Astron. Astrophys. 338, 637-650 (1998)

3. Sodium in the solar photosphere

Na I departure coefficients in the solar atmosphere are best characterized by the overpopulation of the ground state and the first excited states in higher atmospheric layers beyond . The departure coefficients diverge outside . The excited states follow the overpopulation to a lesser extent and only in deeper atmospheric layers; therefore the strongest deviations from LTE in the photosphere should be expected for the subordinate lines arising from the states near 2.1 eV. The origin of the overpopulation of the two lowest levels is the photon suction process described in detail by Bruls et al. (1992). It is triggered by photon losses in the lines. The strong resonance lines are optically thick over a wide depth range. Due to the reduced efficiency of collisions in the higher atmospheric layers the collisional de-excitation rates are small and the photons can be scattered over longer distances without being thermalized. Besides the resonance lines there are other strong lines for transitions with . A cascade of transitions via connects the lowest levels with the continuum. Photon losses in these infrared lines result in a downward flow of electrons producing a statistical equilibrium with a relative overpopulation of the lower levels as demonstrated nicely in the bottom row of Fig. 4. It can be shown that these transitions dominate the statistical equilibrium and that due to this fact the collisional interactions between other transitions are less important.

The rates of the allowed electronic collisions are proportional to the f value. This is the reason why the transitions between only a few levels dominate, and an increase of the collision rates including hydrogen collisions scaled by (see Fig. 5) or even 1.0 does not change the populations very much. Only the very line cores of the NLTE profiles are sensitive to changes in , and their migration towards LTE would require substantial enhancements of the hydrogen collision rates with scaling factors well above 10.

3.1. Line profiles and atomic model iteration

Our approach is similar to that of Baumüller & Gehren (1996). The present analysis of the solar Na spectrum is not intended to demonstrate spectacular changes in our understanding of NLTE line formation; it is rather intended as a replacement for non-existing laboratory experiments, to fix a parameter combination of model atmospheres and model atoms. Laboratory measurements are thus replaced by the observed solar line profiles taken from the Kitt Peak Solar Flux Atlas (Kurucz et al. 1984), and are compared with synthetic line profiles calculated from the level populations according to the NLTE departure coefficients obtained for a large number of different input parameters. The corresponding line data for the final fit are given in Table 1.

Table 1. Na I line data. Results (a) refer to NLTE abundances obtained with a constant km s^-1, = 3.2 km s^-1. (b) denotes best fit results with parameters described in the last two columns

In accordance with Paper I, the synthetic profiles are convolved with a combination of a flux rotation profile of km s^-1 and a radial-tangential macroturbulence which was found to vary for lines of different mean depth of formation between 2.2 and 4.5 km s^-1 . A few results are immediately evident from the solar Na lines reproduced in Fig.6,

• the LTE profiles are generally too weak in the line cores, even in the subordinate lines,

• this defect of the LTE profiles exists for both the semi-empirical model of Holweger & Müller (1974) and our theoretical line-blanketed solar atmosphere. However, due to the flatter temperature gradient the empirical model produces even shallower profiles,

• only the NLTE profiles converge towards an acceptable fit of the observed line cores.

Fig. 6. Solar Na I flux profiles synthesized under both LTE and NLTE conditions, compared with spectra of the KPNO Solar Flux Atlas (Kurucz et al. 1984, -). All profiles are individually fitted at the line wings using van der Waals damping enhancements , solar abundances , appropriate microturbulence velocities , and radial-tangential macroturbulent velocities . All NLTE profiles are calculated with hydrogen collisions of . All plots show LTE profiles calculated with the standard atmospheric model () and with the model of Holweger & Müller (1974, ), and NLTE profiles for the standard model (). Additionally, the 8190 and 5680 doublets include LTE profiles for which the damping constant was substantially enhanced to fit the line equivalent width (). (Top row): D lines at 5890 and 5896 Å . (Second row): transitions at 8193 and 8195 Å . (Third row): transitions at 5683 and 5688 Å . (Bottom row): transitions at 6154 and 6161 Å

Our analysis of the solar line flux profiles is based on the adjustment of the input parameters in the same way as the automatic profile fits published by Takeda (1995). This includes the following data,

1. The appropriate selection of NLTE atomic models with particular emphasis on the collisional interaction with neutral hydrogen atoms. Fig. 5 suggests that in the Sun the influence of electron and hydrogen collisions derived from the corresponding approximations is very similar, at least inside . Therefore it is adequate to adjust only one of the two interaction rates; our decision is to accept the formula for electron collisions given by van Regemorter (1962), and modify only the uncertain formula for hydrogen collisions with the scaling factor . The corresponding NLTE test calculations cover a large range of adjustments from = 0 to = 5, and our choice of the proper scaling factor sets an upper limit to = 0.5 with a best fit at = 0.05, the value we selected for our standard model in Fig. 6. The scaling factor is held fixed for all transitions. In contrast to the results of the analyses of the solar Al I lines (Baumüller & Gehren 1996) and of the solar Mg I lines (Zhao et al. 1998) we see no trend suggesting a variation of the with excitation energy. We note, however, that the D lines may be reproduced with quite large values of since they do not respond to this parameter.

2. The determination of non-thermal small-scale broadening velocities. Granular hydrodynamics of the solar atmosphere suggest that a plane-parallel approach cannot avoid the assumption of a depth-dependent microturbulence velocity. In fact, simple tests show that - depending on the mean depth of line formation - the fit of Na line profile cores requires different values of . In particular the chromospheric rise of the velocity fields is seen in the increased necessary to fit the D line cores and the minimum is seen in the profile width of the Å doublet. Except for this line the resulting fits of case (b) represented in Table 1 are forced to the same value for both lines of a doublet.

3. Van der Waals damping constants are notoriously uncertain since they depend on the description of neutral particle collisions. Holweger (1973) in his classical solar abundance analysis states that the "line broadening of strong solar Na I lines may be accounted for by van der Waals damping with interaction constants increased by over the value given by Unsöld's (1955) approximation". While the exact value of such a correction depends on the atmospheric model, the conclusion itself is questionable since it is based on a fit of the equivalent widths and not on line profiles . The difference is obvious from Fig. 6, where the corresponding LTE fits to the equivalent widths of the and 5680 Å doublets emphasize the failure to fit the profile wings. It is, however, possible to improve the necessary corrections for each Na I doublet by comparing the influence of damping on two lines of different strength. The results are given in Table 1 for the theoretical line-blanketed model atmosphere and they imply .

4. The energy level splittings of Na I are small, so that all transitions to levels with and are broadened mainly by the Stark effect which we have estimated according to Hunger's (1960) approximation of the Lindholm theory. The corresponding data in Table 1 emphasize the dominant role of Stark broadening in all 3P - nD transitions; for Å the value had to be adjusted to fit the observed line wings with instead of -7.52, for Å the theoretical value was increased by a factor 10; all other data were applied as calculated.

5. The adjustment of damping parameters is most strongly coupled to the evaluation of abundances. To determine their solar photospheric value, is treated as a free parameter for each line . We note that the respective contributions of both and (or ) to the line can be disentangled using different portions of the profile. The accuracy of this procedure is between 0.05 to 0.1 for (only for sufficiently strong lines that are sensitive to such corrections), and 0.02 for . The resulting solar abundance will be discussed below.

6. The Na I line flux profiles radiated from the top of the solar photosphere are modified by large-scale motions of which the rotation component can be modeled by a constant value of km s^-1, which accounts for the mean surface rotation of the Sun. The second component, usually termed macroturbulence, is best described with the radial-tangential streaming model of Gray (1977), who found that lines of different strength require corresponding adjustment of the velocities. The optimal determination of this parameter was obtained by fitting the profiles at both line cores and the width of the shoulders between core and line wing. The variation of the resulting roughly follows the depth-dependence of the microturbulence .

The parameter adjustment has been iterated to confirm and eliminate their mutual dependence as much as possible. The final results are given in Table 1 for all lines in the visible and near red spectral range that appear to be sufficiently unblended. For better comparison two cases were calculated; (a) refers to a simple solution with km s^-1, and = 3.2 km s^-1 assumed for all lines. The resulting fits are improved in case (b) leaving and as free parameters for each doublet. The latter case implies some depth-dependence of the velocity fields, as is in fact observed for the Sun. However, comparison with other stars usually requires a simpler parametrization, and it is important to compare both results for the Sun in order to estimate the reliability of the stellar abundances. As a by-product of our NLTE analyses we are able to present solar Na abundances in Fig. 7. Accepting the results for all 15 lines investigated here, the best value for the mean solar abundance would be

Closer inspection reveals that the accuracy with which the different lines are fitted with the standard NLTE model including hydrogen collisions with = 0.05 is not always perfect. If the D line NLTE abundances are used as a reference the lines in the blue yield abundances systematically lower by -0.05. This is a problem encountered also with lines of other elements in that spectral region, and it may be connected with an unidentified line haze that leads to a very uncertain continuum position. However, after removing these lines ( 4497, 4668, and 4751 Å) the mean abundance would be only marginally different,

though the standard deviation would be reduced by a significant fraction. It is nearly impossible to reproduce the proper blend behaviour of Å (see Fig. 6) because the blend component is unidentified; thus its present fit is more a representation of the maximum possible abundance. Moreover, the doublet may be marginally affected by chromospheric parameters that are not considered in our investigation (its mean depth of formation being around ). Note that all these differences are relatively small as are those obtained with respect to LTE abundances. The result of Holweger (1973) obtained with a force-fit to the damping constants of a total of 16 lines is

but the standard deviation is substantially larger.

Fig. 7. Na I line abundances obtained from the best fit NLTE results in Table 1, plotted as a function of equivalent width. The shaded area refers to a error excluding the Å doublet (see text).

The overall fit of the atomic model to the solar lines investigated here is quite satisfactory. Unlike LTE the statistical equilibrium is able to reproduce all parts of the line profiles including the line cores. The sensitivity of most lines to the scaling of hydrogen collisions is small in the solar atmosphere provided is reduced to values below 0.5, in reasonable agreement with the results of Takeda (1995). The small differences in the adopted line parameters (cf. his Table 1) are not due to our respective NLTE line formation results but rather to three differences in our analysis approach,

• Takeda's line data are based on a single doublet, unfortunately the more problematic lines at 8183 and 8194 Å,

• the parameters entering his line formation are chosen with no further constraints. Thus e.g. his hydrogen collision scaling factor is different for the two lines of the same doublet, and the same holds for the other parameters,

• his automatic line fit produces profiles with minimum but wings that are significantly deeper than observed. Such solutions cannot be accepted as realistic.

More recent investigations of the solar sodium lines are restricted to very special problems, predominantly to the reproduction of the D line profiles including their chromospheric and/or umbral contributions. Thus Bruls et al. (1992) have analyzed the response of the D lines to different solar models. Their atomic model ignores hydrogen collisions, and the influence of line-blanketing is simulated by an artificial increase in the H^- continuous opacity. Since photoionization is unimportant the latter approximation is acceptable and the = 0 option does not affect the resonance lines. Therefore their results do not add any new insight to our problem. The work of Caccin et al. (1993) discusses hydrogen collisions, but only to fit the very cores of the D lines in sunspot umbrae. This does not help to determine factors for a complete atomic model such as the one presented here.

The available information about the role of hydrogen collisions is thus extracted from the solar spectrum and the statistical equilibrium equations can be used in the following section to obtain spectroscopic determinations of the sodium abundance in cool metal-poor stars.

© European Southern Observatory (ESO) 1998

Online publication: September 14, 1998
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