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

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12. Conclusions

The convection theory, as it used in the ATLAS9 code, has been reviewed and its effects on the model structures and on the determination of model parameters have been investigated.

As far as the theory for convection is concerned, all the models of the Kurucz (1993a, 1994, 1995) grids with [FORMULA] included between 8750 K and 3500 K have convection computed with a modified mixing-length approach, which allows for the presence of a positive convective flux above the Schwarzschild surface. Although this modification is called "overshooting", it is not the physical "overshooting" as usually defined (Renzini, 1987), but it is rather similar to a smoothing of the convective flux. Because the final convective flux at each layer is defined in ATLAS9 as the maximum value between the mixing-length flux without "overshooting" and mixing-length flux with "overshooting", the final total convective flux of the atmosphere is generally larger than both the unsmoothed and the smoothed fluxes. The effect of the "overshooting" is to decrease the total radiation when it emerges from atmospheric layers deep enough to be affected by the mixing-length convection. In particular for the Sun, the radiation mostly affected by convection lies in the 440-600 nm region. Increasing the mixing-length parameter [FORMULA] from 1.25 (the value assumed for the grid of models) to 2 decreases the emerging radiation to less extent than the "overshooting " modification.

We analyzed the Kurucz solar model (SUNK94) and we showed that the mixing-length theory modified for the "overshooting" yields a solar model which very well fits the observed solar irradiance, the energy distribution from the disk center, the limb-darkening curves, the normalized Mg I strong line at 517.27 nm. However, this solar model does not reproduce the observed [FORMULA], [FORMULA], and [FORMULA] profiles normalized to the continuum level. The computed profiles have wings weaker than the observed ones, so that the solar [FORMULA] should be increased up to about 5900 K in order to fit the observations.

A solar model with the same parameters of the SUNK94 model, but computed without "overshooting" (SUNNOVERC125), also well reproduces the solar irradiance and the normalized Mg I strong line at 517.27 nm, but it does not reproduce the energy distribution from the disk center and the limb darkening curves. Viceversa, it fits very well the Balmer lines.

Neither the SUNK94 nor the SUNNOVERC125 models are able to reproduce the level of the solar true continuum inferred by high-resolution observations, which we found to be lower than the computed one at least in the [FORMULA], [FORMULA], and Mg I 517.27 nm regions. This discrepancy decreases with increasing wavelength so that it is on the order of 6.6% in the [FORMULA] region and disappears in the [FORMULA] region. The difference between the observed and computed continua is much larger than that yielded by the different options for the mixing-length convection. Therefore, if we assume that the Neckel & Labs (1984) calibration for the Sun is correct, we must conclude that either a completely different convection theory has to be used for computing cool models, or that velocity fields have to be considered in computing models (Kurucz, 1996), or that some opacity source has to be revised. In this last case, our analysis has shown that, among the known opacity sources, the most important one for the visible range is the well known H opacity, followed by the less well known line blanketing. H opacity is estimated to be computed with an accuracy better than 1% or 2% (John, 1988), so that future investigations on the line blanketing could further improve our understanding of the discrepancy between observed and computed solar continua.

The final conclusion about solar models is that the SUNK94 model fits more solar data than the SUNNOVERC125 model, but that it is still not able to reproduce all the observations.

For other stars, the comparison of K95 with NOVER models has shown that the convection option affects mostly the models with [FORMULA] included between 8500 K and 5000 K, where the upper limit for [FORMULA] decreases with decreasing gravity. In fact, for solar metallicity, and [FORMULA] =1, only the models with [FORMULA] included between 6000 K and 5500 K show a different structure for the different convections. The number of models affected by the "overshooting" option slightly increases toward the lower temperatures with decreasing metallicity.

Color indices and Balmer profiles may be used to derive stellar parameters. We investigated the effect of the convection on [FORMULA] derived from ([FORMULA]), ([FORMULA]), ([FORMULA]) indices, and from [FORMULA], [FORMULA], and [FORMULA] profiles. We also investigated the effect of convection on the values of [FORMULA] derived from the c1 index. The ([FORMULA]), ([FORMULA]), and ([FORMULA]) indices are almost unaffected by the different convection for [FORMULA] [FORMULA] 6250 K, while for [FORMULA] [FORMULA] 6250 K the differences in the effective temperatures may amount up to 60 K, 100 K, and 180 K respectively. Therefore, [FORMULA] is nearly unaffected by the details of convection. Because it is also nearly independent of gravity and metallicity, it is a very well suited index to fix [FORMULA]. The ([FORMULA]) index depends on metallicity and, for [FORMULA] [FORMULA] 6500 K, it depends also on gravity. The ([FORMULA]) index is the most heavily affected by gravity and for 4500 K [FORMULA] [FORMULA] [FORMULA]  7000 K it depends also on metallicity. For these indices, errors on the order of 0.5 dex in gravity or in metallicity, would yield differences in [FORMULA] of about 200 K, therefore larger than those yielded by a different treatment of the mixing-length convection.

As far as the Balmer lines are concerned, the differences [FORMULA] [FORMULA]  related with the "overshooting" or "no overshooting" option are significant within [FORMULA] =5250 K and [FORMULA] =8750 K, and may amount up to 340 K. The convection option affects the wings of the Balmer profiles more than the level of the continuum.

When the K95 and NOVER models yield different temperatures, [FORMULA] from the K95 models are generally larger than [FORMULA] from the NOVER models.

The comparison of gravities derived from the c1 index has shown that the difference in the value of the gravity as a function of [FORMULA], when K95 and NOVER models are used, grows up to a maximum value for each temperature. The maximum [FORMULA] decreases with increasing temperatures and ranges, for solar metallicity, from 0.7 dex at [FORMULA] =0.5 and [FORMULA] =5500 K to 0.2 dex at [FORMULA] =4.5 and [FORMULA] =8000 K. The behaviour of the plot [FORMULA] versus [FORMULA] is very similar for [M/H]=-3.0.

The conclusion is that, for each model, the "overshooting" and the "no overshooting" mixing-length convection affects the different spectral regions and the spectral features in a different way. The most affected quantities are the Balmer lines, followed by the indices c1, ([FORMULA]), ([FORMULA]), and ([FORMULA]). This last index is almost independent of the convection treatment. Therefore, for solar metallicity, the differences in [FORMULA] when "overshooting" and "no overshooting" models are used, range from about 340 K in the case of the Balmer lines up to 60 K in the case of the ([FORMULA]) index. Gravity differences may amount up to 0.7 dex.

To try to state whether "overshooting" or "no overshooting" models give predictions more close to the observations, we compared, for a sample of stars, [FORMULA] derived from color indices with [FORMULA] derived from the infrared flux method (IRFM); then, we compared for a sample of 7 metal-poor stars, [FORMULA] derived from the [FORMULA] profiles with [FORMULA] derived from the ([FORMULA]) indices. Finally, we used Procyon to test the consistency of [FORMULA] and [FORMULA] derived from different methods when the K95 and the NOVER models are used.

We found that, on average, the ([FORMULA]), ([FORMULA]), and ([FORMULA]) color indices yield [FORMULA] larger from 30 K up to 190 K than those derived from the infrared flux method (IRFM). The differences are smaller for [FORMULA] [FORMULA] 6250 K than for [FORMULA] [FORMULA] 6250 K and they are always larger for the K95 models than for the NOVER models. Therefore, on the basis of the IRFM method, NOVER models should be preferred.

When, for the given sample of stars, [FORMULA] derived from the ([FORMULA]) index is compared with [FORMULA] derived from the ([FORMULA]) index, the general agreement is slightly better for the NOVER models than for the K95 models. The comparison of [FORMULA] from the ([FORMULA]) index with [FORMULA] from the [FORMULA] profiles has shown that for six out of seven metal-poor stars considered the agreement is much better for the NOVER models than for the K95 models. Therefore, in general, NOVER models seem to be more consistent than the K95 models.

Finally, for Procyon, K95 and NOVER models yield average effective temperatures 6744 [FORMULA] 94 K and 6593 [FORMULA] 91 K respectively, when different methods are used for deriving them. [FORMULA] from the NOVER models is more close to [FORMULA] =6560 [FORMULA] 130 K derived by Smalley & Dworetsky (1995) from the angular diameter and the integrated flux than [FORMULA] from the K95 models. The higher [FORMULA] would require an angular diameter much lower than the observed one. As far as the gravity is concerned, the c1 index and NOVER models yield [FORMULA] =4.00 for [FORMULA] =6593 K, while c1 index and K95 models yield [FORMULA] =3.85 for [FORMULA] =6744 K. Therefore the value of gravity from the NOVER models is closer to that from the mass and radius ([FORMULA] =3.95-4.08) than the gravity from the K95 models. However, because for both K95 and NOVER models, the scatter of the average [FORMULA] from the individual determinations is on the order of 90 K, we should conclude that, as for the Sun, none of the models is able to reproduce the whole observed spectrum.

In conclusion, color indices, Balmer profiles, and Procyon analysis have lead us to give more weight to the ATLAS9 models computed with the standard mixing-length theory than to those with the "overshooting" option switched on. Only exception is the Sun, which is better reproduced by the SUNK94 model than by the SUNNOVERC125 model. In any case, the analysis of both the Sun and Procyon has shown that uncertainties in [FORMULA] on the order of 200 K have always to be expected, whether K95 or models without "overshooting" are used.

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© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998