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Appendix A: Pitfalls with the horizontal field

In recent years there has been a controversy concerning the role and importance of the horizontal magnetic fields on the quiet Sun. Starting with the Hinode analysis of Orozco Suárez et al. (2007) and Lites et al. (2008) there have been frequently repeated claims that there is about 5 times larger horizontal than vertical flux density on the quiet Sun. In contrast an in-depth analysis of the identical Hinode data set by Stenflo (2010) finds that the angular distribution of field vectors is strongly peaked around the vertical direction for large flux densities, while the distribution widens as the flux density decreases to become isotropic in the limit of zero flux density. The conclusion is that there is no evidence for any preponderance of horizontal fields.

Such discrepancies can arise since it is vastly more difficult to diagnose transversal than longitudinal magnetic fields. In the following our discussion will refer to the disk center, where the horizontal fields are transversal, while the vertical fields are longitudinal, i.e., directed along the line of sight. One reason for the difference in difficulty is the low sensitivity of the linear polarization to weak magnetic fields as compared with the circular polarization. Thus the same polarization noise, when translated to flux densities, is larger for the horizontal fields by a factor of order 25. Another difficulty is that spatially unresolved substructures that are improperly accounted for lead to much larger errors in the determined horizontal fields than in the vertical ones. In this Appendix we will try to elucidate these pitfalls.

A.1. Meaning of flux densities and filling factors for the horizontal fields

Let us first try to clear up some confusion in the use of the concepts flux density and filling factor for the horizontal fields. Our use here of the concept “flux density” is equivalent to “average field strength”, meaning magnetic flux divided by the area of the resolution element. Since the term “resolution element” appears to refer to the 2D angular resolution element perpendicular to the line of sight, confusion often arises in the understanding of the term flux density for the transverse field, since this field component lies in the plane of the angular resolution element. However, we have to remember that the actual resolution element is a 3D volume, namely the area of the angular resolution element times the depth of spectrum formation along the line of sight. The shape of this volume element does not matter, we can imagine magnetic flux penetrating it in any direction, unrelated to the direction of the line of sight, and define a flux density (or average field strength) for the field component along any such direction. Flux density in the horizontal direction is therefore as well-defined a concept as flux density in the vertical direction.

A similar confusion has arisen around the use of filling factors for the horizontal field, with misled suggestions that one may need to use different filling factors for the horizontal and vertical fields. The source of this confusion again has to do with the distinction between 2D and 3D resolution elements. The magnetic filling factor is often defined as the fraction of the area (of the angular resolution element) that is occupied by the magnetic component, causing confusion as to what this area should be in the case of the horizontal field. This confusion is however a result of a misunderstanding of what filling factor means. To avoid such confusion it should always be seen as a volume filling factor (fraction of the 3D resolution element occupied by the magnetic component), never as an area filling factor. In the special case of vertical flux tubes the concepts of volume filling factor and area filling factor become identical, but in the general case (inclined fields) only the volume filling factor is a meaningful concept. It would be completely unphysical to use separate filling factors for vertical and horizontal fields. Whatever orientation a magnetic component has inside the volume resolution element, the occupation fraction of the volume must be identical for all field components, horizontal as well as vertical.

A.2. Apparent predominance of horizontal fields for an isotropic angular distribution

Another misleading habit in previous literature has been to form the ratio B_⊥/B_∥ and implicitly suggest incorrectly that when this ratio exceeds unity, then the field is preferentially horizontal rather than vertical. However, it is only meaningful to speak of a preference for horizontal fields if the angular distribution of the field vectors is flatter than an isotropic distribution, which is spherical. As explained in Stenflo (2010) the ratio B_⊥/B_∥ is not unity for an isotropic distribution, but π/2 ≈ 1.57. This is so because the horizontal field is Bsinγ, the vertical field Bcosγ, and the ratio between the angular averages of sinγ and cosγ is π/2. The reason for this artificial advantage of the horizontal field is that it is defined as the projection of the field vector onto a plane, while the vertical field is defined as the projection onto an axis. Since one has to compare the B_⊥/B_∥ ratio not with unity but with 1.57, the published ratio of about 5 implies a horizontal dominance with respect to the isotropic case by approximately a factor of 3, not a factor of 5. Still, as shown in Stenflo (2010), this remaining factor of 3 is much too high, and there is no evidence from the Hinode data that it exceeds unity. Next we will discuss the two main pitfalls that can easily lead to large spurious factors: the profoundly different ways in which noise and the filling factor affect the horizontal and vertical fields.

A.3. PDF distributions of measurement noise

In the weak-field limit (below a few hundred G, where the Zeeman splitting is still small in comparison with the line width) the following proportionalities apply in the relations between polarization and field strength: (A.1)where we have added index notations “obs” and “app” to cover the case when the field is not resolved and we have a filling factor f, so that (A.2)and similarly for U and V. If the “true” filling factor were unity, then the apparent field B _{⊥ ,   app} would equal the corresponding “true” field B_⊥. For a given filling factor f the average flux density is (A.3)but it does not equal B _{⊥ ,   app} (see below). In contrast (A.4)To quantitatively explore the influence of noise on the determination of weak magnetic fields we next assume a filling factor of unity (since this was also done in the analysis of Lites et al. 2008) and use the detailed calibrations in Figs. 19 and 21 of Stenflo (2010) that are representative of the uncollapsed flux population, in order to find the constants of proportionality in Eq. (A.1). If the polarizations Q, U, V are expressed in % of the continuum intensity and the field strength in G, then the constant of proportionality is 29.4 in the first equation and 184 in the second.

The polarization noise distributions in Q, U, and V are Gaussian. Since we have direct proportionality between B_∥ and V, the noise distribution in B_∥ is Gaussian as well. In contrast the noise distribution in B_⊥ is entirely different, due to the very non-linear relation between polarization and field strength.

	Fig. A.1 Histograms of the apparent flux densities for the horizontal field B ⊥ ,   app (solid curve) and the vertical field B∥,app (dashed curve), derived from the Gaussian polarization noise with standard deviations 0.035% in Q and U and 0.047% in V (in units of the continuum intensity).
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The measured standard deviations (1-σ errors) in % of the continuum intensity for the Gaussian noise distributions in the measured values of the polarization amplitudes in the Hinode SP deep mode data set are 0.035% for each of Q and U, and slightly worse, 0.047%, for V. These low-noise values are testimony for the excellent quality of the Hinode spectro-polarimeter. With Eq. (A.1) and their respective calibration factors these small polarizations translate into 1.37 G in the case of Stokes V and to the 25 times larger value of 34.5 G for Stokes Q or U.

With the known Gaussian distributions for Q, U and V and with Eq. (A.1) we have used Monte Carlo simulations to derive how the noise propagates into B_⊥. The result is shown in Fig. A.1. The noise distribution for B_⊥ spreads between 5 and 70 G. Due to the noise there will not be any apparent horizontal fields weaker than 5 G, while the analysis in Stenflo (2010) shows that the great majority of pixels have vertical flux densities that are smaller than 5 G. If horizontal flux densities that are only modestly affected by noise are incorrectly assumed to be solar and are combined with the many small vertical flux density values, then we immediately obtain a spurious picture where the horizontal fields appear to dominate. It does not take much of a residual of the noise ratio of 25 between the horizontal and vertical fields to infiltrate the analysis to account for the factor of 3 that has been claimed to represent the predominance of the horizontal fields.

A.4. Influence of incorrect filling factors for the horizontal fields

The second potential pitfall that would always work in the direction of spuriously enhancing the  ⟨ B_⊥ ⟩ / ⟨ B_∥ ⟩  ratio is to use too large a value for the filling factor. As seen from Eqs. (A.1)–(A.4) (A.5)In their analysis Lites et al. (2008) assumed a value of unity for f. If this assumption is incorrect, then the derived ratio between the horizontal and vertical flux densities is too large by a factor of . If for instance the “true” filling factor is 10%, then the average horizontal flux density is spuriously enhanced by more than a factor of 3, enough to account for the previously claimed predominance of the horizontal fields. As we have seen in Sect. 5, the great majority of pixels that represent the collapsed flux population have filling factors smaller than 10%. This illustrates that the filling factor effect alone can fully account for the factor of 3 discrepancy in the literature.

While the claim that there is a predominance of horizontal flux is based on the use by Lites et al. (2008) of filling factor unity at disk center, they also tried to apply Milne-Eddington inversions to determine the actual values of the filling factors. Since such inversions can only be used for observations with sufficient S/N ratio, they could only apply it to 1/3 of the pixels, which are representative of the strongest flux concentrations. They found that for this sample of pixels the typical filling factor was about 20%. This implies that their discrepancy factor of 3 (favoring horizontal fields with respect to an isotropic distribution) would get changed to , which is quite close to the value of 1.0 for the isotropic case. As we find (from our own analysis) that the majority of the 2/3 of the small-flux pixels that could not be analysed by inversion have substantially smaller filling factors, the remaining factor of 1.3 can easily be accounted for by the filling-factor effect alone. For this reason one can say that the Lites et al. (2008) analysis does not lead to any evidence that there is a preponderance of horizontal fields, in contrast to the widespread belief that is based on that paper.

A.5. Empirical tests near the Sun’s limb

In summary we have seen how a small infiltration of noise or an incorrectly assumed filling factor of unity will both artificially enhance the average horizontal flux density with respect to the average vertical flux density. Each effect on its own can easily account for the claimed preponderance of the horizontal fields by the factor of 3, but the combination of both makes the effect potentially much larger.

For those who still believe that there is a preponderance of horizontal fields, there is a straightforward test that may empirically settle the matter, namely to apply the same analysis technique to Hinode data near the solar limb. If we define the Stokes Q direction to be parallel to the radius vector, then near the limb the longitudinal Zeeman signatures in Stokes V diagnose the horizontal component, while the transverse Zeeman signatures of opposite signs in Stokes Q diagnose both the vertical component and the horizontal component that is parallel to the limb.

The best test, which does not depend on any comparison between horizontal and vertical fields, is to compare the apparent magnitudes of the two horizontal components, the one diagnosed by Stokes Q (in the transversal plane) and the one diagnosed by Stokes V (along the line of sight). Statistically these components must be of the same magnitudes on the quiet Sun. It would be totally unphysical if there would be a preference for the transversal plane, since the Sun does not care where the observer is located. If one would find that the apparent field in the transversal plane is systematically stronger, then the effects that we have discussed have infiltrated the analysis and made it faulty.

Limb observations that allow such a test were indeed presented in the paper of Lites et al. (2008). They used a deep mode Hinode recording of March 15, 2007, with the spectrograph slit crossing the south polar limb. Their results (in their Figs. 15 and 17) clearly show that there is always a huge preponderance of the apparent magnetic flux in the transversal plane as compared with the line-of-sight magnetic flux, for all center-to-limb distances, all the way to the limb. This systematic predominance of the transversal plane can hardly be a property of the Sun, since

near the limb the transverse plane is perpendicular to the horizontal plane.

This test can easily be applied to any data set that includes data near the limb and claims to measure the transverse magnetic fields on the quiet Sun correctly. For instance, it could be used to test the data reduction pipeline for HMI on the solar dynamics observatory (SDO) or for SOLIS.

© ESO, 2011