4. Sodium in metal-poor stars
Except for the D lines the analysis of solar sodium lines reveals a significant abundance correction with respect to LTE if abundances are determined from line profile fits instead of equivalent widths (see Fig. 6). Thus it is only natural to expect a more substantial effect in the atmospheres of cool stars that are reduced in metals and thus free electrons. Unlike Al with its extremely large ground-state photoionization the statistical equilibrium of Na will depend more on the efficiency of collisions and, in the absence of free electrons, hydrogen collisions may be the only thermalizing interaction remaining. It is therefore important that the calculations in the solar atmospheric environment indicate that the hydrogen contribution to collisions is best described by a scaling factor with an upper limit corresponding to . Note that increasing far beyond 1 will always drive the system towards LTE. The first step towards a sodium population analysis in cool metal-poor stars is therefore an investigation of the statistical equilibrium as a function of the hydrogen collision rates, parametrized as in the Sun with a scaling factor . Typical changes due to are reproduced in Fig. 8 for two metal-poor stars. It shows that the hydrogen collisions do not enter the results too strongly as long as they are well below = 1 with an estimated upper limit corresponding to an abundance error of [Fe/H] = 0.03 dex. Together with a few more tests this confirms that our choice of is not as important as was anticipated above. In order not to overestimate the influence of deviations from LTE a value of = 0.05 will therefore be used for most of the remaining sodium calculations in metal-poor stars.
Fig. 9 displays the typical variation of level populations with decreasing metal abundance. The corresponding decrease of the electron collision rates should produce the strongest effect. In fact, the departure coefficients change significantly, in particular in the metallicity range 0 [Fe/H] . The departure from thermal populations is driven inwards as is the mean depth of line formation for the strongest lines. Inside all levels are thermalized even in the most extreme metal-poor stars. This is easily understood in terms of identifying the electron donors in moderately cool stars. After reducing the metal abundance to less than [Fe/H] = -2 the remaining electrons are no longer from Mg, Si, or Fe but from hydrogen ionization; although the electron density is now substantially smaller, it is sufficient to thermalize the Na population densities in the inner photosphere. Outside the relative population of the fine structure levels tends to become non-thermal. This effect is even stronger in slightly hotter subdwarfs (see Fig. 10), and it suggests that the forbidden electronic collision rate (assumed to be 500 ) is still too low. On the other hand, the deviation of the fine structure level populations could be real; it does not, however, affect the line formation since even the D lines are formed inside in extremely metal-poor atmospheres. Generally, the NLTE effects are systematically stronger for the hotter models, which is in accordance with the statistical equilibrium of aluminium (see Paper I). The strongest departures from LTE are found for models with high temperature and low gravity (i.e. turnoff stars). The reduction of surface gravity results in a decreased efficiency of collisions by both electrons and hydrogen atoms.
From Figs. 9 and 10 it is evident that NLTE of sodium in metal-poor stars means an increased overpopulation of the ground state. Thus the formation of the D lines should be most affected and it is easy to predict that NLTE sodium abundances obtained from the resonance lines are always smaller than those derived from the LTE approximation. Such predictions based on model calculations follow in Table 2 below for typical solar-type dwarfs, cool subdwarfs and subgiants. As suggested by Fig. 9 the most extreme metal-poor stars will be subject only to small NLTE corrections.
Table 2. Typical abundance corrections necessary when fitting calculated NLTE equivalent widths of Na I lines in cool metal-poor stars with LTE but otherwise the same parameters. Results refer to - . No entries are given for extremely weak lines. The representative atmospheres refer to km s-1 and [Na/Fe] = 0
4.1. NLTE Line formation
In extremely metal-poor stars sodium is represented only by its resonance lines. Therefore the most important result of this analysis will be the formation of the Na D lines in stars of decreasing metal abundance. The discussion above already suggests that these lines change their appearance considerably when varying the metal abundance of a cool star from solar to typical values of halo subdwarfs. The reason for the difference with respect to LTE profiles is the decrease of collisional thermalization, and we have shown that hydrogen collisions do not contribute much to this process if we can trust the calibration based on the solar spectrum. The change of the 5895 Å line with metal abundance [Fe/H] assuming a solar Na/Fe ratio is displayed in Fig. 11. The difference between the NLTE and LTE profiles is striking, not so much at solar metallicities but mostly for metal abundances between [Fe/H] = -1 and -2, where it amounts to 20% of the flux in the line cores.
The overpopulation of at solar abundances decreases slightly in more metal-poor stars (e.g. Fig. 9) while the population of the higher excited levels varies only marginally with metal abundance. Therefore as long as the subordinate lines are sufficiently strong they will also be affected by NLTE line formation. Only in the case of metal abundances below [Fe/H] will the formation of the weak D lines move into the innermost photospheric layers that are sufficiently thermalized by electron collisions, at least for effective temperatures above 5500 K. Consequently, in some of the most metal-poor subdwarfs known today, LTE profiles may be approximately valid again. The same arguments hold for the subordinate lines at slightly higher Na abundances.
4.2. Sodium abundances
The D2 line profiles shown in Fig. 11 make it obvious that the deep NLTE line cores in metal-deficient stars can be compensated in abundance analyses by simply increasing the Na abundance until the observed equivalent width is reproduced. Such results are displayed in Table 2, from which it is evident that LTE abundances can be significantly different from their NLTE counterparts, with differences reaching 0.6 dex in extreme cases.
Since in previous investigations of metal-poor stars no profile analyses were involved, the corresponding errors have not been recognized. Our present investigation starts with spectroscopic data obtained from the ESO 3.6m CASPEC, the ESO 1.5m ECHELEC, and the Calar Alto 2.2m Coudé spectrograph, however, with spectral resolutions mostly around R = 20000, which is still too low to resolve line profiles, at least for the very metal-poor stars. In addition the first observing runs with the FOCES spectrograph at the Calar Alto 2.2m and 3.5m telescopes (Pfeiffer et al. 1998) have brought a wealth of data covering the complete spectra of many subdwarfs from 4000 to 7000 Å at resolutions of R = 40000 or 60000, respectively. We have added a number of these spectra to our list, and we find that the Na line profiles provide some useful constraints to the analyses.
The stellar parameters of these objects are given in Table 3. Some were taken from recent re-analyses of Fuhrmann et al. (1997), Fuhrmann (1997, private communication), and Grupp (1996). These results differ from the Axer et al. data in that their surface gravities have been rederived from the strong damping wings of neutral metal lines (and not from the Fe ionization equilibrium), now closely fitting the HIPPARCOS (1997) data. All other stars have been reanalyzed for the present investigation. For some stars the microturbulence has also been improved using an extended set of Fe lines. The abundances are all obtained from line profile fits to the resonance lines, the doublet at 5680 Å and, whenever possible, the faint doublet at 6160 Å . The external broadening function, mostly determined by the spectrograph slit, has been adjusted to fit the fainter lines in the spectrum. In the most metal-poor stars the Na abundances are obtained exclusively from the D lines. The results are reproduced in Table 3, where both the NLTE profile fit abundances and the LTE abundances yielding the same equivalent widths are listed for comparison. One of the most striking results is the change between LTE and NLTE abundances corresponding to a factor 3.5 in the very metal-poor subgiant HD 140283. This is representative for similarly metal-poor stars. Thus, all stars with [Fe/H] below -1.4 show subsolar NLTE abundance ratios [Na/Fe], whereas their LTE abundances would be substantially above solar.
Table 3. Stellar parameters and sodium abundance ratios [Na/Fe]. Stellar parameters are from Fuhrmann et al. (F, 1997) and Grupp (G, 1996); all other parameters have been obtained from new analyses. refers to the standard deviation
Considering only LTE abundances based on the fit of equivalent widths it is conspicious how the abundances determined from the D lines and the 5680 Å or the 6160 Å doublets differ. Note that in most cases these differences are well above the observational errors. Alternatively, the NLTE abundances based on profile fits produce a satisfactory standard deviation when including lines from all doublets and they compare favourably with the solar abundance errors. Moreover, the standard deviation of the NLTE results is between a factor 5 to 2 lower than for the LTE data. This result is perhaps the most important because it shows that with high S/N spectra of sufficient spectral resolution (R = 40000 or higher) it is possible to reproduce line profiles with a very high accuracy. A few profile fits for the subgiant HD 45282 are shown in Fig. 12. They are representative for the average fit quality obtained for the R = 40000 FOCES spectra.
Table 3 displays the Na abundances of all stars investigated. The NLTE mean abundance data are reproduced together with the nominal standard deviation of a single line, i.e. the error of the mean would be correspondingly smaller. Systematic errors depend on the validity of the plane-parallel model atmosphere concept. It implies that and can be uniquely determined, and that hydrodynamic motions can be modeled with the micro- and macroturbulence velocities, and . The non-thermal velocity parameters, in particular, reflect all the uncertainties of the simple model atmosphere approach. Therefore in some stars the faint 6160 Å lines seem to require a slightly different external broadening parameter than do the D lines. It will be necessary to analyze some of the line profiles with higher resolution, but even at there does not seem to be any alternative to the introduction of a depth-dependent turbulent velocity. Most interestingly, the obvious requirements concern more the external broadening parameters than the microturbulence, and therefore the abundance results should not be very sensitive to this problem.
However, and are important and the accuracy of their determination enters the final NLTE abundances as much as in the LTE data. Since the determinations of and are still mutually dependent, we estimate the systematic errors to be of the order of 0.05 dex for the best spectra. As long as model atmospheres are used differentially both the analysis of the Balmer line wings and that of the damping wings of the strong Mg I and Fe I lines are among the most reliable methods used in spectrum synthesis, i.e. they are much better defined than any synthetic colours. This has been demonstrated recently by Fuhrmann et al. (1997) and Fuhrmann (1998) who found that the surface gravity determined from the iron ionization equilibrium is different from that obtained using HIPPARCOS parallaxes in most of the metal-poor stars. Consequently, for a number of stars for which new spectra were obtained with the FOCES spectrograph (Pfeiffer et al. 1998) new stellar parameters have been determined with surface gravities conforming to the HIPPARCOS results (1997). [Fe/H] has been determined only from Fe II lines; this moved the metal abundance scale towards values substantially more metal-rich than previously accepted and the [Na/Fe] data tend towards correspondingly smaller results. The main effect, however, is due to the adjustment of the surface gravities dictated by the HIPPARCOS observations.
Carlsson et al. (1994) have investigated the formation of the Li I resonance lines under NLTE conditions in cool stars. Although Li and Na have similar atomic structures, photoionization from the 3s and 3p levels of Na I is less important than that of the corresponding Li I levels. Consequently, photon suction plays the dominant role in Na I whereas the Li I population is apparently more sensitive to overionization. Our Na I abundance corrections are all negative and they are particularly important for metal-poor stars. We note that only for the most metal-poor subdwarfs ([Fe/H] = -3) the absolute abundances of Li and Na become comparable, and in that range the abundance corrections of the Na and Li resonance lines are very similar.
© European Southern Observatory (ESO) 1998
Online publication: September 14, 1998