fe I lines.

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© European Southern Observatory (ESO) 1997

Online publication: May 26, 1998