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Astron. Astrophys. 345, 635-642 (1999)

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4. The facular intensities

4.1. Centre-to-limb variation

Although the rough behaviour of the facular contrast as a function of limb angle is well established, there is disagreement concerning the details. The contrast is in the following defined as [FORMULA], where [FORMULA] is the intensity of the faculae and [FORMULA] the intensity of the quiet Sun. In general, the contrast is low at disk centre, (at IR wavelengths sometimes even negative for sufficiently large magnetic filling factors; Foukal et al. 1990; Wang et al. 1998), and increases out to limb angles of at least [FORMULA] (Frazier 1971; Stellmacher & Wiehr 1973; Chapman & Meyer 1986). But there is considerable debate as to whether the contrast continues to increase towards the limb as suggested by the measurements of Lawrence & Chapman (1988) and Taylor et al. (1998) , or whether the contrast peaks at around [FORMULA] to 0.3 and then decreases again towards the limb (Libbrecht & Kuhn 1984; Auffret & Muller 1991).

Fig. 3 shows a selection of contrast observations, as well as our calculations. This figure illustrates some of the problems that arise when trying to use observed contrast values in order to constrain facular models. The contrast not only depends on the wavelength (as can be seen by comparing the two data sets corresponding to different wavelengths of Wang & Zirin 1987), but also on the magnetic filling factor or average field strength, and very significantly on spatial resolution (compare the curve of Auffret & Muller 1991with that of Frazier 1971). Obviously, a good fit to all the data is not possible, or even physically desirable with a single model.

[FIGURE] Fig. 3. Selected facular contrast measurements, [FORMULA], as well as contrasts calculated using our model, plotted vs. [FORMULA], where [FORMULA] is the angle between surface normal and the line of sight. [FORMULA] and [FORMULA] are the facular and quiet-sun intensity respectively. The solid lines linking the different symbols show the measurements of facular contrast. The squares, crosses, diamonds and triangles indicate data by Frazier (1971), Auffret & Muller (1991), Wang & Zirin (1987) and Taylor et al. (1998) respectively. The curve linking the plus signs is the parameterisation adopted by Lawrence (1988) for his measurements taken with a filter centred at 524.5 nm. For the Wang & Zirin measurements, the set with the higher contrast values was taken with a filter centred at 386 nm, the lower set with a filter centred at 525 nm, although the passband was not specified. The measurements by Taylor et al. (1998) are in arbitrary units. We have normalised them so that they agree with the measurements by Frazier at [FORMULA]. The dotted, dashed and dot-dashed curves are the calculated contrasts in different filters. The dotted line is for 572-578 nm, representative of the filter used by Auffret & Muller; the dashed line is for 470-550 nm. Above 500 nm, the contrast hardly depends on the filter width. The dashed line should therefore be representative for the measurements by Frazier (1971), Lawrence (1988) and the lower set of Wang & Zirin (1987) that were all taken with a green filter. The dot-dashed line is for a filter centred at 386 nm. In this wavelength region, the filter width becomes very important - changing the passband from 50 nm to 10 nm predicts contrasts that are almost twice as high.

In order to correctly reproduce most facular contrast measurements, one would have to construct a complex, comprehensive model that includes the fine structure in the faculae, i.e. magnetic flux tubes located at the boundaries of (abnormal) granules. For this, one has to know the geometry (size, expansion with height, Wilson depression, see e.g. Spruit 1976), and the correct temperature structure inside the magnetic features (e.g. Bellot-Rubio et al. 1997, Briand & Solanki 1998, Frutiger & Solanki 1998) as well as in their surroundings. The variation of these quantities, e.g. size (Spruit & Zwaan 1981 , Keller 1992 , Grossmann-Doerth et al. 1994) and temperature (Solanki & Stenflo 1984 , Solanki & Stenflo 1985 , Solanki & Brigljevic 1992), with the amount of flux must be included. Finally, the emerging spectrum from the model for the magnetic filling factor (usually unknown because unmeasured) that is appropriate to the observations must be calculated including the spectral lines and then filtered with the same filter profile as underlies the observations. The magnetic filling factor is the fraction of a given part of the solar surface covered by magnetic field.

This is obviously a daunting task. And even after all this effort, it is likely that success will only be partial, due to the incompatibility between the various observations, and the often unknown magnetic filling factor, spatial resolution and filter function appropriate to the observations.

The facular model we use is one-dimensional and hence neglects all the fine-scale structure. It corresponds to a given (but uncalibrated) magnetic filling factor. Our main aim in this paper is to carry out the last step of the procedure outlined above, namely to calculate the spectral contrast as well as its centre-to-limb variation and compare it with measurements of this quantity (see Sect. 4.2)

The general centre-to-limb variation of the calculated contrast agrees reasonably well with the measurements by Frazier (1971) and Stellmacher & Wiehr (1973) (not shown, but similar to that of Frazier), though our contrast values tend to be higher at disk centre. The absolute value of the contrast probably just reflects the different magnetic filling factors underlying the observations and the model. Frazier (1971) has shown that the facular contrast increases with increasing spatially averaged magnetic field strength, which is equivalent to the magnetic filling factor. His measurements for the faculae with the strongest field strengths are in relatively good agreement with the calculations, albeit still lower. The more recent, high-spatial-resolution measurements by Auffret & Muller (1991) (crosses) indicate much larger contrasts, though the limb-dependence of their contrast values are not in very good agreement with our model. Note, however, that their contrast measurements are of the network bright points and not of spatially averaged faculae, as described by our model. Chapman & Meyer (1986) have parameterized their measurements in terms of [FORMULA] and found b to be about 0.1 when a was taken to be unity. This yields a much steeper gradient than any other measurements presented here or indeed our calculations. Using the same parameterisation, Lawrence (1988) Lawrence 1988 finds coefficients of [FORMULA] and [FORMULA] at 524.5 nm. The resulting curve (indicated by the plus-signs in Fig. 3) is in good agreement with our calculations, in particular if we take into account that his coefficients are for "average" faculae, as they were determined by linear regression to individual measurements that show large scatter. The individual contrast values at [FORMULA], e.g., lie between 2 and 12%. His average curve hence corresponds to a relatively low filling factor.

The steep increase of the contrast near the limb predicted by our model is at least partly the result of the plane-parallel approximation we make. Although this may be an artefact, we do not expect it to seriously influence irradiance reconstructions, since the relative effect of these parts of the solar disc very close to the limb is small, due to their small contribution to the disk area and the limb darkening.

4.2. Colour dependence at a given limb angle

The colour dependence of the facular contrast has been measured by a number of authors, e.g. by Chapman & McQuire (1977) and Lawrence (1988). Chapman & McQuire (1977) measured the facular contrast in five filters between a limb distance of [FORMULA] and [FORMULA], i.e. [FORMULA] and 0.33. After normalising the contrasts by setting the contrast value at 530 nm to unity, they found that their data followed an inverse wavelength dependence. We find excellent agreement between our calculations and their measurements. This is shown in Fig. 4 where their normalised contrast measurements (diamonds), the inverse wavelength law (dotted line) and the contrasts calculated from our facular model (solid line) are plotted. Note that the colour dependence of the contrast does not simply scale with changing limb angle. We found that the curves become slightly steeper towards the blue, and show less variation over small wavelength ranges towards the limb. These small-scale spectral variations are due to spectral lines, which show a heightened contrast relative to the continuum most strongly at large µ. The effect is relatively small, however, as can be seen from Fig. 5.

[FIGURE] Fig. 4. The spectral variation of the facular contrast. The diamonds are the data from Chapman & McGuire (1977). They were obtained from measurements between 16 and 53" (µ = 0.18 to 0.33). The dotted line shows the inverse wavelength fit suggested by Chapman & McGuire.

[FIGURE] Fig. 5. The spectral variation of the facular contrast for different limb angles. The solid line is for the disk centre, the dashed and the dotted lines are for limb angles of [FORMULA] and 0.3 respectively. The contrasts have all been normalised to be unity at 530 nm. At 530 nm, the contrast at the disk centre is about 1.3 and 2.5 times smaller than at [FORMULA] and 0.3 respectively.

Lawrence (1988) measured the contrasts of a large number of faculae, most of them between limb angles of [FORMULA] and [FORMULA]. As pointed out earlier, the contrast values show a large scatter (probably for some of the reasons mentioned in Sect. 4.1). When the contrasts in the different wavelength bands are plotted against each other, the scatter is reduced noticibly, and the ratios between the colour contrasts can be calculated. This confirms our suspicion that a large part of the scatter in the contrast measurements is due to different magnetic filling factors.

These ratios (for three different filters) are shown as the solid lines in Figs 6a to c, along with an indication of the error on the slope (dotted lines). Our calculations of the ratios for the different limb angles are in good agreement with the measurements; they are plotted as the diamonds on Figs 6a to c. We stress that the model (i.e. the temperature stratification) was not in any way optimised by us to fit either of these data sets 1. Note that the inclusion of line blanketing (via the ODFs) in our modelling is of crucial importance for the reproduction of the spectral contrast.

[FIGURE] Fig. 6. Facular contrasts in percent for three different colour filters (green, red and IR, see Lawrence 1988) plotted against each other. The solid lines are the ratios derived from the contrast measurements of Lawrence (1988), along with his error estimates (dotted lines). The diamonds are our calculated contrasts. The diamonds with the highest contrast values are for a limb angle of [FORMULA]. The y-axis offsets of the ratios are arbitrary, as they could not be derived from the measurements by Lawrence (1988).

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Online publication: April 19, 1999