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Astron. Astrophys. 359, 729-742 (2000)

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4. General features of 3D line formation

The convective motions and the atmospheric inhomogeneities leave distinct fingerprints in the spectral lines, which can be used to decipher the conditions in the line-forming layers produced by the convection (e.g. Dravins et al. 1981; Dravins & Nordlund 1990a,b). Line strengths of weak lines are predominantly determined by the average atmospheric temperature structure, while the line widths reflect the amplitude of the Doppler shifts introduced by the velocity fields. Line shifts and bisectors are created by the correlations between temperature and velocity and the details of the convective overshooting, as well as the statistical distribution between up- and downflows. Therefore, obtaining a good agreement between observed and predicted profiles lend strong support to the realism of the simulations.

4.1. Spatially resolved profiles

Although Paper IV and V in the present series of articles will discuss in detail observed and predicted spatially resolved lines, a brief excursion is still warranted here in order to interpret the spatially averaged profiles and bisectors presented in Sects. 5, 6 and 7.

Spatially resolved profiles take on an astonishing range of shapes and shifts spanning several km s-1, as illustrated in Fig. 1. The intensity contrast reversal in the higher layers of the photosphere, i.e. granules become dark while intergranular lanes become bright a few hundred km above the visible surface, is clearly seen in the cores of the resolved profiles. The strengths of spatially averaged profiles are normally biased towards the granules, since the upflows in general are brighter (high continuum intensity), have steeper temperature gradients and have a larger area coverage than the downflows. These trends are also observationally confirmed, which suggests that the combination of 3D hydrodynamical model atmospheres and LTE is appropriate for most Fe lines (Kiselman 1998; Kiselman & Asplund 2000; Paper IV).

[FIGURE] Fig. 1. Spatially resolved profiles and bisectors for the Fe I 608.2 line. The lines are both stronger and have higher continuum intensities in the blueshifted granules compared with the red-shifted intergranular lanes. The largest vertical velocities in the photosphere are encountered in the downflows. The intensity scale is normalized to the spatially averaged continuum level. The thick solid lines correspond to the spatially averaged profile and bisector

Spatially resolved bisectors are not at all typical of the spatially averaged bisectors, which are merely the result of the statistical distribution of individual profiles. Rather than the characteristic [FORMULA]-shape bisectors, individual bisectors normally have inverted [FORMULA]-shape bisectors in granules and [FORMULA]-shapes in intergranular lanes, as seen in Fig. 1. This reflects the in general increasing vertical velocities with depth in the photosphere. However, due to the meandering motion of the downflows, occasionally the line-of-sight passes through both upward and downward moving material which causes large variations for certain columns.

4.2. Spatially averaged disk-center profiles

The widths of spatially averaged spectral lines, which clearly exceed the natural and thermal broadening, predominantly arise from the velocity amplitude of the granules and intergranular lanes and to a lesser extent from photospheric oscillations. A demonstration of the importance of the non-thermal Doppler broadening is presented in Fig. 2, which shows the resulting spatially averaged profile from the 3D simulations but with all convective velocities artificially set equal to zero in the line calculation; thereby the predicted profile closely resemble those calculated with classical 1D model atmospheres. Clearly, without the Doppler shifts the line is much too narrow, which requires additional broadening in the form of micro- and macroturbulence to be introduced. The poor agreement between observations and predictions when not including the self-consistent velocity field as shown in Fig. 2, should be contrasted with the excellent fit shown in Fig. 8.

[FIGURE] Fig. 2. The spatially and temporally averaged Fe I 608.2 line when artificially removing all convective velocities in the simulations (diamonds). In comparison with the solar intensity atlas (solid line, Brault & Neckel 1987) the predicted line is much too narrow and lacks the correct line shift and asymmetry. The result when including the self-consistent velocity field is shown in Fig. 8

The individual line bisectors depends on the details of the line formation and thus on transition properties such as log gf, [FORMULA] and [FORMULA] in an intricate way, as illustrated in Figs. 3, 4 and 5. When discussing mean bisectors (e.g. Gray 1992; Allende Prieto et al. 1999; Hamilton & Lester 1999) it is therefore important to consider only appropriate subgroups consisting of lines with similar characteristics to avoid introducing errors when interpreting the results in terms of convective properties. As a corollary, it follows that reconstructing a mean bisector by averaging bisectors or using shifts of lines of different strengths will in general not recover in detail the individual bisectors, as exemplified in Fig. 3. As expected, for line depths [FORMULA] the bisectors closely coincide for lines of different strengths due to the disappearance of the influence from the convective inhomogeneities. Similarly, decreasing the excitation potential tends to shift the line formation outwards, producing less convective blueshifts for a given line depth (Fig. 4), while decreasing the wavelength increases the brightness contrast and makes the temperature gradients steeper, resulting in more vigorous convective line asymmetries (Fig. 5).

[FIGURE] Fig. 3. The spatially and temporally averaged Fe I 608.2 bisector assuming three different abundances: log [FORMULA], 7.50 and 8.00 (or equivalently with three different log gf-values). Note that the upper parts of the bisectors do not coincide for the different line strengths since the whole region of line formation is shifted outwards for stronger lines

[FIGURE] Fig. 4. The bisectors of the Fe I 608.2 (weak) and 621.9 nm (strong) lines (solid lines) which both have [FORMULA]eV. Also shown are the corresponding (artificial) Fe I lines with [FORMULA]eV (dotted lines) and [FORMULA]eV (dashed lines). The fake lines have all other transition properties the same as the original lines but with the gf-values ([FORMULA]dex) adjusted to return the same line depths. All lines have here been computed for the same 5 min sequence from the full convection simulation, which means that the bisectors differ slightly from the average of the whole 50 min simulation

[FIGURE] Fig. 5. The bisectors of the Fe I 608.2 (weak) and 621.9 nm (strong) lines (solid lines). Also shown are the corresponding (artificial) Fe I lines with [FORMULA] nm (dotted lines) and [FORMULA] nm (dashed lines). The fake lines have all other transition properties the same as the original lines but with the gf-values adjusted to return the same line depths. All lines have here been computed for the same 5 min sequence as in Fig. 4

The temporal evolution of the granulation pattern also introduce changes in the line profiles. Solar granules have typical lifetimes of about 10 min but since the numerical box covers typically [FORMULA] granules at any time, the influence of growing and decaying granules are relatively modest, as illustrated in Fig. 6. Since the emergent [FORMULA] is not enforced but rather is the result of the evolving granulation pattern, the continuum level varies slightly ([FORMULA], corresponding to [FORMULA]K) throughout the simulation, as well as the line strength. The bisector shapes are only slightly modified by the granulation, though the presence of oscillations in the simulation box shifts the bisectors back and forth in a regular fashion with a period corresponding to the solar 5 min oscillations. These numerical oscillations have an amplitude of about [FORMULA] m s-1 in the line-forming layers. The oscillations additionally broaden the line, though without altering the line strength since essentially the whole photosphere oscillates in unison without modifying the atmospheric structure.

[FIGURE] Fig. 6. The temporal evolution of the spatially averaged profile and bisector for the Fe I 608.2 line. The 40 profiles shown here are taken at one minute intervals from the full solar simulation. The intensity scale is normalized to the averaged continuum level for the full simulation sequence

4.3. Spatially averaged off-center intensity and disk-integrated flux profiles

Although not the main emphasis of the present paper a few Fe lines have been computed at different viewing angles (4 µ-angles and 4 [FORMULA]-angles) in order to enable a disk-integration to obtain flux profiles. A disk-integration requires further that the rotational broadening (1.8 km s-1 in the case of the Sun) is taken into account. The line formation of flux profiles is more complex than for intensity profiles due to the contributions from different disk positions. The well-known limb-darkening decreases the continuum intensities towards the limb but the lines also tend to be weaker due to the more shallow temperature gradient in the upper atmosphere. Furthermore, the granulation contrast decreases towards the limb, since higher layers with progressively smaller influence of the granulation is seen. At a given instant the line asymmetries vary significantly more towards the limb as the line-of-sight may pass through both granules and intergranular lanes and also the horizontal velocities may introduce Doppler shifts. Thus the Doppler shifts are less well correlated with the background continuum intensity, which introduces a more random nature of the resulting bisectors for inclined line-of-sights.

Fig. 7 illustrates the variation of the spatially and temporally averaged intensity profiles and bisectors of a weak Fe I line at various viewing angles, which make up the necessary profiles for a disk-integration. Also seen is the limb-effect (Halm 1907): the wavelength of solar lines increases towards the limb such that at small µ the typical convective blueshift seen at disk-center has disappeared completely or even been reversed into a small red-shift when removing the effects of gravitational redshift and rotation. The limb-effect is the same effect as present between strong and weak lines, namely that the core of the lines are formed in high enough photospheric layers where the granulation contrast and velocities have largely vanished. The red-shifted cores at very small µ, which are both observed and predicted with the 3D model atmospheres, are likely due to a bias for receding horizontal velocities to be viewed against the higher temperatures above intergranular lanes while the approaching gas will preferentially be seen against the lower temperatures above granules due to the temperature reversal in the higher photospheric layers (Balthasar 1985).

[FIGURE] Fig. 7. The spatially and temporally averaged Fe I 608.2 profile and bisector at different viewing angles. The line has been computed at 4 µ-angles and 4 [FORMULA]-angles. The profiles have been normalized to the continuum intensity at disk center. The thick solid lines represent the profile and bisector of the line at disk-center. Note the "limb effect" in the bisectors: smaller convective blue-shift toward the limb

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Online publication: July 7, 2000