3 Results

We have analyzed the AR faculae and network contrast dependence on both $\mu$and the measured magnetic signal, B. It is important to recall that the observed magnetic signal drops to zero at the limb, even if strong magnetic field regions are present. This is a straightforward consequence of the fact that magnetograms are only sensitive to the line-of-sight component of the magnetic field and that the magnetic field is mainly vertical. To compensate this effect to first order we have worked with B/$\mu$ (i.e. $\langle \vert\vec B\vert \cos\gamma\rangle$ $/\cos\theta$) instead of B. Second order effects due to radiative-transfer effects or finite thickness of flux tubes remain.

Figure 3: Facular and network contrast as a function of $\mu$ for eight intervals of the strength of the magnetic field, from network values (top left panel) to strong faculae (lower right). A dashed line at $C_{{\rm fac}}=0$ has been plotted. The solid curves represent a second order polynomial least-squares fit to the points. Every dot represents 40 data points. $\mu =1$ is the disk center; $\mu =0$ is the limb.

Figure 4: Dependence of the contrast on the absolute value of the magnetogram signal, corrected for foreshortening effects. The solar disk has been divided into eight bins, from center to limb. Note that some $\mu$-bins overlap (see text for details). As in Fig. 3, a dashed line at $C_{{\rm fac}}=0$has been plotted and solid curves represent a second order polynomial regression. Every dot represents 40 data points.

We have binned the B/$\mu$ values into eight intervals that range from the threshold level of, on average, 15 G to about 600 G. The intervals have been chosen so that they roughly contain the same number of pixels each, with the exception of the highest B/$\mu$ bins which have fewer pixels. By sorting the magnetic field strength into different bins we can distinguish between the CLV of the magnetic features present in regions with different filling factor $\alpha$. Figure 3 displays the contrast, $C_{{\rm fac}}$, as a function of $\mu$, individually for every B/$\mu$interval. A second order polynomial has been fitted (employing least squares) to guide the eye and a dashed line indicating $C_{{\rm fac}}=0$ has been included for clarity. To avoid overcrowding we have binned data points in sets of 40 before plotting.

Figure 3 reveals a clear evolution of the behaviour of the contrast from one B/$\mu$ interval to another. Network features (top left panel) show a low and almost constant contrast, as compared with the very pronounced CLV of the contrast for active region faculae (bottom panels). Intermediate cases show a progressive increase of the contrast towards the limb as well as an increasingly pronounced CLV. When $B/\mu < 200$ G, the contrast peaks at $\mu\sim0.5$, while for magnetic signals B/ $\mu\geq200$ G this maximum shifts to lower values of $\mu$ (see Fig. 8 and its discussion in the text for a more quantitative analysis). Note also that for B/ $\mu\geq200$G the contrast is negative around disk center, while it is positive in the network (i.e. for smaller $B/\mu$). We will return to this point in Sect. 4. The large fluctuations of the contrast near the limb for intermediate and high magnetic signals are due to the distribution of active regions on the ten selected days.

Figure 4 shows the contrast as a function of B/$\mu$, for different positions on the solar disk. The solar disk has been divided into eight bins of $\mu$, centred on $\mu$ values ranging from 0.96 (disk center, top left) to 0.3 (limb, bottom right). Note that to keep the number of points in each bin approximately equal, $\mu$-bins lying closer to the limb are wider than the ones around $\mu =1$, showing therefore some overlap. Each point of the figure is obtained by binning together 40 points of data with similar B/$\mu$, and second degree polynomials have also been plotted, as in Fig. 3.

This figure shows that, in general, the contrast initially increases with B/$\mu$ before decreasing again. We expect it to continue decreasing for even larger values of B/$\mu$ representing pores and sunspots. At large $\mu$ the initial increase is small and the contrast basically decreases with B/$\mu$, while at small $\mu$ it mainly increases. For $\mu =1$, points with large B/$\mu$ show a negative contrast (as in the lower panels of Fig. 3), while points at the limb always have positive contrasts. The bins with $\mu>0.82$ do not display data for high magnetic signals, because their intensity is below the intensity threshold.

Given the regular behaviour of the contrast as a function of $\mu$ and magnetogram signal, $C_{{\rm fac}}(\mu, B/\mu)$, it seems appropiate to search for an analytical expression for this dependence. We have performed a multivariate analysis using a $(\mu, B/\mu)$ grid. The $\mu$ values have been binned linearly, with $\Delta\mu=0.1$. B/$\mu$ bins have been chosen to be equally spaced on a logarithmic scale, with $\Delta\log(B/\mu)=0.05$, in order to compensate for the fact that magnetic signals are mostly concentrated towards lower values (Figs. 3 and 4). The dimensions of the grid are $0.1\leq\mu\leq1$ and $17~\mbox{G}\leq(B/\mu)\leq630~\mbox{G}$, resulting in a $9\times31$bins grid. We do not consider points with B/$\mu > 600$ G to exclude bright features that might belong to pores observed near the limb. We are aware of the fact that 600 G is an arbitrary value for such a cutoff.

Each bin of the grid is defined by the averaged values of the contrast, $\mu$and B/$\mu$, over all the data points of that bin. Although the curves in Fig. 3 are only intended to guide the eye, they do reveal that second order polynomials fit the contrast as a function of $\mu$ well. We have fitted the bidimensional array of contrasts by a second order polynomial function of $\mu$ and a cubic function of B/$\mu$ of the form:

$$C_{{\rm fac}}(\mu, B/\mu) = \sum_{i,j} a_{j,i}\mu^{j}(\frac{B}{\mu})^{i},$$ (2)

where i runs from 1 to 3, j runs from 0 to 2 and a_j,i are the coefficients of the fit. The result of the fit is a surface of second order in position $\mu$ and third order in magnetic signal. The coefficients of the multivariate fit a_j,i are:

 

$$C_{{\rm fac}}(\mu, B/\mu) 10^{-4}\left[0.48+9.12\mu-8.50\mu^{2}\right]\left(\frac{B}{\mu}\right)$$ = (3)

$$+ 10^{-6}\left[0.06-2.00\mu+1.23\mu^{2}\right]\left(\frac{B}{\mu}\right)^{2}$$

$$+ 10^{-10}\left[0.63+3.90\mu+2.82\mu^{2}\right]\left(\frac{B}{\mu}\right)^{3}\cdot$$

The terms of Eq. (2) are grouped in Eq. (3) to make clearer the quadratic dependence of $C_{{\rm fac}}(\mu, B/\mu)$ on $\mu$ and the cubic dependence on $B/\mu$. When B/$\mu$ is small (< 100 G), the first order term in B/$\mu$ provides the dominant contribution. When B/$\mu$ is large ( ${\geq} 200$ G), first and second order terms in B/$\mu$ dominate the contrast, modulated by the contribution of the cubic term which plays a role in this range of magnetic signals. At disk center ($\mu=1)$, those terms result in a dominant negative contribution (Fig. 3). Note that the contrast is constrained to go through zero when B/$\mu =0$, as expected for the quiet Sun.

Figure 5: Polynomial surface of second order in $\mu$ and third order in B/$\mu$ obtained from a multivariate fit performed to the grid of contrasts, covering $\mu$ and B/$\mu$ values. Dashed vertical lines project the corners of the plotted surface onto the $\mu$-B/$\mu$ plane and indicate the region spanned by the fit.

The best-fit surface is shown in Fig. 5. The grid corresponds to the linear $\mu$-bins and the logarithmic B/$\mu$-bins taken for the fit. The shape of this surface is quite congruent with that shown by the observed contrast in Figs. 3 and 4. Note that the function given in Eq. (3) is valid only for the wavelength and spatial resolution of the MDI data, 6768 Å and $2\arcsec$ , respectively. For other values of these parameters we expect another dependence on $\mu$ and B/$\mu$. In particular, the absolute value of the contrast is expected to change.

To better estimate how this analytical surface fits the behaviour of the measured contrasts, we have sliced the surface in both directions, $\mu$ and magnetograph signal, and then compared the result with the measured values. In Fig. 6 the fitted surface is sliced along the $\mu$-axis (solid curve), at three sample magnetic signal ranges, representative of low (top panel), medium (middle panel) and high (lower panel) B/$\mu$ values. Dots represent measured contrasts. To avoid very crowded plots each dot represents 250 (top panel), 100 (middle) and 25 (bottom) data points, respectively. The different amounts of binning reflect the non-uniform distribution of points over the B/$\mu$ range. The multivariate regression surface fits quite well the plotted dependence of the contrast, although minor deviations are visible at small B/$\mu$. Figure 7 shows slices of the modeled surface along the B/$\mu$ axis (solid curves) and the corresponding binned data (dots), at three sample positions on the solar disk, from disk center (top panel) to near the limb (lower panel). The fitted curves now deviate somewhat more from the data points, most significantly for $0.54<\mu<0.66$, where the discrepancy can reach 0.01 in contrast.

Figure 6: Comparison of cuts through the surface (solid curves) returned by the multivariate analysis and the measured contrasts (dots) as a function of $\mu$, for 3 sample bins of corrected magnetic signal. Every dot represents 250 (top), 100 (middle) and 25 (bottom) data points.

Figure 7: The same as Fig. 6, but for cuts along the B/$\mu$-axis (solid curves) made at three positions on the solar disk. Dots represent measured contrasts. The plotted curves represent the same $\mu$ ranges as those of the data points. Every dot represents 200 data points.

The multivariate analysis yields to an expression for the contrast of photospheric bright features, $C_{{\rm fac}}(\mu, B/\mu)$, that cannot be directly compared with previous studies because, to our knowledge, no similar work has been done before. Quadratic functions have already been used by other authors (e.g. Foukal 1981) to fit the dependence of the facular contrast on position over the disk, although most of them use a function of the form $C_{{\rm fac}}(\mu)=b(1/\mu -a)$ (Chapman 1980). A quadratic function agrees quite well with the CLV proposed by the hot wall model. A cubic function has been used for fitting the dependence of the contrast on magnetic strength. In this case we do not have a physical reason, only the goodness of the fit with respect to other bivariate functional dependences tried (see Figs. 6 and 7) and the requirement to force the contrast through zero for a disappearing magnetic signal. We suspect that, in order to obtain a better empirical description of the dependence of facular contrast on $\mu$ and B/$\mu$, a larger number of free parameters is required.

The dependence of the peak of $C_{{\rm fac}}$ on $B/\mu$ is shown in Fig. 8. The $\mu$-values at which $C_{{\rm fac}}$ peaks, $\mu _{{\rm max}}$, have been represented against the corresponding magnetic signal in Fig. 8a. Figure 8b shows the peak $C_{{\rm fac}}$values reached, $C_{{\rm fac}}^{{\rm max}}$ (see Fig. 3), plotted as a function of $B/\mu$. Finally Fig. 8c shows $C_{{\rm fac}}^{{\rm max}}/(B/\mu )$ plotted against $B/\mu$ or, in other words, the dependence on the magnetic signal of the specific contrast per unit of magnetic flux. Errors in $\mu$ are estimated from the difference between the peak of the best-fit curves in Fig. 3 and the peak obtained directly from the data points. Error bars in $B/\mu$ correspond to the size of the $B/\mu$-intervals used in Fig. 3.

Figure 8: Dependence on the magnetic flux per pixel, $B/\mu$, of: a)  $\mu _{{\rm max}}$; b) $C_{{\rm fac}}^{{\rm max}}$ times 102; c) $C_{{\rm fac}}^{{\rm max}}/(B/\mu )$ times 104. See text for details.

A linear regression adequately describes the dependence of $\mu _{{\rm max}}$on $B/\mu$ for the precision achievable with the current data. The best fit straight line (solid line in Fig. 8a) is

$$\mu_{\rm max}=5.60\times10^{-1}-4.97\times 10^{-4} (B/\mu).$$ (4)

Thus, $\mu_{{\rm max}}=0.56\pm0.02$ when $B/\mu$ tends to zero, and $\mu_{{\rm max}}=0$ for $B/\mu\approx1120$ G from extrapolations of this curve. Figure 8c implies that the contrast per unit of magnetic signal decreases strongly with increasing magnetogram signal. Since individual flux tubes are not resolved by MDI, we cannot infer the intrinsic contrast of a flux tube from Fig. 8b, which obviously shows the same pattern as Fig. 4. However, by normalizing by $B/\mu$ we obtain a quantity that is roughly proportional to the intrinsic brightness of the flux tubes (assuming that the field strength of the elemental magnetic flux tubes lies in a narrow range as mentioned in Sect. 2).