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Astron. Astrophys. 363, 1091-1105 (2000)

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8. Comparison with solar observations

The comparison with observed solar spectra requires the use of a photospheric model. In this work we use one-dimensional plane-parallel models which are freely available. Namely these are the semi-empirical solar model of Holweger & Müller (1974) hereafter HOLMUL, a MARCS theoretical solar model (Asplund et al. 1997), and two Kurucz theoretical solar models (Kurucz 1993; Castelli et al. 1997). The two types of Kurucz models used, that with and that without convective overshooting, are hereafter KOVER and KNOVER models respectively. For both MARCS and Kurucz models we use the default mixing length parameters [FORMULA], namely 1.25 for Kurucz and 1.5 for MARCS. We retain the default structure parameters y, namely [FORMULA] for Kurucz and [FORMULA] for MARCS.

It is expected that MARCS and KNOVER models are quite similar as they are both based on essentially the same physics and "standard" mixing length convection theory although with different parameters. For the solar models used here, the computed Balmer line profiles of MARCS and KNOVER were in excellent agreement. Hence below we will only discuss the KNOVER model. However we caution that this agreement may not extend to other stellar parameters. In the KOVER models Kurucz has introduced "approximate overshooting" to the convection treatment. The approximate overshooting assumes "the centre of a bubble stops at the top of the convection zone so that there is convective flux one bubble radius above the convection zone. That flux is found by computing the convective flux in the normal way and then smoothing it over a bubble diameter" (Kurucz 1992).

The purpose of this comparison is to test the broadening theory, not the models or convection treatments. The validity of the theory is tested by comparison of Balmer line results with those of other model predictions such as limb-darkening curves. This situation is clearly not ideal due to uncertainties in the models and the particular sensitivity of Balmer lines to deep layers and convection treatment. However, lack of laboratory data makes this our best option at present. 3D convective models will be investigated in future.

8.1. Profile comparisons

Limb-darkening curves are a powerful test of solar models. On this basis alone HOLMUL is the preferred model as it reproduces limb-darkening curves better than either KOVER, KNOVER or MARCS (Blackwell et al. 1995; Castelli et al. 1997). However Castelli et al. (1997) found that in spite of KNOVER being unable to reproduce limb-darkening curves as well as KOVER it produces a better fit to hydrogen line profiles when Ali & Griem (1966) theory is used.

As HOLMUL is the preferred model on the basis of limb-darkening data and its ability to reproduce the behaviour of a large sample of strong metallic lines, computed synthetic profiles for HOLMUL using both our self-broadening theory and Ali & Griem (1966) theory are compared with the observed solar flux spectrum of Kurucz et al. (1984, NSO/Kitt peak FTS data) in Fig. 8. We do not adjust the H[FORMULA] continuum here as in Paper I. It is seen that for all three lines our broadening theory reduces the discrepancy with observation but the remaining discrepancy is still significant. As line blending is significant, particularly in H[FORMULA] and H[FORMULA] profiles, computations were performed which included all available lines from VALD, the Vienna Atomic Line Database (Kupka et al. 1999) for all three Balmer lines. The predicted residual fluxes with and without blending were found to be in good agreement in the windows between the blending lines, as shown for H[FORMULA] in Fig. 9. However the inclusion of blending lines does not change the conclusion that the synthetic profiles are too weak to match the observations. As there are many blending lines without data or unidentified, particularly for H[FORMULA] and H[FORMULA], there is some element of uncertainty in this conclusion.

[FIGURE] Fig. 8. Comparisons of synthetic flux profiles with observations (NSO/Kitt Peak FTS data) for the sun for H[FORMULA] (top), H[FORMULA] (middle) and H[FORMULA] (bottom). All models lines use the HOLMUL model. The full line uses our self-broadening theory while the dashed line uses the Ali & Griem (1966) theory.

[FIGURE] Fig. 9. Comparison of synthetic flux profiles with observations (double line - NSO/Kitt Peak FTS data) for the sun for H[FORMULA] both with (full line) and without (dashed line) blending lines from VALD, both employing the HOLMUL model and our self-broadening theory. Macroturbulence of 1.6 km s-1 and rotational velocity of [FORMULA] km s-1 are used.

In Fig. 10 predicted Balmer line profiles using our theory for HOLMUL, KOVER and KNOVER solar models are compared with the observed spectrum. KNOVER predicts profiles for all lines that are generally too strong. If blending lines are included the discrepancy is even greater so KNOVER is now a model which fits neither the limb-darkening nor the Balmer line profiles and is therefore strongly ruled out by our line-broadening theory. For H[FORMULA] and H[FORMULA] the synthetic profiles obtained using the KOVER and HOLMUL models are in good agreement with each other but are insufficiently strong to match the observed profiles. In the outer parts of these profiles the discrepancy may be due the failure of the impact approximation (see Table 3) an inadequate temperature structure or both. In the far wings of H[FORMULA] the synthetic profiles obtained using the KOVER and HOLMUL models are again too weak. Within 5 Å of line centre the profile predicted by the KOVER model is too strong which weakly favours the HOLMUL model. The observed core of the line, within 0.7 Å of line centre, the observed profile is much stronger than any of the synthetic profiles. This part of the line is formed in the low chromosphere which is not included in our synthetic modelling.

[FIGURE] Fig. 10. Comparisons of synthetic flux profiles with observations (NSO/Kitt Peak FTS data) for the sun for H[FORMULA] (top), H[FORMULA] (middle) and H[FORMULA] (bottom). Full and dashed lines use the KOVER and KNOVER models respectively, and the dot-dashed lines use the HOLMUL model.

In summary our self-broadening theory is superior to the Ali & Griem (1966) theory because it reduces the discrepancy between the observed and computed Balmer line profiles when the preferred HOLMUL model is used and leads to the KNOVER model being discarded thus resolving the dilemma posed by a model which provides the best match to the Balmer line profiles but fails to match limb-darkening curves when the Ali & Griem (1966) theory is used. In spite of these successes significant discrepancies remain between theory and observation. However the behaviour of the KNOVER model suggests that it may be possible to construct a model with a temperature structure somewhere between the HOLMUL and KNOVER models which provides the best simultaneous match to the limb darkening curves and the H[FORMULA] profile where the validity of the impact approximation is not an issue. The impact approximation is an important issue for H[FORMULA] and H[FORMULA]. Fitting of the profiles of these lines should be confined to the detunings indicated in Table 3 and even then with some caution as these detunings correspond to the extreme limit of validity of the impact approximation.

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Online publication: December 5, 2000