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

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2. Simulations

2.1. Principles

The availability of computing methods, atomic data, and powerful computers may just begin to allow realistic and consistent solutions of 3D NLTE problems in media such as photospheric granulation - meaning large, and essentially complete, model atoms and full coupling between all angles. But such problems are still normally approached with some compromising with regard to atomic models, radiative transfer, or geometry. The aim of Paper I was mainly to study the effects of LTE departures and 3D effects on the Li i lines by comparing different spectral-line treatments in a single solar-granulation-model snapshot, with standard 1D models only used as a qualitative reference. The methods that were introduced, and are used also here, involve the calculation of line profiles and equivalent widths with a hybrid technique using a 1D NLTE code and a 3D background radiation field computed in LTE. This approach was chosen because it allowed a rather simple application of existing codes and because the very low solar lithium abundance makes it likely that the line radiative transfer problem can be treated as a continuum problem.

2.2. 3D photospheric models

The 3D photospheric models are two snapshots from the granulation simulations of Nordlund & Stein (e.g., Stein & Nordlund 1989, Nordlund & Stein 1991) that were also used by Kiselman & Nordlund (1995) - see that paper for illustrations. The snapshots are chosen so that they represent opposite phases in the (internally excited) oscillation that is present in the simulations. The original simulation data consisted of a [FORMULA] grid of thermodynamic quantities. These were cut and resampled to a [FORMULA] grid, corresponding to [FORMULA] Mm on the solar surface. The plots of Fig. 1 only show results for every second x and y grid point, thus [FORMULA] of the vertical columns. The snapshots are periodic in the horizontal directions and this is used in the radiative transfer calculations.

[FIGURE] Fig. 1. Results from line transfer calculations in a granulation snapshot using different treatments and approximations: Li i equivalent widths as function of continuum intensity (left) and the ratio of these to the corresponding LTE values (right). Labels are explained in text. The mean values given are intensity-weighted means

2.3. LTE treatment of lines

Several weak lines observed in the region around the Li i 671 nm line are also included in the analysis for reference. The lines which were found in the VALD data base (Piskunov et al. 1995) or in the Kurucz (1995) line list were modelled in LTE with the routines of Kiselman & Nordlund (1995). The adopted line parameters are given in Table 1.


Table 1. Adopted quantities for the spectral lines and the compilations from which they were taken

2.4. Non-LTE Li i line transfer calculations

The Li i line of interest is the resonance doublet at 671 nm which is very weak in the solar spectrum due to the low lithium abundance - the standard value [FORMULA] (Grevesse et al. 1996; Müller et al. 1975) is used here. The doublet is blended with several other weaker lines all of which are not identified - see Brault & Müller (1975) and Kurucz (1995). The stronger (shorter wavelength) doublet component is less blended. The lines are also subject to hyperfine splitting and isotopic splitting that cause complications when modelling the line.

The Li i line profiles and equivalent widths were computed with version 2.2 of the plane-parallel NLTE code MULTI (Carlsson 1986) which uses the operator perturbation techniques of Scharmer & Carlsson (1985). All calculations of lines in emergent light were done for disk centre, i.e. for [FORMULA].

The 30-level lithium atomic model of Carlsson et al. (1994) was used with one change: the 671 nm resonance line was treated as a single line and not as a blended doublet. This was done because the current version of MULTI will not treat the radiative transfer correctly when the (microscopic) line profile is asymmetric in the presence of macroscopic velocity fields. This treatment of the doublet as a single line will not cause errors of importance for the current study since the line is very weak in almost all cases discussed here, putting it safely on the linear part of its curve of growth. This is confirmed by the fact that the lithium abundance can be increased a factor of ten without changing the qualitative behaviour of the line. The change in line-formation height is apparently not significant for the NLTE behaviour. Thus the equivalent width of each line component will be proportional to the doublet equivalent width. The absolute solar Li abundance will not be a central issue here and line strengths will frequently be rescaled.

The option of providing MULTI with a background radiation field for calculation of the photoionisation rates was used in all calculations that are presented here and in some cases also for bound-bound transitions. The background radiation field was computed using routines written in the IDL data processing language that are similar to those used by Kiselman & Nordlund (1995). The equation of transfer was solved along a set of rays - five inclination angles ([FORMULA]) and eight azimuthal angles ([FORMULA]) - and the resulting intensities along each ray were used to calculate a mean intensity [FORMULA] in each [FORMULA] point. This was done with a pure Planckian source function without any allowance for scattering (i.e. in what is sometimes called "strong LTE "). Some interpolation problems appeared close to [FORMULA] where the horizontal gradients are strong. This caused spurious values of [FORMULA] in a few points but too deep down to have any impact on the resulting line profiles.

The hybrid treatment can make one worry about inconsistencies. It is important that the opacities used for production of the granulation model, the background radiation field, and the non-LTE calculations are compatible since a small difference may cause the optical surface to fall at different depths with large effects on the resulting outgoing intensities. Direct comparisons of the opacities showed that the differences were indeed insignificant within the context of the current study.

The presence of large amounts of spectral lines, especially in the blue and ultraviolet, is known to cause significant depression of the fluxes in these spectral regions. To somewhat allow for this, the OSMARCS plane-parallel LTE model-photosphere code (Edvardsson et al. 1993) was used to produce a solar model from which flux-weighted mean opacities were computed for a wavelength range around the wavelengths used in the computation of photoionising radiation fields. The ratio between these mean opacities, including spectral-line blocking, and the continuous opacities was then used to correct the opacities in the computation via interpolation in temperature. The correction factors range from 1.0 at high temperatures in the deep layers to well above 10 in the low-temperature regions.

2.5. Collisions with neutral hydrogen

The importance of inelastic collisions with neutral hydrogen atoms for photospheric line formation has been debated (e.g., Steenbock & Holweger 1984, Caccin et al. 1993, Lambert 1993). Such processes were not included here. It could, however, be interesting to see if spatially resolved spectroscopy would offer a possibility to decide this issue and pin down the relevant cross sections. Tests where collisional rates were included according to the recipe of Steenbock & Holweger (1984) showed, however, that the changes introduced by the additional collisions were much too small in this case to allow any observational test.

2.6. Results

The left-hand plots of Fig. 1 shows the equivalent width W of the Li i 671 nm line as a function of continuum intensity [FORMULA] for one of the snapshots. Each point in the plots corresponds to one ([FORMULA]) column in the simulation snapshot. The plots in the right column show the departure from the LTE result in the form of [FORMULA] ratios, once again as a function of continuum intensities. From top to bottom, the plots show the results of line calculations with increasing sophistication. This series illustrates the importance of the departures from LTE caused by lateral photon exchange and the impact of ultraviolet line blanketing.

The uppermost pair of plots (LTE) shows the LTE result that was discussed in some detail in Paper I. Here the local temperature sets the line source function and the line opacity according to the Saha and Boltzmann equilibria. The result is that the line generally is strongest in the continuum-bright regions but the [FORMULA] relation shows a significant scatter.

Leaving the LTE approximation gives the result in the second pair of plots (1.5D NLTE). Here the line strengths have been calculated in NLTE, so that each column of the snapshot is treated like a plane-parallel model with horizontal photon exchange neglected. The result is a narrow dependence of equivalent width on continuum intensity since now the atomic level populations are governed by a radiation temperature that is set in the lower regions where the continuum is formed. The experiments in Paper I showed that it is the bound-bound radiative transitions of Li i that matters, ultraviolet overionisation has little importance for the departures from LTE.

The third row (all 3D) shows the results when all lines have been calculated as fixed rates given by the 3D radiation field. Now the value of the mean intensity [FORMULA] in each point is influenced by horizontally neighbouring regions and this gives a larger spread in the plot since the continuum intensity on the plot axis is the intensity coming from straight below.

The final modelling (all 3D LB) - which is to be compared with observations later - is represented by the lowest pair of plots in Fig. 1. Here all transitions are treated as fixed with the 3D radiation field and that field has been corrected for line blanketing as described above. The effect of the schematic line-blanketing correction is to increase the line strength somewhat due to the damping effect this has on the line-pumping-induced overionisation. The difference from the LTE plot is still significant and it seems possible to discern between the two cases observationally as will be tried in the next section.

Uitenbroek (1998) studied Li i line formation with a consistent 2D NLTE treatment, a similar granulation snapshot, but a smaller model atom than the one of this paper. The results are qualitatively similar to those presented here, but the discussion of them differs somewhat in that ultraviolet overionisation is considered to be an important mechanism contrary to what was argued in Paper I.

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© European Southern Observatory (ESO) 1998

Online publication: April 20, 1998