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Astron. Astrophys. 333, 732-740 (1998)

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4. Comparison of observations with simulations

4.1. Smearing and binning of the simulations

In order to allow a fair comparison between simulations and observations, the simulated results must be treated in a way that mimics the effects of the atmosphere, the telescope, and the spectrograph. For this study, a simple model using the convolution with two Gaussians, one narrow and one wider with lower amplitude, was chosen. The FWHM:s and the relative amplitudes of the Gaussians were chosen through experiments. The intensities, equivalent widths, and line-centre positions from the simulation were smoothed with various combinations of parameters until a combination was found that gave reasonable RMS -contrasts and the simulated intensity image looked similar to slit-jaw images. The FWHM of the narrow Gaussian corresponds to 0:005 and that of the broader curve, with an amplitude of 0.25 of the narrow one, to 4:003.

The experiments showed that the basic feature used here as a diagnostic, the [FORMULA] diagram, did not change dramatically in shape when smoothed in different ways. Thus the detailed smoothing procedure used is not crucial to the results (though not unimportant).

Finally, the simulation results were binned in the same way as the observational spectra. Since the simulated data is noiseless, the lines that show simulation results in the figures represent the mean quantities for each bin.

4.2. Discussion

4.2.1. All lines

Fig. 3 shows the observed and predicted line-centre velocities as function of continuum intensity, measured from binned spectra. The theoretical data come from the LTE simulations. A constant has been added to the velocities so that the intensity-weighted mean velocities for all data sets are zero. The correspondence is generally good, giving some confidence in the methods and models used here.

[FIGURE] Fig. 3. Comparison of line-centre velocities v and their dependence on continuum intensity [FORMULA] as measured from binned observed spectra (crosses) with LTE simulations in two granulation snapshots (dashed and solid lines) for several weak spectral lines in the same spectral region as Li i

Fig. 4 shows similar plots for equivalent widths. The theoretical values have been multiplied by a constant so that their intensity-weighted mean values coincide with that of the observations. The observed equivalent widths (binned) are well described by the simulations for some lines but not for others. The Fe I lines seem to be reasonably well described by the LTE modelling though with some deviations. The behaviour of the Si I line is well reproduced. The CN line shows what appears to be a discrepant behaviour, but the weakness of the line calls for cautiousness in the interpretation. The discrepancy between observations and LTE theory for the Li i seems more significant. The observed [FORMULA] relation shows a weak slope in the opposite direction to the predicted LTE curves.

[FIGURE] Fig. 4. Comparison of equivalent widths W measured from binned observations (crosses) with LTE simulations in two granulations snapshots (dashed and solid lines) for several weak spectral lines in the same spectral region as Li i

The central idea of this paper is that discrepancies between the theoretical LTE results and the observations of Fig. 4 are due to NLTE effects. Such conclusions are valid if we can trust the granulation models and the observational data. The following discussion will concentrate on the Li i line because of the (purported) reliability of the NLTE modelling for this light atom, which can be represented with an essentially complete atomic model. For heavier atoms like Fe I, the problem of assembling sufficiently complete and precise atomic data makes NLTE spectral-line modelling difficult and always questionable. The Li i line is thus a good test case in theory, but its weakness introduces observational uncertainties.

4.2.2. Excitation-potential dependence

The different slopes in the diagrams just discussed are highlighted in Fig. 5 where the normalised slopes from the observational data are plotted against the lower-level excitation energies of the respective lines. The normalised measures of the slopes are defined as [FORMULA] for the line-centre velocities and [FORMULA] for the equivalent widths. The derivatives come from second-degree polynomial fits to the observational points and their values at mean intensity are given.

[FIGURE] Fig. 5. Plots showing the sensitivity on continuum intensity of observed line parameters to the excitation energies of the lines' lower levels. [FORMULA], the slope of the line-centre velocities. [FORMULA], the normalised slope of the equivalent widths.

The upper plot of Fig. 5 shows that the line-core velocity slopes correlate well with excitation potential. This is most easily understood as caused by the high-excitation lines being formed at greater depths (and temperatures) where the vertical velocities are greater than higher up.

The equivalent-width slopes plotted in the lower plot do not show a clear correlation with excitation potential. The Fe I lines show slopes that do not increase for high-excitation lines. This is similar to what was found for the Fe I lines in Fig. 5 of Kiselman (1994), and could be a manifestation of NLTE effects.

4.2.3. The Li i line

Fig. 6 compares the observational line-centre-velocity and equivalent-width results for Li i with the theoretical predictions. The LTE results are the same as those in Figs. 3 and 4, the NLTE results are those labelled (all 3D LB) in Sect.  2.6 and Fig. 1.

[FIGURE] Fig. 6. Comparison of line-centre velocities (upper panel) and equivalent widths (lower panel) of the Li i 671 nm line measured from binned observational spectra (crosses) with LTE and NLTE simulations of the Li i 671 nm line. Each cross comes from one of seventeen exposures. Thick dotted and full-drawn lines are LTE simulations for the two granulations snapshots, lines with diamonds are NLTE. The theoretical values have been adjusted so that their mean values coincide with that of the observations

The difference seen before between the theoretical LTE and NLTE equivalent-width results is now reduced because of the binning and smoothing operations. The spread among the observational points (crosses) at a specific intensity value is probably mostly due to random errors, though there are also inherent differences from frame to frame. Note how this spread increases towards the limits of the plots due to fewer points from very bright or very dark regions contributing to each binned spectrum. (Remember that results from bins with fewer than 20 contributors are not shown.) In essence the observed equivalent widths fall in between the LTE and the NLTE curves, though closer to the latter. The NLTE curves fall inside the range of the observational points for most of the intensity range, but this is not the clear confirmation of the NLTE results one could have hoped for. The possible explanations for the discrepancy between observations and simulations can be summarised as follows.

The NLTE treatment. The discrepancy could be explained by underestimated collisional cross sections. This explanation would be the natural choice according to the central hypothesis of this paper. If we believe in the granulation models and the observational results, the most important (or uncertain) cross sections could in principle be determined by tuning them until there is correspondence between simulations and observations. The NLTE results are sensitive to the input [FORMULA] values, which are computed in a simplified way with a coarse angular resolution. The discrepancy could be explained if [FORMULA] in the visual region is underestimated in the dark regions relative to the bright ones. There is, however, no evident reason for this to be the case.

The granulation snapshots. While these kind of granulation simulations have demonstrated their realism, one must remember that many approximations have been employed in their computation - e.g. the finite spatial resolution that precludes the small structures which are certainly there in the real Sun. Possibly the two snapshots used are unrepresentative, but it should be noted that they are chosen to be independent and of opposite oscillatory phases. Also the apparent success in the LTE modelling (of both line strengths and line-centre velocities) for most of the other weak lines observed here is evidence for the snapshots being essentially realistic. Another piece of evidence for this is the general result that the (NLTE) Li i line strength depends more on the temperature structure at greater depths than on the kinetic temperature in the higher layers where the uncertainty of the granulation simulations can be expected to be greatest.

The smearing procedure. The smoothing of the simulations is made in a rather schematic way, but as discussed earlier, this is not likely to cause significant changes in the [FORMULA] diagrams.

Blends. The Li i feature is contaminated by CN lines, as demonstrated by Brault & Müller (1975). According to their analysis, the contribution is relatively small. It should, however, make the measured [FORMULA] dependence flatter according to the behaviour of the CN line in Fig. 4. If there is contamination of some other line, or if the CN contribution was much underestimated by Brault & Müller, this could explain the observed behaviour. This would mean that the solar lithium abundance is significantly lower than previously thought. That would seem to be at odds with the behaviour of the Li i line in sunspots (e.g. Barrado y Navascués et al. 1996).

Systematic errors in the measured equivalent widths. These are difficult to estimate. A direct integration gives insignificant differences compared to the Gaussian profile fits, so the latter procedure is probably not a major source of error. In the same way, making the experiment of subtracting 10 % of the mean light level from the spectral frames did not result in significant changes in the [FORMULA] diagrams. Straylight is therefore not likely as an explanation for the discrepancy. The continuum placement is, however, crucial and expected to be the most important source of errors in the observational equivalent widths. An erroneously placed continuum will lead to an error in the [FORMULA] slope that is not corrected when equivalent widths are rescaled with a constant factor. This was clear from the preliminary reduction work, though it should be noted that the algorithm for continuum fitting was chosen on what was considered to be objective grounds with the continuum level placed close to that of Brault & Müller (1975). The size of these errors is very difficult to estimate, but there is a clear possibility that they could be significant.

Were it not for the possibility of observational errors due to the continuum placement, the discussion would leave errors in the atomic data as the most probable explanation for the discrepancy. Problems with the granulation simulations, with the NLTE treatment or the possibility of blends would also be interesting to investigate further. The likely presence of systematic errors due to continuum-placement problems makes it, however, somewhat uncertain that the discrepancy is real and significant.

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© European Southern Observatory (ESO) 1998

Online publication: April 20, 1998