© ESO, 2013

1. Introduction

The measurement of magnetic fields in the quiet solar photosphere is a complicated task because it relies on the measurement of weak polarization signals and on inversion methods to relate the observed Stokes profiles to mostly unresolved magnetic field vectors. It has been stressed for a long time that this last operation is an ill-posed problem, where in principle, the uniqueness of the solution is not guaranteed. In these cases, it is of crucial importance to carefully select spectral lines providing us with so-called good diagnostics of the magnetic field, in order to avoid ambiguities in the inversion. It is also often necessary to add a priori information about the solution.

High-resolution spectro-polarimeric measurements performed with the spectropolarimeter at the Solar Optical Telescope (SOT/SP) onboard the satellite Hinode in the FeI 630 nm line pair are now widely available (Lites et al. 2013). They have been used extensively to determine solar photospheric magnetic fields through Milne-Eddington inversions of both the linear and circular polarization in the two lines. Using simulated Stokes profiles derived from 3D MHD models of the solar photosphere, Orozco Suárez et al. (2010a,b) have shown that a simultaneous inversion of the Stokes profiles in both lines, carried out with one single magnetic field vector and identical thermodynamic parameters (and a prescribed depth-independent line opacity ratio) even increases the accuracy of the inversion. Based on this inversion method, several authors found that the internetwork magnetic fields of the quiet Sun are weak and highly inclined, in contrast to network fields which are mostly strong and vertical (Orozco Suárez & Bellot Rubio 2012; Lites et al. 2008). The same kind of inversion method, but with a different treatment of the magnetic filling factor, was applied to Themis observations in Bommier et al. (2009) and Bommier (2011). The analysis suggests that in the photospheric internetwork strong kGauss fields are more likely vertical with small filling factors, and that the weakest fields are more horizontal with larger filling factors. This picture is compatible with scattered narrow fluxtubes consisting of strong vertical fields expanding and weakening with height like in small-scale magnetic loops.

Fig. 1 Polarization images in the FeI 630.15 nm line. The 1024-pixel spectrograph slit was orientated along the north-south solar axis and scanned 380 positions along the east-west direction. Left panel: southern solar limb, right panel: solar disk center. The colors are coded according to the logarithm of the total polarization normalized by its mean value over the image. Color code: the mean polarization in the image is coded in black. The logarithmic polarization range in each image is divided into four intervals indicated by four colors, red: [Mean + (Max − Mean)/2;Max], magenta: [Mean;Mean + (Max − Mean)/2], cyan: [Mean − (Mean − Min)/2;Mean], blue: [Min;Mean − (Mean − Min)/2]. Max and Min denote, respectively, the maximum and minimum of the normalized polarization in the image on a logarithmic scale. At the limb, Max/Mean = 17.01 and Min/Mean = 0.52; at disk center Max/Mean = 24.32 and Min/Mean = 0.44.

On the other hand, line-ratio technics applied to Hinode observations of the FeI 630 nm line pair have been used by Stenflo (2010, 2011) who detected two magnetic populations in the quiet Sun, with respectively strong and weak intrinsic strengths. The strong fields are mainly vertical and found in the intergranular lanes, whereas the weaker fields have an isotropic distribution and are preferentially located above the bright granules. The physical mechanism of convective magnetic collapse could lead to this magnetic field distribution (Parker 1978; Spruit 1979; Spruit & Zweibel 1979). There is, however, an on-going debate about the validity of the line-ratio method when applied to the two FeI 630 nm lines that are not formed at the same heights in the photosphere (see Martínez González et al. 2006).

In this paper we do not intend to measure the magnetic field vector in the quiet Sun. Our aim is to characterize the magnetic regions by considering polarization images in the two FeI 630 nm lines, and by inspecting their cross-correlations with images of the granulation and of the reversed granulation at varying limb distances. We consider the total polarization signal in a line, defined by (1)where the wavelength λ₀ is the line center wavelength and 2Δλ is the spectral range where the line absorption is detected. At each pixel along the spectrograph slit we determine λ₀ from the location of the minimum of the line intensity profile. The total linear and unsigned circular polarization images, obtained by summing and over the lines, are also presented briefly. The total polarization signal is mostly dominated by the circular polarization, it is stronger in the network elements, but it is detected on the whole solar surface. It thus allows us to explore both the network and internetwork regions.

Fig. 2 Upper-left panel: linear polarization image in the FeI 630.25 nm line at the southern solar limb. Upper-right panel: linear polarization images in the FeI 630.25 nm line at solar disk center. Lower-left panel: unsigned circular polarization image in the FeI 630.25 nm line at the southern solar limb. Lower-right panel: unsigned circular polarization images in the FeI 630.25 nm line at solar disk center. The colors are coded according to the logarithm of the polarization normalized by its mean value over the image. For the linear polarization, at the limb Max/Mean = 16.00 and Min/Mean = 0.56; at disk center, Max/Mean = 10.05 and Min/Mean = 0.52. For the unsigned circular polarization, at the limb Max/Mean = 23.14 and Min/Mean = 0.26; at disk center Max/Mean = 35.75 and Min/Mean = 0.23. Color code: same as in Fig. 1.

In the absence of noise, the total polarization signal arises from magnetized regions only. It may thus be used as a tracer of the spatial distribution of these regions. However, in the unavoidable presence of noise, a weak polarization signal may also arise from non-magnetic regions. In order to study a possible effect of noise patterns on our investigations we computed polarization-noise images with the same integral as in Eq. (1), but in the continuum of the spectral domain where a spurious polarization signal is found due to the presence of noise. Then we examined the 2D correlation and cross-correlation of the noise-images with granulation images to estimate the impact of any noise spatial patterns on our data.

As shown in the following, the correlation of noise images is not a Dirac function, as it would be for a simple white noise. However, the amplitude of the correlation peak is very small compared to the amplitude of the peak in line-polarization correlations, and its width is half the width of line-polarization correlation peak. The origin of the polarization noise patterns is unclear, but we can say that they appear on spatial scales which are significantly smaller than the granulation scale and with a very tiny contrast. In consequence, they cannot significantly affect the cross-correlation method that we use in this paper.

We stress that we used statistical methods that take advantage of the very good spatial resolution of the Hinode data on quite large images. We computed the cross-correlation of polarization images with the granulation on 60′′ × 20′′ regions where more than 1000 granular structures were detected. Furthermore, we took advantage of our center-to-limb set of images to interpret the behavior of the cross-correlation signals.

2. Polarization images

We used high-resolution spectroscopic scans of the granulation obtained in the FeI 630 nm line pair with SOT onboard Hinode, on December 19, 2007. The 1024-pixel spectrograph slit parallel to the north-south polar axis of the Sun was successively located at 20 latitudes allowing us to continously scan the full center-to-limb variations of the solar disk image. At each location a 380-step west-east scan was performed, with a step size of 0.16 arcsec. The pixel size along the slit is 0.16 arcsec, which limits the spatial resolution of the images to 0.32 arcsec. We note that on December 19 the geometrical distance on the Sun surface, seen under one arcsecond, is 714 km. The pixel size was then 114 km on the solar surface. Because each 380 × 1024 image covers a large part (60.8′′ × 164′′) of the solar diameter, we divided it into eight 380 × 128 images (60.8′′ × 20′′), where we assumed that the value of the heliocentric angle remains approximately constant (we considered its value at the central pixel). The exposure time was 4.8 s per slit position. The spectral resolution of the SOT spectrograph is 2.15 pm/px in our spectral domain between 630.08 and 630.32 nm. The observational data we analyzed consist of (380 × 128) images at 160 successive heliocentric angles along the solar diameter and at 112 wavelengths scanning both lines. The noise level on polarization is on the order of 0.1% of the intensity in the continuum.

At each spatial pixel of the scans we computed the total polarization signal in the two FeI 630 nm lines defined in Eq. (1), together with the total linear polarization signal and the unsigned circular polarization signal. We performed the analysis described in this paper for both lines of the FeI 630 nm pair, but in the following we only show some examples of the cross-correlation images obtained for a few limb distances and one of the two lines, as these figures are very similar in both lines. Figure 1 shows an example of the FeI 630.25 nm total polarization images obtained at the center of the solar disk and at the south solar limb, and Fig. 2 shows linear and unsigned circular polarization images in this line for the same scans as in Fig. 1.

We see in Figs. 1 and 2 that the total polarization signal is mainly dominated by circular polarization. It is larger in the network patches, but we do measure circular polarization (and total polarization) all over the solar disk, and outside network elements as well. This is even clearer in images taken outside the solar disk center where the line of sight is inclined with respect to the vertical.

2.1. Correlations of polarization images

Fig. 3 Comparison of total polarization and granulation images. Southern hemisphere. Upper panels, from left to right: 2D cross-correlation of FeI 630.15 nm polarization image with the granulation, 2D correlation of the granulation image seen in the wings of the FeI 630.1 nm line, and 2D correlation of the polarization image in the FeI 630.1 nm line. Lower panels, from left to right: north-south cut along the polar axis of the 2D cross-correlation of FeI 630.15 nm polarization image with the granulation, north-south cut along the polar axis of the 2D correlation of the granulation image, and north-south cut along the polar axis of the 2D correlation of the polarization image. The images are at μ = 0.66 (sinθ = 0.75) in the southern hemisphere.

Fig. 4 Comparison of total polarization and granulation images. Northern hemisphere. Same as in Fig. 3, but for images at μ = 0.66 (sinθ = 0.75) in the northern hemisphere.

In Faurobert et al. (2012) images of the granulation were obtained at 25 line-cord levels in the FeI 630 nm lines, from level 1 at line center, to level 25 close to the continuum level. Reversed granulation appeared in the line-core images, at line-cord levels smaller than 8. In the following we have used, as granulation images, the sum of the line-wing images from level 25 to level 18; in Faurobert et al. (2013) it was shown that all these levels are formed at altitudes below 30 km at the bottom of the photosphere.

Figures 3 and 4 show examples of the 2D correlations of the line-wing granulation and of the total polarization, for images at μ = 0.66 in the southern hemisphere and in the northern hemisphere, respectively (μ denotes the cosine of the heliocentric angle), together with the 2D cross-correlation of the FeI 630.15 nm total polarization and of the line-wing granulation, that we will comment on later. The amplitude of the line-polarization correlation peak is smaller than the amplitude of the granulation correlation peak by a factor of about 2 × 10^-4; this is consistent with the order of magnitude of the polarization signal, which is at most a small percentage of the continuum intensity. It should be noticed here that before computing the correlations and cross-correlations, we always subtract the mean of the images so that we work with zero mean intensity (or polarization) images. It then follows that negative values for the cross-correlation actually means negative correlation. The average value of the image correlation is also zero; this is the reason why the positive correlation peaks are surrounded by regions of negative values.

Fig. 5 Left-hand panel: radius of the correlation peak of the granulation seen in the continuum (pink curve), and in the wings of the FeI 630.15 nm (blue curve) and FeI 630.25 nm lines (red curve), as functions of the image position. For images in the southern hemisphere we use − sinθ as the coordinate, whereas in the northern hemisphere we use sinθ (θ is the heliocentric angle). Right-hand panel: same as the left-hand panel but for the correlation peaks of the polarization images.

In Fig. 5 we show the center-to-limb variations of the equivalent radius (in arcseconds) of the correlation peaks of the granulation seen in the continuum and in the wings of the two FeI lines, and of the correlation peaks of polarization images. We derived the equivalent correlation radius in the following way. We first computed the volume of the 2D correlation peak over a circular integration domain of radius 21 pixels; then we divided this volume with the amplitude of the correlation peak and the equivalent radius was computed assuming a circular shape for this surface. The radius of the integration disk was chosen to limit the integral to the significant part of the correlation images, as no significant structures were found farther away from the origin. The characteristic scale of the correlation is twice its radius. We note that for the granulation we recovered a characteristic scale around 1.2′′, on average. The correlation sizes obtained for line-wing granulation images are in good agreement with the results for continuum images. We obtained roughly the same characteristic scale for the correlation of polarization images in the lines (1.3′′), but it showed large fluctuations. We did not measure significant center-to-limb variations of its average value.

2.2. Noise patterns

To rule out the effect of noise on our morphological study of the polarization images, we also computed images of the polarization obtained in the continuum of the spectrum. As there should be no polarization signal in the continuum, we used this means to investigate possible patterns of the polarization noise.

At first sight, continuum polarization images did not seem to show any structures; however, we did compute the correlation of continuum-polarization images and their cross-correlation with the granulation seen in the continuum. An example of the results obtained at μ = 0.66 in the southern hemisphere is shown in Fig. 6. The amplitude of the continuum-polarization correlation peak is smaller than the amplitude of the line-polarization correlation peak by a factor of about 100; this is consistent with the order of magnitude of the polarization rms-noise which is approximately ten times smaller than the maximum of the line-polarization signal. However, the origin of the continuum-polarization correlation peak is not clear. A residual cross-talk between the intensity and the other Stokes parameters could lead to spurious polarization in the continuum that could be correlated with the granulation, but in that case, the width of the continuum polarization correlation peak should be on the same order as the width of the granulation correlation peak. Figure 5 shows that this is not the case. The width of the correlation peak of the continuum polarization is on the order of 0.6 arcsec, which is half the characteristic scale of the granulation.

Whatever the origin of the noise, it seems to us that its effect on line-polarization patterns at the granulation scale is negligible, both because of the low amplitude of the peaks due to noise patterns and because of the difference in their characteristic scales.

3. Cross-correlation of polarization images with the granulation

3.1. Total polarization images

Fig. 6 Noise patterns. Upper panels: from left to right, cross-correlation of polarization noise patterns with the granulation, correlation of the granulation image in the continuum, and correlation of the polarization noise patterns. The images are at μ = 0.66 in the southern hemisphere. Lower panels from left to right: north-south cuts along the polar axis of the 2D images shown in the upper panels.

In order to characterize the spatial distribution of magnetized regions with respect to the granulation pattern, we studied the 2D cross-correlation of line-polarization images with granulation images. We noticed that the cross-correlation signal increased significantly when we used for the granulation image the sum of the images obtained in the line wings between level 25 and level 18.

We observe both a positive and a negative peak in the 2D cross-correlation of polarization images with the granulation. Examples are shown in Figs. 3 and 4 for images at μ = 0.66 in the southern and northern hemispheres. Figure 7 shows the center-to-limb variations of the amplitudes of the maximum and of the minimum of the cross-correlation, for both lines and for the polarization noise seen in the continuum.

Fig. 7 Left panel: value of the maximum of granulation and polarization cross-correlation. Blue curve: polarization in the FeI 630.15 nm line, red curve: polarization in the FeI 630.25 nm line, pink curve: positive peak in the cross-correlation of granulation with continuum polarization. Right panel: value of the minimum of the cross-correlation; the curves have the same meaning as in the left-hand panel. The horizontal axis gives the value of − sinθ for images in the southern hemisphere and of sinθ for images in the northern hemisphere, where θ is the heliocentric angle.

Fig. 8 Comparison of total polarization images and granulation images at disk center. Left panel: cross-correlation of granulation and polarization images for a scan close to the center of the solar disk. Right panel: auto-correlation of the polarization image at disk center.

The two peaks are present for all the images of our data set, except at disk center, with varying amplitudes. The presence of these two peaks shows that the polarization signal of the quiet Sun is due to two spatially different components. One component is mainly located over the bright granules, so it gives rise to the positive cross-correlation peak with the granulation pattern, whereas the other one is mainly located in the intergranular-lanes and it gives rise to the negative cross-correlation peak. We observe that close to the center of the solar disk, the amplitude of the negative peak (in absolute value) is approximately 3 times larger than the amplitude of the positive peak, but at intermediate limb distances the amplitude of the negative peak decreases sharply; both peaks show large fluctuations from one image to the other, but their amplitudes are quite often on the same order of magnitude. We also remark that the positive peak is at the level of the positive peak of noise patterns for many scan positions, whereas the negative peak is always well above the noise level.

When observed out of disk-center the two peaks are separated because they are not centered at the center of the cross-correlation image, but are shifted, with different shifts. Instead at disk center the two peaks are superimposed and the strongest peak, i.e., the negative one, dominates (see Fig. 8). The shifts of the peaks have roughly opposite values for images taken at the same heliocentric angle in opposite hemispheres (see Figs. 3 and 4). We assign the origin of these shifts to the perspective effect arising when images formed at different depths in the solar atmosphere are viewed at an angle. This indicates that the two magnetic components that we have detected in the FeI line-polarization are not localized at the same depth in the photosphere. But because the two peaks are blended, we cannot accurately measure the location of their respective centers.

3.2. Linear and unsigned circular-polarization images

We also computed linear and unsigned circular-polarization images and their cross-correlation with the granulation. As expected, the results for the unsigned circular polarization are very similar to the results we discussed previously for the total polarization. However, unexpectedly, we also find measurable cross-correlation signals with linear polarization images. This is illustrated in Fig. 9 where we show the cross-correlation images for both the linear and unsigned circular polarization obtained in the FeI 630.25 nm line. We obtain similar results with the other less magnetic-sensitive line, but with smaller amplitudes of the cross-correlation peaks. Only a positive peak is detected in the cross-correlation of the granulation with linear polarization images. The amplitude of the peak (1.9 × 10⁵ and 2.8 × 10⁵ for the images shown in Fig. 9 on the northern and southern hemisphere, respectively) is often only slightly larger than the amplitude of the peak in the cross-correlation of the polarization noise with the granulation (1.5 × 10⁵) because only a small fraction of the pixels in the images have a linear polarization signal larger than the noise level. However, the peak is slightly displaced toward the north in the northern hemisphere and toward the south in the southern hemisphere, like the positive peak that is observed in the cross-correlation of the granulation with the unsigned circular polarization.

Fig. 9 Comparison of linear and unsigned circular polarization with granulation images. Upper-left panel: cross-correlation of granulation and linear polarization in the FeI 630.25 nm line at μ = 0.66 (sinθ = 0.75) in the southern hemisphere. Upper-right panel: cross-correlation of granulation and unsigned circular polarization in the FeI 630.25 nm line at μ = 0.66 in the southern hemisphere. Lower-left panel: cross-correlation of granulation and linear polarization in the FeI 630.25 nm line at μ = 0.66 in the northern hemisphere. Lower-right panel: cross-correlation of granulation and unsigned circular polarization in the FeI 630.25 nm line at μ = 0.66 in the northern hemisphere.

We measured the displacement of the positive peak in the linear polarization and granulation cross-correlation. The results are shown in Fig. 10. We observe large statistical fluctuations of the measured shifts from one scan to the other. This shows that we would need to accumulate more signal to improve the statistics of this measurement. This is not surprising because the linear polarization signal is very weak in the quiet sun, long exposure times are necessary to increase the number of pixels where a significant linear polarization signal can be measured (see Bellot Rubio & Orozco Suárez 2012). In the following section we study the cross-correlations of the polarization images with the reversed granulation seen at line centers, and we show that we are able to measure the perspective shift of the polarization component located in the intergranular lanes.

Fig. 10 Center-to-limb variations of the displacement (in arcsec) of the cross-correlation peak of linear polarization and granulation images. Left panel: for the FeI 630.15 nm line. Right panel: for the FeI 630.25 nm line. The displacements have been divided by sin(θ) to correct the perspective effect from the projection on the plane of the sky. Results obtained in the northern (southern) hemisphere are shown as a function of sinθ (− sinθ).

4. Cross-correlation of polarization images with the reversed granulation

Fig. 11 Comparison of total polarization with reversed granulation images. Upper-left panel: cross-correlation of FeI 630.15 nm polarization image with the reversed granulation at μ = 0.66 (sinθ = 0.75) in the northern hemisphere. Upper-right panel: cross-correlation of FeI 630.15 nm polarization image with the reversed granulation at μ = 0.66 in the southern hemisphere. Lower-left panel: north-south cut along the polar axis of the 2D cross-correlation shown in the upper-left panel. Lower-right panel: north-south cut along the polar axis of the 2D cross-correlation shown in the upper-right panel.

Fig. 12 Center-to-limb variations of the displacement (in arcsec) of the cross-correlation peak of polarization and reversed granulation images. Left panel: for the FeI 630.15 nm total polarization. Right panel: for the FeI 630.25 nm total polarization. The displacements have been divided by sin(θ) to correct the perspective effect from the projection on the plane of the sky. Results obtained in the northern (southern) hemisphere are shown as a function of sinθ (− sinθ). The thick blue lines show least-squares fittings of the measurements by constant values on each hemisphere.

Figure 11 shows examples of the 2D cross-correlation of line-polarization images with the reversed-granulation images at line centers, for the same heliocentric angles as in Figs. 3 and 4. We note that only a positive correlation peak is now observed. It is due to the polarization component which is located in the intergranular lanes (seen bright in the line-core intensity). The other component is not detected; it should give rise to a negative peak, which is probably too small to be detectable here. The correlation peak is displaced along the north-south axis with respect to the origin; the displacements observed at symmetrical positions with respect to the solar disk center are opposite, as shown in Fig. 12.

We note here that we can only measure displacements along the north-south direction because we are using cross-correlations between images which must be observed simultaneously. So we determine the shifts on 1D brightness distributions along the slit for each slit location in the scan. The measured shift is then obtained from an ensemble average over the 380 slit positions of the scan (for more details about the method, see Grec et al. 2007). The averaging eliminates statistical fluctuations and we obtain a measurement of the systematic effects on the ensemble of the granular structures of the images.

We have measured the displacement of the cross-correlation peak for all 160 positions of the images along the north-south axis, both for the FeI 630.15 nm and for the FeI 630.25 nm line. The results are shown in Fig. 12. We were able to measure the shifts even quite close to disk center, up to μ = 0.95. Then we derived the corresponding height difference by correcting the shift for the projection on the plane of the sky, i.e., by dividing it with the sinus of the heliocentric angle. This enhances the errors at the center of the solar disk where we obtain quite a large dispersion of the measurements. However, we observe that on average the displacements have opposite values in the southern and in the northern hemispheres, so this confirms that we may assign these north-south displacements to a perspective effect due to the difference of image formation-heights. The signs of the displacements indicate that the polarization image is formed lower in the photosphere than the reversed granulation image. The formation-depth difference increases slightly toward the limb. It is on average 0.85 pixels for the FeI 630.25 nm line and 1.36 pixels for the FeI 630.15 nm line.

We notice that even though granular and intergranular regions have quite different formation depths, the method we have briefly presented above allows us to measure formation depth differences between the bright features of the reversed granulation image and of the polarization image. We obtain values of 100 km and 150 km, for the FeI 630.25 nm and FeI 630.15 nm lines, respectively. Faurobert et al. (2012, 2013) have shown that the bright features observed at line centers are formed, respectively, around 220 km and 250 km above the bright granules observed in the continuum, so it seems that the polarization signal arising from the intergranular lanes is formed roughly at the same heights for the two lines, i.e., around 100–120 km above the level of the bright granules seen in the continuum.

5. Conclusion

In this work we have investigated the distribution of the magnetic regions of the quiet Sun revealed by total polarization images in the two magnetic sensitive FeI 630 nm lines and their center-to-limb variations. We stress that the total polarization, which is dominated by the circular polarization signal, is detected all over the quiet-Sun surface, and not only in the network patches so it allows us to explore the magnetic field distribution both in the network and in the internetwork. We find that the polarization image correlation peaks have approximately the same widths as the granulation correlation peaks; this shows that the granular scale is a characteristic scale of the quiet Sun magnetized regions distribution. We may interprete this result in the following way. In the polarization maps the stronger signals are found preferentially in the intergranular lanes, thus leading to a charateristic scale on the order of the granulation scale. The weaker signals found over the granules are likely to play a secondary role in the correlation analysis.

The cross-correlations of total polarization images with images of the granulation show both a positive and a negative peak, with different perspective shifts when the images are taken out of the solar disk center. At disk center the perspective effect vanishes and the two peaks are superimposed; only the strongest, negative peak can then be detected. The presence of these two peaks is the signature of two spatially separated magnetized components. One of them is mainly localized over the bright granules and gives rise to the positive peak, whereas the second one is mainly localized in the intergranular lanes and gives rise to the negative peak. The differential perspective effect between the two components indicates that the granular polarization is formed deeper in the photosphere than the intergranular. However, because the two cross-correlation peaks are blended, it is not possible to accurately measure their height difference.

We also made a tentative study of the linear and unsigned circular-polarization images. The latter show qualitatively the same behavior as the total polarization image, whereas the linear polarization images show only a positive correlation peak with the granulation. The amplitude of the peak is slightly larger than the effect of the noise; furthermore, it is shifted toward the north in the northern hemisphere and toward the south in the southern hemisphere. However, the precise measurement of the perspective shift is not possible because of large statistical fluctuations

of this signal from one image to another. One would need either longer exposure times or to accumulate more images in order to increase the statistics of the linear polarization signal.

We also computed the cross-correlations of total polarization images with reversed granulation images obtained at the FeI line centers. They show a positive cross-correlation peak only, which is due to the intergranular magnetized component. The granular component, which should give rise to a negative cross-correlation peak with the reversed granulation is not detected. This could be due to the absence of the relevant scales in the spatial spectrum of the reversed granulation observed above the granulation contrast inversion layer. The positive cross-correlation peak due to the intergranular component is shifted by a perspective effect when the images are taken out of the solar disk center. We were able to measure this perspective effect and to derive that the intergranular total polarization signals in the FeI 630.15 nm and 630.25 nm are formed, respectively, 150 km and 100 km below the bright features seen at line centers.

The two magnetic components that we have detected could be related to the strong and weak magnetic components previously detected by Stenflo (2010, 2011) and/or to the vertical and horizontal fields intranetwork fields discovered by Lites et al. (2008) and Bellot Rubio & Orozco Suárez (2012) with completely different approaches based on inversion of the FeI 630 nm line Stokes profiles. We used total polarization images, without distinguishing between linear and circular polarization, as tracers of the magnetic field distribution all over the solar disk surface in order to follow its center-to-limb variations. This does not give any diagnostics on the angular distribution of the magnetic field vector, so we cannot here debate the question of the isotropy or anisotropy of the magnetic fields in the quiet Sun. This issue will be investigated in a future work.

Acknowledgments

Hinode is a Japanese mission developed and launched by ISAS/JAXA, collaborating with NAOJ as a domestic partner, NASA and STFC (UK) as international partners. Scientific operation of the Hinode mission is conducted by the Hinode science team organized at ISAS/JAXA. This team mainly consists of scientists from institutes in the partner countries. Support for the post-launch operation is provided by JAXA and NAOJ (Japan), STFC (UK), NASA, ESA, and NSC (Norway).

References

All Figures

Fig. 1 Polarization images in the FeI 630.15 nm line. The 1024-pixel spectrograph slit was orientated along the north-south solar axis and scanned 380 positions along the east-west direction. Left panel: southern solar limb, right panel: solar disk center. The colors are coded according to the logarithm of the total polarization normalized by its mean value over the image. Color code: the mean polarization in the image is coded in black. The logarithmic polarization range in each image is divided into four intervals indicated by four colors, red: [Mean + (Max − Mean)/2;Max], magenta: [Mean;Mean + (Max − Mean)/2], cyan: [Mean − (Mean − Min)/2;Mean], blue: [Min;Mean − (Mean − Min)/2]. Max and Min denote, respectively, the maximum and minimum of the normalized polarization in the image on a logarithmic scale. At the limb, Max/Mean = 17.01 and Min/Mean = 0.52; at disk center Max/Mean = 24.32 and Min/Mean = 0.44.

In the text

Fig. 2 Upper-left panel: linear polarization image in the FeI 630.25 nm line at the southern solar limb. Upper-right panel: linear polarization images in the FeI 630.25 nm line at solar disk center. Lower-left panel: unsigned circular polarization image in the FeI 630.25 nm line at the southern solar limb. Lower-right panel: unsigned circular polarization images in the FeI 630.25 nm line at solar disk center. The colors are coded according to the logarithm of the polarization normalized by its mean value over the image. For the linear polarization, at the limb Max/Mean = 16.00 and Min/Mean = 0.56; at disk center, Max/Mean = 10.05 and Min/Mean = 0.52. For the unsigned circular polarization, at the limb Max/Mean = 23.14 and Min/Mean = 0.26; at disk center Max/Mean = 35.75 and Min/Mean = 0.23. Color code: same as in Fig. 1.

In the text

Fig. 3 Comparison of total polarization and granulation images. Southern hemisphere. Upper panels, from left to right: 2D cross-correlation of FeI 630.15 nm polarization image with the granulation, 2D correlation of the granulation image seen in the wings of the FeI 630.1 nm line, and 2D correlation of the polarization image in the FeI 630.1 nm line. Lower panels, from left to right: north-south cut along the polar axis of the 2D cross-correlation of FeI 630.15 nm polarization image with the granulation, north-south cut along the polar axis of the 2D correlation of the granulation image, and north-south cut along the polar axis of the 2D correlation of the polarization image. The images are at μ = 0.66 (sinθ = 0.75) in the southern hemisphere.

In the text

	Fig. 4 Comparison of total polarization and granulation images. Northern hemisphere. Same as in Fig. 3, but for images at μ = 0.66 (sinθ = 0.75) in the northern hemisphere.
In the text

Fig. 5 Left-hand panel: radius of the correlation peak of the granulation seen in the continuum (pink curve), and in the wings of the FeI 630.15 nm (blue curve) and FeI 630.25 nm lines (red curve), as functions of the image position. For images in the southern hemisphere we use − sinθ as the coordinate, whereas in the northern hemisphere we use sinθ (θ is the heliocentric angle). Right-hand panel: same as the left-hand panel but for the correlation peaks of the polarization images.

In the text

Fig. 6 Noise patterns. Upper panels: from left to right, cross-correlation of polarization noise patterns with the granulation, correlation of the granulation image in the continuum, and correlation of the polarization noise patterns. The images are at μ = 0.66 in the southern hemisphere. Lower panels from left to right: north-south cuts along the polar axis of the 2D images shown in the upper panels.

In the text

Fig. 7 Left panel: value of the maximum of granulation and polarization cross-correlation. Blue curve: polarization in the FeI 630.15 nm line, red curve: polarization in the FeI 630.25 nm line, pink curve: positive peak in the cross-correlation of granulation with continuum polarization. Right panel: value of the minimum of the cross-correlation; the curves have the same meaning as in the left-hand panel. The horizontal axis gives the value of − sinθ for images in the southern hemisphere and of sinθ for images in the northern hemisphere, where θ is the heliocentric angle.

In the text

	Fig. 8 Comparison of total polarization images and granulation images at disk center. Left panel: cross-correlation of granulation and polarization images for a scan close to the center of the solar disk. Right panel: auto-correlation of the polarization image at disk center.
In the text

Fig. 9 Comparison of linear and unsigned circular polarization with granulation images. Upper-left panel: cross-correlation of granulation and linear polarization in the FeI 630.25 nm line at μ = 0.66 (sinθ = 0.75) in the southern hemisphere. Upper-right panel: cross-correlation of granulation and unsigned circular polarization in the FeI 630.25 nm line at μ = 0.66 in the southern hemisphere. Lower-left panel: cross-correlation of granulation and linear polarization in the FeI 630.25 nm line at μ = 0.66 in the northern hemisphere. Lower-right panel: cross-correlation of granulation and unsigned circular polarization in the FeI 630.25 nm line at μ = 0.66 in the northern hemisphere.

In the text

Fig. 10 Center-to-limb variations of the displacement (in arcsec) of the cross-correlation peak of linear polarization and granulation images. Left panel: for the FeI 630.15 nm line. Right panel: for the FeI 630.25 nm line. The displacements have been divided by sin(θ) to correct the perspective effect from the projection on the plane of the sky. Results obtained in the northern (southern) hemisphere are shown as a function of sinθ (− sinθ).

In the text

Fig. 11 Comparison of total polarization with reversed granulation images. Upper-left panel: cross-correlation of FeI 630.15 nm polarization image with the reversed granulation at μ = 0.66 (sinθ = 0.75) in the northern hemisphere. Upper-right panel: cross-correlation of FeI 630.15 nm polarization image with the reversed granulation at μ = 0.66 in the southern hemisphere. Lower-left panel: north-south cut along the polar axis of the 2D cross-correlation shown in the upper-left panel. Lower-right panel: north-south cut along the polar axis of the 2D cross-correlation shown in the upper-right panel.

In the text

Fig. 12 Center-to-limb variations of the displacement (in arcsec) of the cross-correlation peak of polarization and reversed granulation images. Left panel: for the FeI 630.15 nm total polarization. Right panel: for the FeI 630.25 nm total polarization. The displacements have been divided by sin(θ) to correct the perspective effect from the projection on the plane of the sky. Results obtained in the northern (southern) hemisphere are shown as a function of sinθ (− sinθ). The thick blue lines show least-squares fittings of the measurements by constant values on each hemisphere.

In the text