Estimating the longitudinal magnetic field in the chromosphere of quiet-Sun magnetic concentrations

The WFA can be applied when the Zeeman splitting is much smaller than the Doppler width of the spectral line under consideration and the longitudinal component of the magnetic field is constant along the LOS (see Sect. 9.6 in Landi Degl’Innocenti & Landolfi, 2004). Under these circumstances, the observed Stokes $V$ profile is related to the observed Stokes $I$ profile as

We compute $B_{\|}$ as the maximum-likelihood estimation from Eq. (1) assuming Gaussian noise with zero mean and variance $\sigma^{2}$, which leads to

The Ca II 854.2 nm line covers a large range of formation heights (e.g., Uitenbroek, 2006; Cauzzi et al., 2008; Quintero Noda et al., 2016). Its core is formed in the low- to mid-chromosphere, while its wings have a photospheric contribution. The variation from the LTE wings to the nonLTE chromospheric core occurs at $\pm$0.03–0.04 nm from the line center (Cauzzi et al., 2008). Therefore, the estimation of $B_{\|}$ values from the low- to mid-chromosphere is possible if the spectral range is restricted to the line core (Carlin & Asensio Ramos, 2015). Given its wide formation region, delimiting spectral ranges in the $\lambda$8542 line is a common method to infer $B_{\|}$ at certain heights (e.g., de la Cruz Rodríguez et al., 2013; Esteban Pozuelo et al., 2019; Morosin et al., 2020). Recently, this approach was also applied to synthetic data in the Mg II h and k lines (Afonso Delgado et al., 2022). In this paper, we applied Eq. (2) to different spectral ranges across the line to inspect how the $B_{\|}$ estimates differ. Specifically, we selected almost all the spectral sampling ($\Delta\lambda=\pm 1.040$ Å), a limited range including both lobes of the $V$ profile ($\pm 0.325$ Å), and the inner-core positions ($\pm 0.130$ Å). Hereafter, we denote these ranges as $\mathrm{\Delta\lambda_{1}}$, $\mathrm{\Delta\lambda_{2}}$, and $\mathrm{\Delta\lambda_{3}}$, respectively.

We identified 19 cases as small concentrations. They appear in intergranular lanes and are highly influenced by neighboring granules. We focus on the case labeled ‘3’ in the first row of Fig. 1. During its temporal evolution (see the movie attached to Fig. 3), small positive-polarity field concentrations appear between granules and move closer to each other until they form a single structure. After that, the structure changes its shape in reaction to the motion of the surrounding granules.

Figure 3 also displays the variation of the blue-wing magnetogram (at –0.39 Å) and $B_{\|}$ with the scanning time $\mathrm{\Delta t}$. At the original cadence, estimates usually range from 100 to 360 G for $\mathrm{\Delta\lambda_{1}}$, and decay to 80–250 G and to 40–150 G for $\mathrm{\Delta\lambda_{2}}$ and $\mathrm{\Delta\lambda_{3}}$. Due to its dynamic behavior, the structure appears larger in maps at long $\mathrm{\Delta t}$. Furthermore, $V$ signals and $B_{\|}$ estimates vary with $\mathrm{\Delta t}$ depending on the individual $I$ and $V$ profiles that are summed and the number of scans involved. Despite pixel-by-pixel differences, the estimates obtained for each $\mathrm{\Delta\lambda}$ do not differ significantly with $\mathrm{\Delta t}$. We remind the reader that $\mathrm{\Delta t}$ refers to the time used to calculate the averaged Stokes $I$ and $V$ profiles (see Sect. 3).

We assessed the validity of the WFA conditions by comparing the observed Stokes $V$ profiles to the $I$ derivatives scaled by the estimates for each $\Delta\lambda$. Figure 4 displays these comparisons at two different locations as $\mathrm{\Delta t}$ increases. The two upper and lower rows show the $I$ and $V$ profiles from a pixel at the center and from another closer to the edge of the concentration, respectively. The plus symbols ($+$) and crosses ($\times$) in Fig. 3 indicate these two specific locations. Although these locations do not share the same features, their $I$ and $V$ signals rely on the imprints left at these pixels during the instants involved in the signal addition. Consequently, all the estimates vary with the scanning time. We note that the gradient between the estimates obtained for each $\mathrm{\Delta\lambda}$ changes very little with $\mathrm{\Delta t}$.

From a qualitative comparison, we observe a systematic divergence between the Stokes $V$ profiles and the scaled $I$ derivatives at the position of the blue lobe. Both curves only match in some pixels for $\mathrm{\Delta\lambda_{3}}$ (as in the fourth row of Fig. 4), but they slightly differ in others due to a relative shift between Stokes $I$ and $V$ at the line core (see the second row). This relative shift may be due to the coexistence of multiple components in the same resolution element. Using wider spectral ranges, the divergences between the $V$ profiles and the scaled $I$ derivatives remain regardless of $\mathrm{\Delta t}$.

At this point, we note that we only describe the goodness of the fits from a qualitative point of view. A quantitative description is also possible. In that case, we would conclude that the quality worsens with $\mathrm{\Delta t}$ because the noise level for $V$/$I$ decreases with the square root of the number of scans involved in each sum. This is mostly a consequence of the WFA not being able to explain the observations independently of $\mathrm{\Delta t}$. As such a conclusion seems counterintuitive to what we observe from a naked-eye comparison, we opted for a qualitative description of the quality of the fits.

In addition, we observe that the fit quality between Stokes $V$ and the scaled $I$ derivatives at the outer-wing positions improves close to the concentration edge (fourth row in Fig. 4). To examine this further, we compared the synthetic Ca II 854.2 nm $V$ profiles resulting from two model atmospheres with the STiC code (de la Cruz Rodr´ıguez et al., 2019). The model atmospheres are based on the FALC model (Fontenla et al., 1993) to which we added $B_{\|}$ values monotonically decreasing between log($\mathrm{\tau_{500}}$) = 1.3 and $-$5.3: (1) from 1 000 to 100 G, and (2) from 300 to 100 G. We observe similarities in the outer-wing positions of the synthetic $V$ profiles (shown in Fig. 5) and the observed ones, which indicates the high sensitivity of the $V$ signals of the outer-wings to $B_{\|}$ in the photosphere. Therefore, the fit quality in the outer-wing positions relies not only on the spectral range but also on the pixel location, as the field concentrations fan out with height. Other features may involve gradients in other parameters or the presence of an unresolved magnetic component not included here for simplicity.

In view of the above findings, we compared the estimates for $\mathrm{\Delta\lambda_{3}}$ and the outer-wing positions (between $\pm$1.040 and $\pm$0.520 Å), from which we expect mainly chromospheric and photospheric contributions, respectively. We denote the latter range as $\mathrm{\Delta\lambda_{4}}$. As it may change with height, we delimited the structure again using intensity maps at –0.520 Å, where we still discern the structure despite the presence of fibrils. However, Fig. 6a–b shows that the structure barely varies with height, whereas $B_{\|}$ varies conspicuously. Central and outer regions of the concentration show significant height gradients in $B_{\|}$ that resemble those used in the synthesis with STiC. Moreover, these gradients are consistent with time and are independent of the profile smoothing and of the noise decrease, as they appear regardless of $\mathrm{\Delta t}$ (see Fig. 4). Therefore, WFA conditions cannot be satisfied by adding signal.

Finally, we computed the magnetic flux from the chromospheric and photospheric contributions using the expression

We classified ten cases as medium-sized field concentrations; they have a practically fixed position and are observed as stable structures. In this subsection, we show an analysis of the case labeled ‘3’ in the second series (see Fig. 1), which is portrayed in Fig. 7. This positive-polarity case undergoes fragmentations and mergings (see the animation attached to Fig. 7). At the original cadence, the $B_{\|}$ estimates for $\mathrm{\Delta\lambda_{1}}$ and $\mathrm{\Delta\lambda_{2}}$ are usually above 200 G and exceed 350 G at the center of the structure. These values are weaker for $\mathrm{\Delta\lambda_{3}}$ overall, reaching $\sim$250–300 G. Because of its stable behavior, $B_{\|}$ maps barely change for $\mathrm{\Delta t}$ $<$ 5.25 minutes.

The intensity profiles at the center of the concentration are different from those emerging closer to the edges, as shown in the upper and lower rows of Fig. 8. Locations close to the edge usually show line asymmetries and shifts, which may indicate that the physical conditions at the center of the structure are firmer than in the outer regions. In particular, the intensity profiles represented in the lower panels display slight asymmetries compared to those in the upper rows. Stokes $V$ profiles also differ as amplitudes weaken away from the center. Aside from the $V$ amplitude variations, we do not find changes with $\mathrm{\Delta t}$ worth mentioning. Regarding the $B_{\|}$ estimates, the central and outer locations of the concentration show values of $\sim$420 and 260 G for $\mathrm{\Delta\lambda_{1}}$ at the original cadence, respectively. These $B_{\|}$ weaken by about 60 G and 100–200 G for $\mathrm{\Delta\lambda_{2}}$ and $\mathrm{\Delta\lambda_{3}}$, respectively. Moreover, the gradient between the estimates for each spectral range is stable with $\mathrm{\Delta t}$ and comparable to the individual values.

We also compared the mostly chromospheric and photospheric $B_{\|}$ estimates in this structure. The inferred values diverge significantly (see Fig. 9a–b), having median values of 105 and 524 G, respectively. Thus, the magnetic flux values for $\mathrm{\Delta\lambda_{3}}$ and $\mathrm{\Delta\lambda_{4}}$ differ again by one order of magnitude (Fig. 9c and Table 2) but, unlike the small case, they are stable with time. The height gradient in $B_{\|}$ also explains the poor fit quality between the $V$ profiles and the scaled $I$ derivatives for $\mathrm{\Delta\lambda_{1}}$ and $\mathrm{\Delta\lambda_{2}}$. In contrast, fits are usually better for $\mathrm{\Delta\lambda_{3}}$, although they are affected by relative shifts between Stokes $I$ and $V$ at some positions. All in all, the estimates for wider spectral ranges are unsafe as the WFA conditions are unfulfilled.

We identified two cases of large field concentrations; they are labeled ‘1’ and ‘2’ in the last row of Fig. 1 and may be an example of enhanced network. We focus on case 2, shown in Fig. 10, the polarity of which is negative.

The movie attached to Fig. 10 shows that the $B_{\|}$ values seem to oscillate with time, mostly at the center of the structure. As these oscillations coincide with overall intensity fluctuations, we think they are related to seeing instabilities instead of actual changes in the structure. At the original cadence, we infer estimates that progressively increase from the edges to the center of the structure, where they are of about $-$750 G for $\mathrm{\Delta\lambda_{1}}$. The $B_{\|}$ maps for $\mathrm{\Delta\lambda_{2}}$ and $\mathrm{\Delta\lambda_{3}}$ are similar to that obtained for $\mathrm{\Delta\lambda_{1}}$, though estimates are weaker reaching up to $-$650 and $-$550 G, respectively. In addition, the concentration grows with $\mathrm{\Delta t}$ due to interactions with nearby structures and to the variable seeing conditions, whereas $B_{\|}$ values weaken globally.

At the original cadence, asymmetries and blueshifts are apparent in the intensity profiles from the center of this concentration and from a pixel closer to the edge (see upper and lower rows of Fig. 11, respectively). These features persist with the scanning time until they soften at the longest $\mathrm{\Delta t}$ due to the profile smoothing. At the same time, $V$ amplitudes decrease with the distance to the center of the structure and barely change with $\mathrm{\Delta t}$. As a result of these changes along the concentration, $B_{\|}$ values differ significantly depending on the pixel location. We find that, at the original cadence, estimates range from $-$500 to $-$700 G and from $-$180 to $-$260 G for the different $\mathrm{\Delta\lambda}$ at the center and closer to the edge, respectively. These $B_{\|}$ gradients are stable with $\mathrm{\Delta t}$, and so the conditions along the formation region related to the used spectral ranges hardly vary with time.

Although not displayed, the difference between the mostly chromospheric and photospheric estimates in this concentration is not as significant as in smaller structures. The median values in this case for $\mathrm{\Delta\lambda_{3}}$ and $\mathrm{\Delta\lambda_{4}}$ are $-$160.7 and $-$360.3 G, respectively. Consequently, the photospheric magnetic flux is three to four times greater than the chromospheric flux (see Table 2). Similarly to the medium-sized case, the magnetic flux values inferred in this concentration have a stable temporal evolution. The reduced difference between the photospheric and chromospheric estimates may explain a better match between Stokes $V$ and the scaled $I$ derivatives regardless of the spectral range, scanning time, and location.

MOMFBD is a successful image-correction technique. However, its application comes with high computational cost and may lead to amplified noise levels (Puschmann & Beck, 2011).

Alternatively, images can be restored by adding the accumulations acquired per wavelength position. In this case, most of the reduction process is carried out using the CRISPRED pipeline (de la Cruz Rodríguez et al., 2015) and the effects of seeing in observations are partially corrected by applying the destretching technique. During this process, differential motions in small subfields of the wide-band images, which are previously derotated and aligned, are measured using cross-correlation. After combining accumulations, the polarimetric calibration is performed as in van Noort & Rouppe van der Voort (2008). Finally, each pair of images with orthogonal polarization states, recorded simultaneously by the narrow-band cameras, are combined to remove the seeing-induced polarization, and the residual crosstalk is corrected. We also corrected each monochromatic polarization image for high-frequency polarimetric interference fringes in a similar way to Pietrow et al. (2020). As each image is apodized in this process, we clipped the blurred edges of the output polarization maps, obtaining smaller images. The noise level of the resulting Stokes $V$ images is in the range of 1.8–2.5$\times$10^-3 measured on the $V$/$I_{0}$ maps at the first observed wavelength for each series. We refer to the datasets restored with and without the MOMFBD technique as M and NM datasets, respectively.

We examined 27 of the cases shown in Fig. 1 because some of them are partially or completely beyond the image boundaries of the NM datasets. Each case has been delimited using thresholds of intensity at $-$1.755 Å  and $V(\lambda)/I(\lambda)$ at $-$0.39 Å  of 0.95 $I_{0,QS}$ and 4$\times$10^-3, respectively. After that, we estimated $B_{\|}$ by applying the WFA in pixels with two-lobed $V$ profiles and analyzed their variation with the scanning time, $\mathrm{\Delta t}$ (see Sect. 3).

Figure 12 shows the $B_{\|}$ estimates obtained for the cases depicted in Sect. 3 using the NM datasets. The physical quantities in this figure appear smeared in comparison to those in Figs. 3, 7, and 10. We observe not only blurrier intensity images but also smoother $B_{\|}$ estimates. Indeed, the latter are generally 50–100 G weaker for all $\mathrm{\Delta t}$. These differences between both datasets appear because the intensity signal in each pixel is spread out over its surroundings in the NM datasets, as neither the blurring effect of the telescope point spread function (PSF) nor the low-altitude seeing is corrected. Nonetheless, the smearing relies on the concentration size. This correlation is clear in Fig. 13, where the divergence of the $I$ and $V$ profiles from the NM and M datasets increases as the concentration size decreases.

Furthermore, in Fig. 13, we observe that the fit quality between the Stokes $V$ profiles and the scaled $I$ derivatives from the NM datasets is similar to that described for the M datasets. However, the smearing in the NM datasets leads to better fits in all concentrations, particularly for $\mathrm{\Delta\lambda_{3}}$. In the case of wider spectral ranges, the fit quality again relies on the concentration size, because height gradients in $B_{\|}$ are more abrupt in small and medium-sized concentrations. Magnetic flux variations between the photosphere and chromosphere are therefore of the same order as in the M datasets (Table 3). Specifically, the median values of the estimates for the photospheric $\mathrm{\Delta\lambda_{4}}$ range (and the chromospheric $\mathrm{\Delta\lambda_{3}}$ one) in the small, medium-sized, and large concentrations are 461.3 (33.7), 475.3 (174.3), and $-$404.9 ($-$248.2) G, respectively.

Finally, Fig. 14 shows the histograms of the number of pixels with two-lobed $V$ profiles in the concentrations analyzed using the NM datasets and their $B_{\|}$ values for each spectral range. The former distributions are normalized to the number of scans showing each group whereas the latter are normalized to the number of pixels per group in all sequences ($\sim$46$\times$10³, 150$\times$10³, and 510$\times$10³ pixels for small, medium-sized, and large structures), respectively. Although all distributions resemble those for the M datasets (Fig. 2), they are affected by the value smearing, as fewer pixels meet the threshold used to delimit each case (Fig. 14a) and the peaks of some distributions are shifted to stronger estimates (Fig. 14b–d).

Over the past few years, investigations based on high-spatial-resolution data have reported a weakening of the quiet-Sun magnetic fields with height (e.g., Gošić et al. 2018; Robustini et al. 2019; Morosin et al. 2020). We also observe a transition to weaker longitudinal fields as the spectral range narrows down around the line core.

Close to the limb, Robustini et al. (2019) found longitudinal magnetic fields of $\sim$400 G in the photosphere decreasing to 200 G in the chromosphere of quiet-Sun areas. To estimate the latter, these authors applied the WFA to a spectral range of $\pm$1 Å  from the Ca II 854.2 nm line center. At a heliocentric angle of 37^∘, Pietrow et al. (2020) inferred $B_{\|}$ values of about 250 G outside a plage by performing STiC inversions of full-Stokes data in the $\lambda$8542 line (sampled within $\pm$880 mÅ  from the line center). Also outside a plage but closer to the disk center, Morosin et al. (2020) reported a weakening of $B_{\|}$ from $\sim-$600 G in the photosphere to $-$200 G in the low-mid chromosphere by applying a spatially constrained WFA method to the wings of the Na I 589.6 nm line and the Ca II 854.2 nm core ($\pm$110 mÅ). Recently, Gošić et al. (2021) found maximum $B_{\|}$ values of 400–800 G in internetwork clusters in Fe I 617.3 nm data that weaken by more than 200 G when applying the WFA to a spectral range of $\pm$200 mÅ  in the Ca II 854.2 nm line core.

We report $B_{\|}$ values of hG in quiet-Sun magnetic concentrations. Our mostly photospheric and chromospheric estimates in small and medium-sized structures are compatible with those previously found. However, our results for wider ranges across the line are stronger. Large concentrations also show stronger estimates, which is likely due to a departure of their physical properties from those typically found in the quiet Sun. Finally, we note that differences in the heliocentric angle may also lead to divergence of the results from those of other studies.

Furthermore, an interesting aspect of our study is the comparison of the photospheric and chromospheric estimates in concentrations of different size. Small concentrations usually host compact height variations of $B_{\|}$ exceeding 400–-500 G, where the maximum can be stronger than 800 G. Medium-sized and large structures show similar height variations but more extended over the concentrations. However, inspection of the average height variation of $B_{\|}$ reveals that small and medium-sized concentrations have stronger average height variation (of about 400 and 425 G, respectively) than large concentrations (of $\sim$300 G). Corroborating this result by analyzing other large concentrations will be of great interest because our sample only has two large structures, which were observed under poor seeing conditions. In addition, the height variations of $B_{\|}$ can be considered weaker or stronger than the chromospheric $B_{\|}$ estimated in each structure. In this regard, when compared to large concentrations, small and medium-sized concentrations display stronger height gradients in $B_{\|}$ than their chromospheric estimates. Consequently, large structures usually show better fits between the observed $V$ profiles and the scaled $I$ derivatives for wider spectral ranges.

Investigations of the stratification of the magnetic field with height in sunspots have revealed discrepancies depending on the approach used to determine the magnetic gradients (see the review of Balthasar 2018). Undoubtedly, similar studies in the quiet Sun are also of great importance as they can shed light on the topology of its magnetic field. Nonetheless, the elusive detection of the weak polarization signals emerging from the quiet-Sun chromosphere hinders these investigations. This impasse will be overcome by using observations acquired with the next-generation solar telescopes, such as DKIST (Rimmele et al., 2020) and EST (Quintero Noda et al., 2022).