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Astron. Astrophys. 318, 841-869 (1997)

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4. The solar model and the value of the mixing-length

Owing to the availability of absolute values for the solar central disk radiation and for the irradiance, the Sun is usually adopted to test the model predictions for cool stars. The variation of the solar opacity with frequency allows looking to different physical depths in the atmosphere, so that disk-center absolute intensity and irradiance can be used over an extended range of wavelengths in order to test the run of the temperature versus optical depth in a computed solar model. In the Sun, the peak of the emitted radiation as a function of wavelength is in the range from 410 to 510 nm, so that this is one of the best suited regions for getting information on the deepest layers where convection occurs.

In the next subsections we compare some observed quantities, such as the disk central intensity, the irradiance, the limb-darkening curves, ([FORMULA]), ([FORMULA]), ([FORMULA]) color indices, Balmer profiles, and the Mg I line at 517.27 nm, with the quantities computed from solar models differing only in the way that ML convection is handled. We will show that the SUNK94 model yields the closest agreement with the observations. The only exception are Balmer profiles which would require a [FORMULA] 100-150 K higher than that of the Sun in order to fit the observed profiles.

4.1. Comparison between observed and computed disk central intensities and irradiances

The parameters of the SUNK94 model are [FORMULA] =5777 K, [FORMULA] =4.4377, microturbulent velocity [FORMULA] =1.5 km s-1 for the line blanketing, and abundances from Anders & Grevesse (1989). We adopted these same data for all the solar models we computed for this paper. We use elemental abundances relative to the total atomic number density. On this scale, the hydrogen and helium solar abundances are [FORMULA] / [FORMULA] =0.911 and [FORMULA] / [FORMULA] =0.089.

Observed low-resolution spectra are those from Neckel & Labs (1984) for the visible region and from Labs et al. (1987) for the ultraviolet region. The spectra for the visible region, from 329.8 to 1247.0 nm, consist of absolute 20 Å integrals for both the central intensity [FORMULA] (0) and the disk-averaged radiation (mean intensity) F([FORMULA]). We associated to the central wavelength of each 20 Å interval the values of [FORMULA] (0) and F([FORMULA]) averaged over 20 Å. F([FORMULA]) was converted to irradiance in order to be consistent with the ultraviolet data. In fact, Labs et al. (1987) tabulated irradiances, averaged over each passband, from 200 to 358 nm. These solar spectra are the same data used by Kurucz (1992).

The comparison of the observed [FORMULA] (0) with that computed with the ML convection modified only for the horizontally averaged opacity (Sect. 2.2) has shown that the computed [FORMULA] (0) is larger than the observed one in the 410-510 nm region; the discrepancy slightly decreases with increasing [FORMULA]. It goes from about 7% for [FORMULA] =1.25 to about 5% for [FORMULA] =2.0 (Fig. 2a and 2b). This small variation of [FORMULA] (0) with the mixing-length parameter shows that the emerging radiation is weakly affected by the specific value assumed for the [FORMULA] parameter.

[FIGURE] Fig. 2. Comparison between observed (dashed line) and computed (full line) intensities [FORMULA] (0) from the center of the solar disk. Observations are from Neckel & Labs (1984). a and b Computed intensities correspond to [FORMULA] =1.25 and [FORMULA] =2.0 respectively. In both cases solar models are computed with the "overshooting" option switched off. c [FORMULA] (0) is from the SUNK94 model ([FORMULA] =1.25,"overshooting" option switched on). d [FORMULA] (0) is from a solar model computed with the only ML, [FORMULA] =2.0, and k=1 in formula (1); e [FORMULA] (0) is from the solar model No. 743 of the Kurucz (1979b) grid computed with ATLAS6

The "approximate overshooting" (Sect. 2.3) is much more effective in reducing the computed [FORMULA] (0) than the simple increasing of the [FORMULA] value. Kurucz (1992) assumed [FORMULA] =1.25 as giving the best fit between the observed and computed irradiances when convection is represented by the ML theory modified for both the horizontally averaged opacity and the "approximate overshooting". Actually [FORMULA] (0) yielded by the SUNK94 model ([FORMULA] =1.25) is generally larger than the observed one by about 3% in the 465-480 nm region (Fig. 2c). A mixing-length value [FORMULA] =2.0 would still improve the fit.

Because the mixing-length theory is far from being a rigorous theory, it is easy to obtain for the Sun the same results yielded by the SUNK94 model by dropping the two modifications to the mixing-length and by changing some of the free parameters. For instance, the same difference of about 3% between the observed and computed disk central intensity can be obtained by assuming k=1 in the flux formula (1) and [FORMULA] =2.0 (Fig. 2d).

Finally, Fig. 2e shows the solar central intensity [FORMULA] (0) computed from the solar model No. 743 of the old ATLAS6 grid (Kurucz, 1979b). For these ATLAS6 models the mixing-length parameter was assumed to be [FORMULA] =1.0. The comparison of Fig. 2e with Fig. 2c points out the big improvement attained by the SUNK94 model.

When the observed irradiance is compared with that computed from the SUNK94 model the result is the same as that already shown by Kurucz (1992) and obtained with a previous solar model (SUNK92). Observed and computed solar irradiances agree very well in the whole 200-1100 nm region (Fig. 3a). The SUNK94 model differs from the SUNK92 model mostly for the number of layers (72 instead of 64) which extend toward lower optical depths, for a better treatment of the radiation emerging from the uppermost layers, and for a few changes in some opacities routines, as that for H .

[FIGURE] Fig. 3. a Comparison between observed (dashed line) and computed (full line) solar irradiances. Observations are from Neckel & Labs (1984) for the visible region and from Labs et al. (1987) for the ultraviolet region. Solar model is SUNK94 given in Table 1. b Comparison between irradiances computed from models which differ only for the "overshooting" option. It is "on" in the SUNK94 model (full line) and "off" in the SUNNOVERC125 model (dashed line)

Irradiances from the SUNK94 model and from a model with the "overshooting" option switched off and [FORMULA] =1.25 (SUNNOVERC125), differ each from the other not more than 2.2% in the 400-500 nm region (Fig. 3b). For a mixing-length parameter [FORMULA] =2.0 the model without "overshooting" (SUNNOVERC20) yields an irradiance which differs not more than 1% from that yielded by the SUNK94 model. The SUNK94 irradiance fits better the observations than the SUNNOVER irradiances. In conclusion, when the SUNNOVERC125 model is used, the difference between the observed and computed irradiances is smaller than that between the observed and computed central-disk intensities (Fig. 2a-2b)

4.2. The limb-darkening curves

Observational profiles of the continuum solar limb darkening at a number of wavelengths from 303.327 to 729.775 nm and from 740.4 to 2401.8 nm can be derived from the tables of Pierce & Slaughter (1977) and Pierce, Slaughter & Weinberger (1977) respectively. Blackwell et al. (1995) compared the observed limb-darkening curves with those predicted by the SUNK92 model. They concluded that for [FORMULA] 500 nm the observed [FORMULA] (cos [FORMULA])/ [FORMULA] (0) are larger than the computed ones, while for wavelengths shorter than 500 nm computations fit observations very well. We obtained the same results from the SUNK94 model. However, Fig. 4a shows that the agreement between observation and computations is not too bad up to nearly 600 nm. The difference increases in the whole wavelength range from 400 to 2400 nm when SUNNOVERC125 or SUNNOVERC20 models are used to compute the solar intensities [FORMULA] (cos [FORMULA]) (Fig. 4b).

[FIGURE] Fig. 4. Comparison between observed (points) and computed (full line) solar limb-darkening curves [FORMULA] (cos [FORMULA])/ [FORMULA] (0). Observations are from Pierce & Slaughter (1977) and Pierce, Slaughter & Weinberger (1977). Computed limb-darkening curves are from models which differ only for the "overshooting" option. a it is "on" (SUNK94 model) b it is "off" (SUNNOVERC125 model). The different curves correspond to different values of cos [FORMULA]

A fact to be considered when observed limb-darkening curves are compared with the computations is that the line opacity is negligible only for few wavelengths, as we can infer from the analysis of high-resolution spectra (Kurucz et al, 1984). This implies that the continuum windows selected by Pierce & Slaughter (1977) and Pierce et al. (1977) may not always correspond to the real continuum at several wavelengths.

4.3. Color indices

Table 2 compares ([FORMULA]), ([FORMULA]), and ([FORMULA]) observed color indices with those derived from the SUNK94 and the SUNNOVERC125 models. Owing to the uncertainty in the solar colors and to the small differences between the SUNK94 and the SUNNOVERC125 colors we cannot state which model has to be preferred.


[TABLE]

Table 2. Observed and computed color indices for the Sun


4.4. The Balmer profiles

Comparison of Balmer profiles from BALMER9 with those from SYNTHE has shown that the metallic lines do not affect the shape of the wings of the [FORMULA] and [FORMULA] profiles. The violet wing of [FORMULA] predicted by the synthetic spectrum is a little bit broader than that predicted by BALMER9, owing to the presence of a strong Fe I line at 432.576 nm.

Observed high-resolution spectra were taken from the Solar Flux Atlas of Kurucz et al. (1984). We lowered the continuum by 1% in the [FORMULA] region, because a close analysis of the way the continuum was drawn has shown that the continuum could have been drawn too high in this spectral range.

Fig. 5 compares the normalized observed profiles with BALMER9 profiles computed both with the SUNK94 model (thin line) and the SUNNOVERC125 model (thick line) respectively. The SUNK94 model yields computed wings weaker than the observed ones for all [FORMULA], [FORMULA], and [FORMULA] profiles. The SUNNOVERC125 model reproduces very well all the observed Balmer lines.

[FIGURE] Fig. 5. Comparison between the solar observed spectrum and computed Balmer profiles: from top to bottom: [FORMULA], [FORMULA], [FORMULA] normalized to the continuum level. Computed profiles are from the SUNK94 model (thin line) and from the SUNNOVERC125 model (thick line). The models differ for the "overshooting" option which is switched "on" and "off" respectively

Fuhrmann et al. (1993, 1994) extensively discussed the comparison between observed and computed Balmer lines in cool stars. They showed that the wings of [FORMULA] are independent of the value of the mixing-length parameter for stars with solar metallicity and for metal poor stars with [FORMULA]   [FORMULA]  5500 K. Viceversa, the wings of [FORMULA] and of the higher series members depend on it. Therefore they adopted [FORMULA] wings to fix the effective temperature and the wings of [FORMULA] to fix the value of [FORMULA]. Finally they were able to compute the model (with [FORMULA] from [FORMULA] and [FORMULA] from [FORMULA]) which well reproduces the observed and computed [FORMULA] and [FORMULA] profiles at the same time. They derived [FORMULA] =0.5 for the Sun.

Our results for the Sun are different from those of Fuhrmann et al. (1993). The SUNNOVERC125 model, in which the convection is similar to that adopted by Fuhrmann et al. (1993), well reproduces all the profiles without any need of changing the [FORMULA] value. When ATLAS9 models with "overshooting"are used, [FORMULA] must be increased up to 5875 K or 5900 K to fit the Balmer profiles, but also in this case all the profiles are well reproduced by the same [FORMULA] =1.25. By the way, the mixing-length parameter should be [FORMULA] 1.25 instead of 0.5, as we have derived from the comparison of observed and computed energy distributions.

Our results for the Sun are different from those of Fuhrmann et al. (1993) probably because a different solar model has been used by us and by Fuhrmann et al. (1993) when computing the profiles. In particular, the solar model used by Fuhrmann et al. (1993) is based on continuum and line opacities different from those adopted in the SUNK94 and SUNNOVERC125 models. Also, the ML convection is different. We already showed in Fig. 2e the effect on the disk-center intensity of different ATLAS versions.

By using our same codes and input data, but different observations, Van't Veer & Mégessier (1996) found for the Sun results very similar to those derived by Fuhrmann et al. (1993) and therefore different from the ours. We also found, as Van't Veer & Mégessier (1996), that [FORMULA] computed from the SUNK94 model is weaker than the observed one. However, we derive an effective temperature about 100 K higher than that of the Sun, while Van't Veer& Mégessier (1996) require for the Sun 6200 K in order to fit at best both [FORMULA] and [FORMULA] profiles. We found that the same model, regardless of whether the "overshooting" option is switched on or off, fits both [FORMULA] and [FORMULA] profiles. In contrast with us, Van't Veer & Mégessier (1996) showed that the SUNNOVERC125 model ([FORMULA] =1.25) well fits [FORMULA], but yields too weak computed wings for [FORMULA], so that they lowerd [FORMULA] up to 0.5, in analogy with Fuhrmann et al. (1993). Therefore, they suggested to adopt for the Sun a SUNNOVERC05 model ([FORMULA] =0.5), because they found that this solar model yields agreement between the observed and computed profiles for both [FORMULA] and [FORMULA] lines. This suggestion of Van't Veer & Mégessier (1996) of lowering [FORMULA] up to 0.5 is inconsistent with the need of a value for [FORMULA] equal to 1.25 or even larger in order to fit observed and computed energy distribution (Fig. 2a and 2b). We note that the mixing-length parameter should be fixed from both the energy distribution and wings of [FORMULA] profiles. In fact, normalized Balmer profiles change with [FORMULA] mostly for the effect of [FORMULA] on the continuum level. Fig. 6 compares both absolute [FORMULA] profiles and continuum levels corresponding to two solar models differing only for the value of [FORMULA], which is 0.5 and 1.25 respectively. The level of the continuum increases when [FORMULA] decreases and it is much more affected by the convection than the wings, which become only a little bit broader when [FORMULA] decreases. Finally, because different [FORMULA] yield different levels for the continua, but almost the same wings, it follows that when we normalize the absolute profiles to the continuum level, the total effect of decreasing the mixing-length is to yield wings remarkably broader than those of the absolute profiles, mostly because the continuum level has increased.

[FIGURE] Fig. 6. Comparison between [FORMULA] profiles in absolute flux units computed from solar models which differ only for the value of the mixing-length parameter [FORMULA]. Thin line is for [FORMULA] =1.25 (SUNK94 model), thick line is for [FORMULA] =0.5. The ordinate is the flux [FORMULA] in 9.5 106 erg cm-2 sec-1 nm-1

A possible explanation for the disagreement between our results and those of Van't Veer & Mégessier (1996) could be the different way of placing the level for the continuum. To further investigate this hypothesis, we analyzed Balmer profiles in absolute flux units. We converted the normalized spectrum of the Kurucz et al. (1984) Flux Atlas in absolute flux units by using the irradiance values of the pseudo-continuum tabulated in Kurucz et al. (1984). It came out that absolute profiles were not helpful for clarifying the disagreement because the solar computed continuum results higher than that inferred from the observed spectrum, so that a direct comparison of the wings of the observed and computed profiles is impossible (Fig. 7). In the wings of [FORMULA] the continuum from the SUNK94 model is about 6.6% higher than that deduced from the observed spectrum. The discrepancy decreases with increasing wavelength, so that the difference nearly disappears at [FORMULA]. The SUNNOVERC125 model yields a still larger difference between the observed and computed continuum levels, because, as we already discussed above, the "overshooting" effect is to decrease the emergent radiation.

[FIGURE] Fig. 7. From top to bottom: comparison between observed (thick) and computed (thin) solar [FORMULA], [FORMULA], and [FORMULA] profiles in absolute flux units. Computed profiles are from the SUNK94 model. The computed continuum is also drawn on the figure. The ordinate is the flux [FORMULA] in 7.0 106 erg cm-2 sec-1 nm-1 for [FORMULA] and in 9.5 106 erg cm-2 sec-1 nm-1 for [FORMULA] and [FORMULA]

The comparison of Fig. 6 with Fig. 7 shows that there are no reasonable mixing-length values able to decrease the computed continuum to the level of the observed one. We obtained the same results by replacing the profiles from the Kurucz et al. (1984) Atlas by those from the Neckel (1987) Atlas. The disagreement between the observed and computed continua does not change when profiles averaged over the disk are replaced by profiles from the disk-center.

In conclusion, the discrepancy between our results and those of Van't Veer & Mégessier (1996) could be due to the different observations and/or to different continuum levels.

4.5. The Mg I b line

Holweger (1967, 1979) pointed out that the strong Mg I line at 517.27 nm is well suited to test the structure of the solar model atmosphere, because the profile emerges from several depths in the atmosphere and it is almost independent of microturbulent velocity and radiative damping. Therefore, once the [FORMULA] value, the van der Waals damping constant [FORMULA], and the Mg abundance are known, the comparison of the computed and observed profiles may give information on the quality of the model structure. Line data in the Kurucz (1993b) line lists for Mg I 517.27 nm are: [FORMULA] (from Anderson et al., 1967) [FORMULA]  sec-1, and [FORMULA]  sec-1. The Van der Waals damping constant is derived from the classical value [FORMULA]  sec-1 scaled for the correction factor [FORMULA] log C6 =1.2 (Gigas, 1988). However, [FORMULA]  sec -1 yields a computed profile everywhere broader than the observed one, so that we preferred to fix [FORMULA] from the comparison of the observed and computed profiles. This empirically derived value well agrees with [FORMULA]  sec -1 obtained for T=10000 K from the line broadening cross-sections from Anstee & O'Mara (1995). The temperature dependence of the Anstee & O'Mara (1995) computed linewidths is T0.38, while it is T0.3 in Kurucz' codes. Fig. 8 shows the comparison between the observed and computed profiles for [FORMULA]  sec -1 from Anstee & O'Mara (1995) and by assuming a temperature dependence equal to T0.38 for it. The results do not change in an appreciable way when the temperature dependence is T0.30. The agreement between the observed and computed normalized profiles shows that, within the error limits for the Van der Waals constant, the structure of the SUNK94 solar model is correct.

[FIGURE] Fig. 8. Comparison between solar observed (thick line) and computed (thin line) normalized Mg I profiles at 517.27 nm.
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© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998
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