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,
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,
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.
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,
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