5.2 Spatially averaged lines

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Figure 28: The predicted spatially and temporally averaged 3D LTE solar line profile of a typical Fe i line (solid line) compared with the corresponding calculation when ignoring all Doppler shifts arising from the photospheric velocity field (dashed line), demonstrating the importance of convective line broadening. The latter profile closely resembles 1D line profiles without application of the fudge parameters micro- and macroturbulence.

The widths of spatially averaged photospheric lines are much wider than predicted solely from natural and thermal broadening. As explained in Asplund et al. (2000bJump To The Next Citation Point), most of the line broadening instead arises from the Doppler shifts associated with granular scale convective motions, with a minor contribution also coming from the photospheric oscillations (the solar p-modes, see Section 6). Since 1D model atmospheres by definition do not predict the photospheric velocity field, the resulting 1D line profiles are too narrow compared with observations; a similar effect is obtained by artificially setting all velocities to zero in the 3D model atmosphere before computing the line formation, as illustrated in Figure 28View Image. To rectify this problem, two fudge parameters, microturbulence and macroturbulence, are introduced in all 1D line formation calculations. The former supposedly represents small-scale velocities while the latter tries to encapsulate the effect of motions occurring on length-scales larger than one optical path-length (e.g., Gray, 2005Jump To The Next Citation Point). The value for the microturbulence parameter is then determined by requiring that the derived elemental abundance is independent of line strength, while the macroturbulence is estimated from the overall widths of spectral lines. However, even with the luxury of having two additional free parameters to tune, 1D calculations cannot explain the detailed shape of the observed lines (nor, as will be demonstrated below, the shifts and asymmetries of lines). Naturally this division into micro- and macroturbulence is too simplistic since the convective motions occur on a range of spatial scales. Furthermore, the names are misleading as the line broadenings have very little to do with turbulence in its classical meaning (Asplund et al., 2000bJump To The Next Citation Point). Small-scale energy cascades and turbulence associated with a high-Reynolds-number plasma as the solar atmosphere are present but their intensities are very small in the upflows to which the line strengths are strongly biased and thus they do not affect the line formation significantly. As emphasized above, the dominant factor is instead the velocity gradients and the corresponding Doppler shifts stemming from convection and its overshoot in the photosphere.

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Figure 29: The predicted temporally and spatially averaged 3D profile (blue solid line) compared with the observed solar disk-center line (red diamonds). Note the excellent agreement as seen in the residuals (the discrepancies in the far red and blue wings are due to unaccounted for blends). Also shown is the best-fitting 1D line profile after having optimized the micro- and macroturbulence (green solid line), which clearly has the wrong shape, asymmetry, and shift.

In sharp contrast to the 1D case, 3D line formation calculations taking into account the self-consistently predicted velocity field in 3D hydrodynamical solar model atmospheres reproduce the observed line profiles exceedingly well for most lines so far considered, as exemplified in Figure 29View Image. The agreement is equally good for disk-center and flux profiles (Asplund et al., 2000bJump To The Next Citation Point). Indeed, a comparison between observed and theoretical 3D line shapes is a very powerful tool to identify previously unknown blends or erroneous rest wavelengths for the transition in question.

The correlation of velocity and temperature that is an inherent property of convection causes correlations of radiation intensity and Doppler shifts. This results in both a net blueshift of spectral lines, and a characteristic line asymmetry, which may be quantified by measuring the line shifts (as defined by the wavelength with the minimum intensity/flux in the line profile) or bisectors (the midpoint of the profile at different line depths) (e.g., Dravins et al., 1981Dravins, 1982Dravins and Nordlund, 1990a,bAllende Prieto and García López, 1998Asplund et al., 2000a,bJump To The Next Citation Point,cJump To The Next Citation PointAsplund, 2005Jump To The Next Citation PointGray, 2005). In the Sun, most spatially averaged photospheric lines have a characteristic ⊂-shape, which arises because the upflowing (blueshifted) granular gas is both warmer and have a larger area coverage than the downflowing (redshifted) material in the intergranular lanes. In general, the predicted asymmetries of solar Fe i and Fe ii lines agree very well with the observed bisectors (Asplund et al., 2000bJump To The Next Citation Point); a few examples are shown in Figure 30View Image. It should be noted that both the theoretical and observed profiles are on an absolute wavelength scale for the Sun and that no arbitrary wavelength shifts have been applied to improve the agreement. Of course, 1D profiles are all completely symmetrical without any line shift. Weak Fe i lines have line shifts of ≈ –500 m s–1 (after correcting for the solar gravitational redshift of 633 m s–1) while Fe ii lines have even larger blueshifts due to their in general greater depths of formation where the granulation contrast and convective velocities are larger. The amount of convective blueshift decreases as the line strength increases when the line-forming region is shifted upwards in the atmosphere until eventually convective (overshoot) motions become unimportant. The theoretical 3D line shifts agree very well with the observed shifts for weak and intermediate strong lines but becomes progressively worse for even stronger lines. This may signal departures from LTE, particularly in the line cores of strong lines, which are formed in high layers with low densities. However, it may also be due to the limited height extension and missing chromospheric physics of the employed 3D simulation (Asplund et al., 2000bJump To The Next Citation PointScott et al., 2006Jump To The Next Citation Point). Towards the limb the convective blueshifts become smaller or vanish due to the smaller granulation intensity contrast compared with at disk-center (Balthasar, 1988).

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Figure 30: A comparison between the predicted and observed (solid lines with error bars) line bisectors for a few Fe i lines on an absolute wavelength scale. In 1D models all lines are perfectly symmetric with no line shift.

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