4 Discussion

4.1 Comparison with previous observations

Comparison with other contrast observations is not easy because of the differences in the selected wavelength, spatial resolution, range of studied heliocentric angles, magnetic filling factor and size of the analyzed features. All these factors contribute to the scatter between the existing contrast measurements.

Our results differ from earlier observations of the contrast of bright features, specially when considering magnetic signals $B/\mu > 200$ G at disk center. Previous measurements of disk center facular contrast have frequently yielded positive values, although they usually were close to zero. Thus, from multi-color photometric images, Lawrence (1988) measures $C_{{\rm fac}}\sim0.005$ at disk center, and Lawrence et al. (1988) find $C_{{\rm fac}}=0.007\pm0.001$. In fact, our results agree better with those of Topka et al. (1992, 1997) and Lawrence et al. (1993) despite the difference in spatial resolution and studied wavelength. For $200~\mbox{G}\leq(B/\mu)\leq600~\mbox{G}$ the agreement is also surprisingly good. Nevertheless, these authors distinguish between active regions and quiet Sun. For active regions they always measure a negative contrast around disk center (for $\mu=0.97$ and $\mu=0.99$) irrespective of the magnetogram signal, while we get slighly positive contrast values for $B/\mu\leq200$ G in agreement with their results for the network. Since at these field strengths most of our signal originates in the network, this agreement is probably not surprising.

Chapman & Klabunde (1982) claim that the contrast shows a sharp increase near the limb (and even fit a $\mu^{-1}$ dependence). We find that $C_{{\rm fac}}$ peaks between $\mu=0.5$ and $\mu=0.2$, depending on the magnetic strength of the signal, and then decreases towards the limb (Fig. 3). Libbrecht & Kuhn (1984, 1985) also find this behaviour; however, they give $\mu\leq0.2$ for the peak of the contrast. Wang & Zirin (1987) and Spruit's hot wall model also give a similar value for the $\mu$ at which $C_{{\rm fac}}$ peaks. It is worth noting that Libbrecht & Kuhn (1984, 1985) and Wang & Zirin (1987) do not take into account the magnetic field of the observed feature, which makes the comparison between our results and theirs more difficult, as the CLV obtained is different when features are selected according to their brightness rather than the magnetogram signal. In the former case there is a bias towards brighter features. Our results indicate that the higher the magnetic signal, the smaller the $\mu$-value at which the contrast peaks (see Fig. 8a) so that network-like features dominate at disk center and features with increasingly large $B/\mu$ closer to the limb. This should move the peak of the contrast to smaller $\mu$ when the brightest features are searched for, than when magnetograms are used to identify faculae. Finally, it should be pointed out that, for increasingly smaller $B/\mu$ values the contrast becomes increasingly independent of $\mu$; this agrees with the conclusion of Ermolli et al. (1999) that the network contrast is almost independent of $\mu$.

It is remarkable that an expression as given by Eq. (2) reproduces the dependence of the contrast of bright features on their position ($\mu$) and on the magnetic flux per pixel (B/$\mu$), within the range $0.1\leq\mu\leq1$ and $17~\mbox{G}\leq(B/\mu)\leq630~\mbox{G}$. A relative accuracy of better than 10% is achieved almost everywhere within this domain. However, this multivariate analysis is only a first step and considerable further work needs to be done, since two other relevant parameters for the contrast, namely the wavelength and the spatial resolution, are kept at fixed values (those prescribed by MDI) in our analysis. A 4-dimensional data set is thus needed. A first step was taken by Lawrence et al. (1993), who compared observations from different instruments. At least some further progress in this direction can be achieved by employing MDI high resolution data, although off-center pointing is required.

4.2 Comparison with flux-tube models

MHD models including self-consistent energy transfer predict that small flux tubes (diameters smaller than 300 km) appear bright at disk center but with decreasing contrast near the limb; somewhat larger tubes are predicted to appear dark at disk center but bright near the limb, and finally, very large flux tubes (pores and sunspots; not considered in this study) are predicted to be dark everywhere (e.g., Knölker & Schüssler 1988, 1989). In such models the contrast at $\mu\approx0.1$ is largely determined by the brightness of the bottom of the flux tube (and the brightness of its surroundings, e.g. granular down flow lanes), while the CLV of the contrast is strongly influenced by the visibility of the hot walls. The bottom of a flux tube is defined as the horizontal optical depth unity surface in the interior of a flux tube.

Our results are qualitatively in accordance with this prediction if we make two reasonable assumptions. First, the network and facular features are composed of a mixture of spatially unresolved flux tubes of different sizes. Second, the average size of the flux tubes increases with increasing magnetogram signal or filling factor. Under these assumptions the upper panels of Fig. 3 refer to, on average, small flux tubes which dominate the network, while the lower panels of that figure refer to larger tubes mostly present in AR faculae. In our study the contrast always has a minimum at $\mu =1$ and increases with decreasing $\mu$ (as part of the hot wall becomes visible), until a maximum when the contrast peaks (the maximum surface of the hot wall is seen). Closer to the limb the contrast decreases as less wall surface is exposed. There are, however, clear differences between small $B/\mu$ network flux tubes and tubes found in AR faculae, i.e. regions with large $B/\mu$. Network tubes are bright everywhere on the solar disk and exhibit a low contrast (Fig. 8b), but a high specific contrast (Fig. 8c). This implies that network flux tubes are brighter than AR flux tubes and partly reflects the fact that network flux tubes are hotter than AR tubes (e.g., Solanki & Brigljevic 1992; Solanki 1993). The greater brightness at large $\mu$ implies that network flux tubes have a hotter bottom than larger flux tubes. Since this is also true at $\mu\leq0.6$, it suggests that the walls of smaller tubes, or of tubes in regions with lower filling factor, are hotter as well. This is in agreement with the theoretical finding of Deinzer et al. (1984b) that the inflow of radiation into the tube leads to a cooling of the surroundings and a lowering of the temperature of the walls. This temperature reduction is indeed predicted to be greater for larger flux tubes (Knölker & Schüssler 1988).

A mixture of flux-tube sizes at a given $B/\mu$ is needed because the CLV of $C_{{\rm fac}}$ at small $B/\mu$ does not agree with the predictions for any size of flux tube. The model flux tubes are all bright over only a relatively small range of $\mu$ values. Hence the mixture of flux tube sizes is needed in order to produce a relatively $\mu$-independent contrast, as exhibited by magnetic features at small $B/\mu$. As can be seen in Fig. 3, the contrast shows a more pronounced CLV as tube size increases, in accordance with the hot wall model, and larger tubes have a negative contrast at disk center, as predicted. The high specific contrast of small $B/\mu$ features (Fig. 8c), and the fact that their contrast is positive over the whole solar disk indicates that a change in the magnetic flux of the network has a much larger contribution to the change of the irradiance than a similar change in flux in active regions.

From Fig. 8a we can determine the heliocentric angles that make the contrast peak, $\theta_{{\rm max}}$. For the intervals displayed on Fig. 3, $\theta_{{\rm max}}$ is $63\degr$, $55\degr$, $58\degr$, $62\degr$, $66\degr$, $68\degr$, $72\degr$ and $77\degr$, respectively. Assuming the hot wall model with a simplified cylindrical geometry for the flux tubes, a Wilson depression $Z_{{\rm W}}$ of 150 km (Spruit 1976) and the derived $\theta_{{\rm max}}$ values, it is possible to roughly estimate the average value of the tube diameter for each magnetic range. Taking into account that the maximum depth of the wall $Z_{{\rm W}}$ is seen when the angle between the local vertical to the tube and the line of sight is equal to the heliocentric angle, then the diameter D should be $D=Z_{{\rm W}}\tan(\theta_{{\rm max}})$. Applying this approximation to our observations, we obtain diameters of 290, 210, 240, 281, 334, 365, 460, and 650 km respectively, for the mentioned $\theta_{{\rm max}}$ values and their respective magnetic ranges. These diameters are estimated to be uncertain by approximately a factor of two. For example, uncertainties in $Z_{{\rm W}}$translate into proportionate relative uncertainties in D.

Finally, we wish to draw attention to the wiggle of the measured contrasts around $\mu=0.95$ in Figs. 3 and 6. This can be observed at all magnetic strengths. At $\mu =1$ the contrast has a minimum value, then increases, descending a bit later for still smaller $\mu$'s before finally increasing slowly towards the limb. Topka et al. (1992) show some of these variations very close to disk center in Fig. 5 of their paper. They argue that such variations are partly due to the inclination of the flux tubes of opposite polarities toward each other in the active region they observe. However, we average over many network elements on multiple days and the persistence of such a structure is surprising, in particular also for small $B/\mu$ values where the statistics are extremely good.