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Astron. Astrophys. 317, 942-948 (1997)

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4. Results and discussion

4.1. Details for line properties for Fe I 617.3

To illustrate observable effects of the presence of small magnetic elements, we first give some detailed results for the 617.3 line from single spectrograms. From the smallest features seen in such spectrograms, we estimate a spatial resolution of 0[FORMULA]6-1[FORMULA]0, i.e. we certainly do not yet resolve single flux tubes. The data extracted from the I profiles are plotted in Fig. 2. At the positions with sufficiently strong V signal the curves are fully drawn, they are dotted otherwise. At spatial resolution not better than 0[FORMULA]5, the magnetic elements do not show up near disc centre by conspicuous continuum intensities. This is known (cf. Kneer & von Uexküll 1991 and references there), but still of some relevance, since Solanki & Briglevic (1992) from low spatial resolution V profiles came to the conclusion that magnetic elements, i.e. the flux tubes proper, in plages are noticeably darker than average. Such low intensities were later modeled by Grossmann-Doerth et al. (1994). We shall come back to this point below.

[FIGURE] Fig. 2. Properties of the I profiles of the Fe I  617.3 nm line along the spatial position. From top to bottom: continuum intensity [FORMULA] relative to the average; line centre intensity [FORMULA] ; and the full width at half depression FWHD. Solid lines: at positions with V signal; dotted lines: otherwise. The large deviations of [FORMULA] at the slit position [FORMULA]  stem from the fiducial mark at the entrance slit of the spectrograph.

Close to the limb, the photospheric facular points in plages appear as bright features. Our observations from the position cos [FORMULA] = 0.28 (not shown here) show enhancements by 10-15% in the continuum intensity at the same spatial resolution. The maximum continuum intensity always appears about 0[FORMULA]5 closer to the limb than the maximum V signal.

The relative line centre intensities along the spatial direction are plotted in the second panel of Fig. 2. Obviously, magnetic signals are accompanied by substantial line centre intensity enhancement, which is known as the "gap phenomenon" (cf. Kneer & von Uexküll 1991). The FWHD of the I profiles is increased at positions where magnetic signal occurs and where the line centre intensity is increased (that of the average quiet Sun profile at disc centre (from the flat fields) amounts to [FORMULA] 9.0 pm).

Fig. 3 shows the data obtained from the V profiles. The uppermost panel shows the signed extremal amplitude [FORMULA] at positions where it is larger then 0.02 [FORMULA]. The highest values are above 0.10 [FORMULA]. Comparing these data with the first panel of Fig. 2, one obtains the impression that the strongest V signals occur at positions where [FORMULA] is lower than average. In the present observations by about 5% at the positions [FORMULA]  and [FORMULA]. However close inspection of this and other spectrograms shows that this is not regularly the case. Again, we shall expand on this point further below.

[FIGURE] Fig. 3. Properties of the V profiles of the Fe I  617.3 line along the spatial position. From top to bottom: signed extremum of V, separation of the V extrema [FORMULA], amplitude asymmetry [FORMULA], velociy [FORMULA] of the zero crossing of the V profile (solid lines) in comparison with the velocities [FORMULA] (dotted lines) determined from the I profiles. Large-scale motions are subtracted from [FORMULA] and [FORMULA].

In the subsequent panel, the separation [FORMULA] of the position of the V extrema is plotted. The error of the individual values can be estimated from the differences between the fitted and the observed profiles. It amounts to [FORMULA] pm. The average [FORMULA] is 12.0 pm with fluctuations between 10.0 and 14.0 pm. Calculations by Amer & Kneer (1993) and Kneer et al. (1996) have shown that this [FORMULA] is much larger than the limit set by the weak field approximation. Using this separation to measure the field strength, one arrives at 135.0 mT on average with fluctuations between 112.5 and 157.5 mT. These values are in good agreement with previous determinations (e.g. Beckers & Schröter 1968, cf. also the references given in Sect. 1). We note however that calculations of both static flux tube models (Kneer et al. 1996) and dynamic models (Grossmann-Doerth et al. 1994) fail to reproduce the separation of the extrema: some additional broadening of the profiles is always needed to obtain the observed [FORMULA].

In most cases the V profiles of Fe I  617.3 possess blue-asymmetries near disc centre (panel 3 of Fig. 3), with an average value [FORMULA] of 0.11 and fluctuations between 0.0 and 0.25. This confirms earlier results by Amer & Kneer (1993; see also Solanki 1993, 1995, and references therein). The error of individual values is found from [FORMULA], where [FORMULA] is the estimated error of the amplitudes [FORMULA] and [FORMULA]. With [FORMULA] and [FORMULA] in the range of 0.05-0.10, we obtain errors of the asymmetry [FORMULA] of about 0.14-0.07, i.e. decreasing with increasing [FORMULA] and of the same order as [FORMULA] itself.

The bottom panel of Fig. 3 depicts the velocity [FORMULA] together with the small-scale granular motions after subtraction of large-scale flows (positive velocities are directed away from the observer). From the difference between the observed and fitted zero crossings, one obtains an error estimate of [FORMULA] 130 m s-1. The rms value of [FORMULA] is 410 m s-1 in agreement with the finding of Martnez Pillet et al. (1996), while that of [FORMULA] is 280 m s-1. Multiplying the rms value with [FORMULA] (assuming a Gaussian velocity distribution), one arrives at a macroturbulent parameter [FORMULA] = 580 m s-1 for the zero crossing velocity. This falls short by factors of 3-5 of the value needed to fit the V profiles observed with low spatial resolution (cf. Keller et al. 1990, Rüedi et al. 1992). Thus, the high velocities in the magnetized plasma are not (yet) seen. A similar concern was already expressed by Fleck & Deubner (1991). We note, however, that within one magnetic patch of a few arcsec extent, [FORMULA] may vary by 1.0-1.5 km s-1 (see also Fleck & Deubner 1991 and Volkmer et al. 1995). Also, [FORMULA] may exhibit large differences from [FORMULA] at the same location. The latter measures, to a large extent, the velocity in the non-magnetic atmosphere surrounding the small-scale flux tubes.

Recently, Steiner et al. (1996) have given a possible solution to the discrepancy between the rather low [FORMULA] seen at high spatial resolutions and the high "macroturbulent" velocity derived from low spatial resolution data. Their numerical simulations of the dynamics of small-scale magnetic fields embedded in the granular flow exhibit strong vertical velocity gradients and shocks within the magnetic element. In such scenarios, the velocities are intrinsically hidden from high angular resolution observations: radiative transfer effects average out the contributions to [FORMULA] from widely differing velocities along the line of sight and the V profile exhibits a large separation [FORMULA] of the extrema.

4.2. Correlations between profile parameters of the Fe I 617.3 line

We now combine the data from several Fe I  617.3 nm line spectrograms and give some examples of correlations among profile parameters in regions with sufficiently strong V signal. This gives typical, average line properties and their fluctuations from high resolution observations. We present these data because they give constraints for modeling small-scale magnetic structures together with their ambient atmosphere.

In Fig. 4, the pairs [FORMULA] - [FORMULA], [FORMULA] - [FORMULA], and [FORMULA] - FWHD are plotted. With the achieved resolution, the presence of small-scale flux tubes does not noticeably change the continuum intensity [FORMULA] at disc centre. The upper panel of Fig. 4 shows that, at positions with V signal, [FORMULA] fluctuates by [FORMULA] % about the average [FORMULA] calculated from the continuum intensities at all positions. The mean continuum intensity, taken only from positions with V signal, amounts to [FORMULA]. However, a weak relation between [FORMULA] and [FORMULA] with a correlation coefficient -0.57 is present: strong V signal occurs preferentially in "dark" areas. Our limited spatial resolution does not allow us to discriminate between the two possibilities: either the "low" continuum intensity stems mainly from the ambient medium, e.g. from intergranular lanes, or the "low" intensity is an intrinsic property of the flux tubes in plages, as deduced by Solanki & Briglevi (1992). Kneer et al. (1996) estimate from their flux tube models that the area filling factor (at [FORMULA], or z = 0) at positions with highest V signal in the present observations is 25-50%. Thus, a substantial part, if not most, of the continuum radiation does indeed come from the ambient, non-magnetic atmosphere. However, Grossmann-Doerth et al. (1994) have shown that models with low intensity are feasible. It would be of interest to know how such models compare with observation when the continuum intensity is averaged over areas that include the surrounding, non-magnetic atmosphere.

[FIGURE] Fig. 4. Relationships between various profile parameters of Fe I  617.3

[FORMULA] depends on the internal flux tube structure, essentially its run of temperature (cf. Kneer et al. 1996), and on the total flux within the resolution element, i.e. the filling of the considered area with magnetic structure (middle and lower panels of Fig. 4). Likewise, the line width FWHD is a function of the temperature and density structure inside and outside the flux tubes and of the magnetic flux within the resolution element. On the other hand, the separation of the V extrema, [FORMULA], does not depend on the amount of light from non-magnetic regions, i.e. the filling factor: It is determined solely by the flux tube properties and their fluctuations within the resolution element. Thus, [FORMULA] is not expected to be related to the other parameters.

Parameter pairs with fair to high correlations are depicted in Fig. 5. These are the combinations FWHD - [FORMULA], [FORMULA] - [FORMULA], and FWHD - [FORMULA]. All three parameters have similar dependencies on temperature/density, total flux, and contribution from non-magnetic areas. At the extreme of low spatial resolution and low detected flux only averaged V profiles carry information about flux tubes, again averaged over various thermal, magnetic, and dynamic states. With increased resolution, the properties of the I profiles may serve, in addition to the V profiles, to deduce the structure of flux tubes and of their neighbourhood.

[FIGURE] Fig. 5. Relationships between the profile parameters [FORMULA], FWHD, [FORMULA] of Fe I  617.3.

The flux tube velocities [FORMULA] are uncorrelated or only weakly correlated with [FORMULA], [FORMULA], and [FORMULA], so we comment upon this point only briefly. It appears reasonable that the amplitude of the V signal, i.e. essentially the magnetic filling, has no influence on the measurement of the velocity of the magnetic plasma. Remarkably, [FORMULA] does not depend on the separation of the V extrema [FORMULA]. For strong magnetic fields, [FORMULA] may be used as a (zeroth order) proxy of the field strength. It does depend, however, on the line width and thus on unresolved velocities, as noted above. The more vigorous the dynamical processes are, the higher are their influence on the separation [FORMULA]. Dynamic model calculations by Steiner et al. (1996) give strong blue-shifts of the V profile simultaneously with wide separations of the V extrema.

There is a very weak tendency (correlation coefficient 0.43) for increased blue-asymmetry with increased downflow. This contradicts the siphon flow model of Degenhardt & Kneer (1992) in which an enhanced blue-asymmetry of the V profiles is produced by upflows in flux tubes. Yet data with higher signal/noise are needed to settle this point.

Consistent with the results of Fig. 5, parameters as FWHD and [FORMULA] are uncorrelated with [FORMULA] in the same way as [FORMULA].

4.3. Correlation of line parameters among different lines

Some relations of line parameters among different lines appear reasonable and expected, so we will not show them as figures. Some typical parameters of the I and V profiles may be extracted from Fig. 1. First, the values of [FORMULA], [FORMULA], and FWHD of two simultaneously observed lines are correlated: in any pair of the observed spectral lines the same magnetic elements are seen and essentially the same mixture of magnetic and non-magnetic areas contributes to the signals. Second, the velocities [FORMULA] are correlated within the error bars of the individual measurements. Finally, the asymmetries [FORMULA] of two lines are uncorrelated, simply because the [FORMULA] errors of the individual values are as large as the values themselves.

The separation of the V extrema [FORMULA] of the two line pairs Fe I  617.3 - Fe I  615.1 and Fe I  617.3 - Fe II  614.9 are shown in Fig. 6, along with the [FORMULA] error bars of the individual determinations (large crosses). The correlation coefficients are low: 0.42 for 617.3/615.1 and 0.20 for 617.3/614.9, due to the errors of the individual determination. However, the wide fluctuations of the [FORMULA] values of Fe I  617.3 and Fe I  615.1 are significant and indicate a large variability of magnetic flux tubes, or of clusters of flux tubes within the spatial resolution element. Further work is required to settle this point. The Fe II  614.9 line exhibits the smallest fluctuations of [FORMULA], presumably due to its low Lande factor, which is about 1/2 of that of the Fe I  617.3 line.

[FIGURE] Fig. 6. Separation [FORMULA] of V extrema of the line pairs Fe I  617.3 - Fe I  615.1 and Fe I  617.3 - Fe II  614.9. The large crosses indicate the 1 [FORMULA] errors of the individual determinations of [FORMULA].
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© European Southern Observatory (ESO) 1997

Online publication: July 8, 1998