EFFECTS OF UNRESOLVED MAGNETIC FIELD ON Fe i 617.3 AND 630.2 nm LINE SHAPES

It is well established that both the total and spectral solar irradiance (SI henceforth) are modulated by the emergence and evolution of the magnetic field in the solar atmosphere. In particular, SI variations measured on timescales from days to several solar rotations can be explained by accounting for the radiative effect of the presence of magnetic flux in sunspots, faculae, and network regions (e.g., Domingo et al. 2009). However, there is still uncertainty in measurements of SI variations on timescales of the activity cycle and longer (e.g., Harder et al. 2009; Fröhlich 2011), as well as in the determination of individual contributions of different magnetic features to the measured variations (e.g., Krivova et al. 2011; Fontenla et al. 2011). The role of the quiet Sun, that is those areas on the solar surface away from magnetically active and network regions, the so-called intranetwork regions, is particularly debated. For instance, long-term SI variations unexplained by some recent models have been attributed either to a 0.2 K global change in the Sun's effective temperature (Fröhlich 2011) or to quiet-Sun variations associated with small-scale magnetic field changes during the activity cycle (Fontenla et al. 2011).

Two main difficulties hinder the verification of these hypotheses. First, investigations of temperature variations would be achievable only by measurements of absolute radiometric contribution of the quiet Sun, and such measurements are not available yet. Second, the small-size magnetic features (100 km) that are seen permeating the whole solar surface in high-resolution observations (Bonet et al. 2012 and references therein) are unresolved in full-disk observations usually employed for SI reconstructions (Domingo et al. 2009), so that their contribution to SI variations is hidden. Note that the origin of this so-called quiet-Sun magnetic field is also controversial, as it is not yet clear whether and to what extent it results from the decay of active regions, turbulent dynamo effects, or both (e.g., Lites 2011; López Ariste & Sainz Dalda 2012). Variations of quiet-Sun magnetic flux over the cycle would indicate a global dynamo origin of the field; nevertheless, most of such measurements seem to indicate no temporal variations of quiet-Sun field (Sánchez Almeida & Martínez González 2011 and references therein), thus suggesting a local dynamo origin. Whether this result is due to the lack of sensitivity of instrumentation and the observational techniques employed, or to a true constancy of the field over the cycle is still an open issue.

The presence of magnetic field influences the emitted radiation in magnetically sensitive lines directly, by the Zeeman effect, and indirectly, by changing the thermodynamic properties of the plasma (e.g., Fabbian et al. 2010). Variations of line properties have therefore been employed to study the properties of the quiet Sun. For instance, Stenflo & Lindegren (1977) derived, by the investigation of the subtle broadening of a large sample of Fe i lines, an upper limit for the magnetic field amplitude of quiet-Sun network of 1000–2000 G and an upper limit of 90 G for intranetwork regions. Moreover, line parameters of a set of Fraunhofer lines were observed from 1974 to 2006 (Livingston et al. 2007), with the aim of inferring long-term variations of the photospheric temperature. Analysis of these data leads, however, to inconclusive results (Penza et al. 2004; Livingston et al. 2007).

In this study we investigate the effect of magnetic features on the shape of two Fe i magnetically sensitive photospheric lines, the 630.2 and 617.3 nm, by analyzing high-resolution spectro-polarimetric data of NOAA 11172 observed on 2011 March 17 at the Dunn Solar Telescope (DST) with the Interferometric Bidimensional Spectrometer (IBIS; Cavallini 2006; Reardon & Cavallini 2008). Qualitative comparison with results from numerical synthesis has been performed to disentangle the direct and indirect effects of the magnetic field on the measured line shapes variations. We then investigate the effects of unresolved magnetic features on the shapes of the two lines by analyzing IBIS data spatially degraded to mimic medium-resolution observations. The Fe i 630.2 and 617.3 nm lines were chosen because they have been frequently employed for photospheric magnetic field studies. Moreover, data at these spectral ranges are currently acquired with several instruments, such as the HINODE/SOT (Tsuneta et al. 2008), SOLIS/VSM (Keller et al. 2003), and SDO/HMI (Scherrer et al. 2012).

The article is organized as follows: in Section 2 we describe the observations, the data reduction, and the numerical synthesis; in Section 3 we discuss the line formation properties of the two lines; in Section 4 we present the results derived from analyses of observational data; in Section 5 we describe results obtained from synthesis; in Section 6 we investigate the effects of unresolved magnetic features; and in Section 7 we discuss the results and draw our conclusions.

We investigated the behavior of the two Fe i lines at 617.3 and 630.2 nm in data acquired at the NSO DST with IBIS. In order to interpret the results derived from observations in terms of properties of the observed solar plasma, we also synthesized the two lines in semi-empirical atmospheric models representative of quiet Sun and magnetic regions.

2.1. Observations and Data Analysis

The IBIS data set employed for this study was taken on 2011 March 17 from 14:19 UT to 15:19 UT. It consists of 60 spectro-polarimetric scans of the two lines, each sampled at 22 spectral positions; the nearby 0₂ 630.28 nm telluric line was also sampled at six spectral positions and served for line position calibration purposes. A full scan of the two lines took ≈60 s, so that each line was scanned in ≈30 s. This temporal cadence is short with respect to typical evolution timescales of solar photospheric phenomena, so that the integrity of each line during a scan can be assumed to be preserved. The field of view (FOV) was approximately 45 × 90 arcsec and included a facular region and a small pore located at [16.5N,10.3E] in between the leading and following parts of NOAA 11172. White light (WL hereafter; 621.3 nm) frames imaging the same FOV were acquired contemporaneously to spectro-polarimetric data. The spatial sampling of IBIS and WL frames was 017 pixel⁻¹.

The High Order Adaptive Optics System of the DST (Rimmele 2004) was tracking on a small pore in the FOV during the acquisition, thus improving image quality and stability. The data set was reduced and compensated for instrumental blueshift, and for instrument and telescope polarization by standard procedures (i.e., Cauzzi et al. 2008; Viticchié et al. 2009; Judge et al. 2010). Residual aberrations introduced by the atmospheric seeing were estimated by applying Multi Frame Blind deconvolution technique (Van Noort et al. 2005) to WL images and were employed to destretch the spectro-polarimetric images. The residual crosstalk between the Stokes parameters and the rms noise level were estimated to be 2 × 10⁻³ and of the order of 10⁻³ of the intensity in the nearby continuum, respectively. An infrared blocker in the optical path reduced the spectral stray-light contamination, so that its residual, estimated as described in Criscuoli et al. (2011), is below 8% and 4% for 617.3 nm and 630.2 nm, respectively. For the purpose of this study, we analyzed a subsample of 10 scans having the highest rms contrast in WL images. Figure 1 shows an example of the analyzed data.

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We estimated the magnetic flux density along the line of sight in the FOV by applying the Center Of Gravity method (COG; Rees & Semel 1979) to the measured right- and left-circularly polarized signals. Uitenbroek (2003) showed that the COG provides good estimates of the magnetic flux at the formation height of typical photospheric line core for fields lower than 1.5 kG. To estimate uncertainties of magnetic flux density values derived with the COG, we applied standard error propagation to the COG formula. We found that the uncertainty is on average equal to 8 G for flux density values smaller than 150 G; it increases monotonically for larger flux density values, although the relative uncertainty is approximately constant and equal to 3%. The scatter of values increases with the decrease of the estimated flux. The average value of the uncertainties computed over the whole FOV is 9 G, with a standard deviation of 6 G. Note, however, that the finite polarimetric sensitivity of IBIS sets a lower limit of confidence on magnetic flux density estimates. To estimate this limit, we computed the average absolute flux over those pixels in the FOV whose total polarization signal³ is lower than 3 mÅ and found a value of 20 G with a standard deviation of 10 G. We therefore defined as a quiet Sun all those pixels whose absolute magnetic flux density is smaller than 30 G (that is, below one standard deviation from the mean).

Next, in order to characterize the shapes of the lines, we computed the core intensity (IC hereafter), the equivalent width (EQW hereafter), and the FWHM of the two lines at each image pixel of the FOV. In particular, the IC and FWHM were computed by fitting the line sampling with a Gaussian function. We computed the three line shape parameters only for those pixels in the FOV with a magnetic flux below 1000 G, thus excluding the small pore and the area around it from the analysis, but including network and facular elements. This allowed us to investigate the dependence of line shape parameters on magnetic flux density variations.

2.2. Spectral Synthesis

In order to discuss the results obtained from the two Fe i lines we first investigated their formation heights. To this end we analyzed results derived from spectral synthesis of the two lines through FAL-C and FAL-P models, the latter with a uniform magnetic field of 1000 G. We then computed the response function (RF hereafter) of IC of both lines to perturbations of either the temperature (T_RF) or magnetic field (B_RF) on both atmosphere models.

Figure 2 shows the T_RF of IC for both lines and models. We find that the 617.3 nm line core in FAL-C forms at a wide range of photospheric heights, from 0 to 600 km, with a peak at 200–300 km above optical depth unity at 500 nm (in agreement with Norton et al. 2006). The core of the 630.2 nm line instead forms in a slightly narrower range of photospheric heights, from 180 to 600 km, and peaks slightly higher in the atmosphere at approximately 300 km. Figure 2 also shows that the core of the two lines forms in the deepest layers of the photosphere in magnetic features (in agreement with Khomenko & Collados 2007). This is the result of the lower temperature of FAL-P with respect to FAL-C at heights below optical depth unity in the latter (Fontenla et al. 1999), which is also found in computations of magnetic flux tube models (e.g., Pizzo et al. 1993; Steiner 2005) and full three-dimensional magnetohydrodynamic simulations (e.g., Carlsson et al. 2004). Both computed T_RF peak at 100 km, with contributions from higher photospheric layers for the 630.2 nm line and from layers below optical depth unity for the 617.3 nm line. The bumps of both T_RF at approximately 200 km reflect the change of the temperature gradient with height in FAL-P model.

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Figure 3 shows the B_RF of IC for both lines in the FAL-P model. The two B_RF are quite similar, as both peak at approximately 190 km, with the B_RF of the 617.3 nm line having again a slightly larger contribution from the deepest layers than the one calculated for the 630.2 nm line.

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The results derived from the synthesis therefore indicate that both lines form in similar regions of the solar atmosphere, with the 630.2 nm line sampling slightly higher atmospheric heights than the 617.3 nm line.

In this section we investigate the dependence of the three line shape parameters on magnetic flux density.

Note that we found that the flux density values estimated from the 630.2 nm data are on average lower than those obtained from the 617.3 nm set, and that the discrepancy increases with the magnetic flux density, with a maximum of approximately 40% at 1000 G. Because, as shown in Section 3, the two lines form in similar regions of the solar atmosphere, this discrepancy is unlikely to reflect a stratification of the magnetic field with height. To verify this hypothesis, we applied the COG to synthetic spectra of the two lines computed in the FAL-P model with a stratified magnetic field and obtained magnetic field estimates in agreement within 4%. We conclude that the discrepancy that we found in our data is most likely due to the telluric blending affecting especially the right circular polarization signal in the 630.2 nm line. In the following, we therefore refer to flux density estimates derived from the 617.3 nm data only. Note that the blending may also lead to overestimates of EQW and FWHM (but not of IC), especially at the largest flux values investigated. To reduce these effects, the Gaussian fit was performed on a spectral range that did not include the two spectral points in the far right wing of the line and the nearby continuum. Similarly, the EQW was computed substituting the values at the two spectral points in the far right wing and the nearby continuum with those at the two spectral points in the far left wing and nearby continuum. It is difficult to estimate the residual effects due to the blending in these estimates. Nevertheless, the good qualitative agreement with results obtained with the 617.3 nm line (discussed below) suggests that the trends of the FWHM and EQW with the magnetic flux are not significatively affected by the telluric blending.

Results from our measurements are presented in Figure 4. The dots show the dependence of IC, EQW, and FWHM on the magnetic flux density for both lines, while asterisks indicate the average value of the same line parameters computed over bins of magnetic flux. Note that the bins were increasingly larger in order to guarantee sufficient statistics. We found that the IC and FWHM increase with the magnetic flux for both lines, whereas the EQW shows the opposite trend. All three line shape parameters show saturation. For both IC and EQW this saturation occurs at approximately 300 G for both lines, while the FWHM saturates at flux density values above 700 G and 500 G for the 617.3 nm and the 630.2 nm line, respectively. Note that the maximum relative variations with magnetic flux density (i.e., relative difference between values obtained at the largest flux and those obtained for quiet regions) are approximately 40% for IC for both lines, 55% and 36% for FWHM in the case of 617.3 nm and 630.2 nm, respectively, and −15% and −8% for EQW, respectively.

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The measurements present large scatter, especially at the lowest flux density values, even though the standard deviations of the values distributions in each bin (error bars in the plots) are rather small. The observed scatter of values reflects the sensitivity of line shape parameters to temperature and velocity variations of the plasma. The larger the magnetic flux, the more the temperature and velocity fluctuations of convective plasma motions are inhibited (e.g., Ishikawa et al. 2007; Romano et al. 2012), so that the dispersion of the line shape parameters becomes smaller and smaller with the increase of the magnetic flux density. Note that the dispersion for the 617.3 nm line is larger than that obtained for the 630.2 nm data, especially when comparing IC. This is due to the lower excitation potential of the 617.3 nm line, which makes the IC of this line more sensitive to temperature and velocity fluctuations (Gray 1992, p. 285–287).

The IC in particular depends on both temperature and magnetic field in a manner that is difficult to disentangle. The results from numerical models and simulations of magnetic flux tubes in fact show that the photospheric temperature within a tube rises following an increase of magnetic field due to the intensification of the lateral heating effects from the surrounding plasma (e.g., Pizzo et al. 1993). As also qualitatively shown by the results obtained from numerical synthesis reported in Section 5, an increase in either of these quantities, the field strength via the Zeeman effect, or the temperature via irradiation, leads to an increase of IC, so that it is difficult to discern between the two effects.

The trend with the magnetic flux density of FWHM found for both lines instead suggests that this line shape parameter is more sensitive to magnetic field variations and velocity effects than to temperature. In fact, from the rise of temperature within the flux tubes mentioned above, we would expect the FWHM to decrease with the magnetic flux. This decrease is compensated on one hand by the fact that magnetic flux concentrations tend to be located in or nearby downflow regions, which are characterized by shallower and broader profiles, and on the other hand, to the increase of the splitting of the lines with the magnetic flux. This latter effect becomes more and more important with the increase of the field (and therefore of the flux), due to the inhibition of the convective flows mentioned above. This interpretation is confirmed by the results obtained from numerical synthesis of line profiles discussed in Section 5.

We also note that at low magnetic flux density values, the FWHM of 617.3 nm line increases slower with the magnetic flux density than for the 630.2 nm. This indicates that for the former line the effect of temperature counteracting Zeeman broadening is larger than for the 630.2 nm. Note that the maximum difference between the FWHM at saturation and the FWHM of average quiet Sun agrees within the errors for the two lines (i.e., is approximately 0.06 Å), as expected from Zeeman splitting theory for lines with equal Landé factor.

Finally, the dependence of EQW on the magnetic flux reflects variations of both the width and depth of the studied lines. The rapid saturation of EQW with the magnetic flux is due to the fact that the line IC increases more rapidly than the FWHM for a given increase of magnetic flux. As formally shown in the Appendix and confirmed by results obtained from numerical synthesis shown in Section 5, the EQW of the two iron lines is almost independent of the magnetic field and is mostly sensitive to temperature variations. Figure 4 therefore suggests that the magnetic features in our data are characterized by similar temperatures.

In the previous section we have analyzed the dependence of the three shape parameters of the two iron lines on the magnetic flux density. As already noted, this dependence is not explicit because of the indirect effects of the magnetic field on other physical parameters. In particular, the indirect effect on the temperature stratification highly affects the shape of a line. In order to separately investigate the effects of magnetic field and temperature on line shape parameters, we have therefore analyzed results obtained from numerical synthesis through various one-dimensional models, as described in Section 2.2. The results are presented in this section.

To investigate the effects of temperature on the shapes of line profiles we have compared results obtained from the various FAL models, which are characterized by different temperature stratifications. The plots in Figure 5 show the relative differences of the line parameters computed for FAL-E, FAL-H, and FAL-P models with respect to line parameter obtained with FAL-C model. Since the three models are characterized by increasing temperature at the formation heights of the two lines, the plots show that the increase of temperature causes IC to increase and the FWHM and EQW to decrease. While for IC and EQW we found changes up to 50% and more, the FWHM changes do not exceed 10%.

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To investigate the effects of the magnetic field, we considered line profiles obtained with FAL-C model in the case of fields constant with geometric height and whose strength varied from 50 to 1000 G. The results reported in Figure 6 show the relative difference between line parameters obtained for the different field strength values with respect to those obtained in the case of no magnetic field. We note that the increase of the magnetic field strength leads to a large increase of IC and FWHM (more than 100% in the case of IC), especially at the largest magnetic strength values, while EQW stays fairly constant (variations are smaller than 1% for both lines), thus confirming the analytical derivation presented in the Appendix.

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Both plots show that the two lines are affected differently by changes of plasma physical properties, mainly because of their different atomic parameters. This is also confirmed by plots in Figure 2, which shows that the difference of the shapes of the T_RFs obtained for FAL-C and FAL-P models is larger for the 617.3 nm line than for the 630.2 nm line. In particular, while the T_RFs of the 630.2 nm line obtained with the two models appear mostly shifted in depth with respect to each other, the two T_RFs obtained for the 617.3 nm line also show different shapes. The 617.3 nm line therefore experiences larger changes due to temperature effects than the 630.2 nm.

A direct comparison of the plots shown in Figure 5 and in Figure 6 is not straightforward, as they show the variations of the line parameters with respect to two different physical quantities, the temperature and the magnetic field strength, respectively. Nevertheless, results from the magnetic flux tube models show that within magnetic flux tubes in radiative equilibrium the temperature at the formation heights of the two investigated lines is positively correlated with the field strength, so that, for instance, model FAL-P can reasonably have associated a field strength of 1000–1500 G. This allows a comparison of the trends of the line parameters with respect to the two physical quantities. The plots show that IC increases with the increase of both temperature and field strength, in a manner difficult to disentangle. Opposite trends are instead found in the case of the FWHM, as an increase of temperature makes this line parameter to decrease, while an increase of magnetic field strength causes it to increase. A comparison of the value of FWHM variation obtained for FAL-P with the value obtained for 1000 G shows that magnetic field effects are more likely to dominate. Variations of the EQW found in the case of the increase of temperature are more than one order of magnitude larger than those obtained when increasing the magnetic field strength, again confirming that the direct effect of Zeeman splitting on this line parameter is negligible for the two lines.

Finally, it is important to note that variations found from the numerical synthesis presented here are meant to qualitatively interpret observational results presented in Section 4, as they are representative of the ideal case of unitary magnetic filling factor and observational effects like finite spatial and spectral resolution and scattered light have not been taken into account.

In previous sections we have investigated the direct and indirect effects of the magnetic field on the profiles of the two iron lines and we have shown that different shape parameters are affected differently by temperature and magnetic field variations. In this section we investigate whether and to what extent line shape parameters are affected by small amounts of magnetic flux not resolved in the observations.

To this purpose, we analyzed average line shape parameters obtained in images that were synthesized by replacing line profiles of pixels whose magnetic flux density exceeds a certain threshold with the average quiet-Sun profile. By varying the magnetic flux threshold, images of successively lower average magnetic flux density (in particular from approximately 90 to 10 G) were constructed. Note that the quiet-Sun profile was computed averaging profiles of those pixels of the original images whose measured magnetic flux density is lower than 30 G (see Section 2). The line shape parameters were then determined from the average profiles of each of these synthetic maps, thus mimicking observations that do not resolve the remaining magnetic features. Figure 7 shows the relative differences between line shape properties from the synthetic frames (characterized by different values of the average magnetic flux) and those of the synthetic frame with the lowest magnetic flux density (i.e., 10 G). In the case of IC and FWHM the error bars are derived from the uncertainties of the fitting of the line to a Gaussian function (see Section 2), while for the EQW the errors are estimated by the Cayrel (1988) formula. We found that the average line shape parameters vary by a few percent with the increase of the average flux density. The maximum variation that we found is approximately 6% in the case of IC for both lines, 4% and 2.5% in the case of FWHM for the 630.2 nm and the 617.3 nm, respectively, and −1.8% and −3.5% in the case of EQW. Variations are larger in 617.3 nm in the case of EQW, as a result of the previously mentioned larger sensitivity of this line to temperature. For the same reason, larger variations of FWHM with the magnetic flux density are instead found for the 630.2 nm. As explained above, in fact, for the 617.3 nm line the effect of temperature counteracting the Zeeman line broadening is larger, thus reducing the increase of the FWHM of this line with the flux.

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The measured line shape parameters of the two Fe i lines vary with magnetic flux because of the direct effect of the Zeeman splitting and the indirect effect of temperature variations induced by suppression of convective motions. IC and FWHM increase with the magnetic flux, while the opposite trend was found for EQW (Figure 4). The results from numerical synthesis show that in the case of IC the trends reflect both the Zeeman splitting and the temperature effects in a manner that is difficult to disentangle, whereas the FWHM is mostly sensitive to the Zeeman effect, in particular for larger field strength. We have also shown that the dependence on the magnetic field of the EQW is negligible at the values of magnetic flux investigated, so that the observed trend must be ascribed to temperature variations induced by the presence of magnetic field concentrations.

We also found that the three line shape parameters show saturation with the magnetic flux density (Figure 4). Interpretation of these results in terms of magnetic flux tube models suggests that features characterized by small magnetic flux values are mostly spatially unresolved in our observations and that the pixels characterized by the largest magnetic flux values (above the saturation threshold) belong to spatially resolved features that may result from the clustering of magnetic elements. According to flux tube models, in fact, an increase of the magnetic field leads to an increase of the temperature within the tube in line-forming layers. Instead of a saturation we should therefore observe IC and FWHM to monotonically increase and EQW to monotonically decrease with the flux. In particular, the saturation of EQW suggests that the magnetic features that we observed are mostly characterized by very similar photospheric temperatures. This is expected from models of clusters of magnetic flux tubes (e.g., Criscuoli & Rast 2009; Fabiani Bendicho et al. 1992), and is in agreement with results recently reported by Viticchié et al. (2010). It is worth noting that Harvey & Livingston (1969a, 1969b) also concluded that the profiles of the Fe i 525.0 and 525.3 nm lines suggest the "quantization" of the magnetic flux in the photosphere.

We also investigated the effects of an unresolved magnetic field on the shapes of line profiles. The results reported in Figure 7 show that variations of a few percent in the average shapes of the investigated lines are produced for modest (below 100 G) values of average magnetic flux. Note that the average flux values investigated are compatible with those employed in irradiance reconstruction techniques to identify quiet-Sun regions (Fligge et al. 2000; Ball et al. 2011 and references therein). In the case of the FWHM, larger variations with the average magnetic flux are found for the 630.2 nm than for the 617.3 nm, whereas in the case of the EQW larger variations are found for the 617.3 nm. The 630.2 nm line is therefore more favorable to investigate magnetic field variation, while the 617.3 nm is more suitable to investigate the effects of temperature variations.

It is interesting to note that our estimates of the EQW variations are in agreement with values reported by Fabbian et al. (2010) for the Fe i 608.27 nm and 624.07 nm lines. These authors analyzed snapshots from magnetohydrodynamic simulations to investigate the variations of average EQW values of some iron lines with the increase of the average magnetic flux in the simulations. Comparing of our Figure 7 with their Figure 4 reveals that the agreement is better for the 617.3 nm line, as this has an excitation potential similar to that of 608.27 nm and 624.07 nm lines. Their plots also indicate that the variations of EQW mostly reflect variations of the average temperature in their simulations. The small dependence in their plots on the average magnetic flux is due to the wavelength shift of the opacity discussed in the Appendix. For completeness, we note that those authors also reported a large dependence of EQW of the Fe i 1.5 μm line on the magnetic flux. This is due on one hand to the fact that this line forms in regions where the variation of temperature induced by the magnetic field is negligible, and on the other hand to its large Landé factor so that this line is not in the weak field regime.

The amount of variations of line shape parameters due to the presence of unresolved magnetic flux that we found is measurable with instruments such as HINODE/SOT, SOLIS/VSM, and SDO/HMI. Line shape variations of the investigated lines observed at the disk center can be employed for studies of long-term temporal variations of physical properties of the quiet Sun. In particular, the results presented here show that it would be possible to disentangle thermal and magnetic effects by comparing variations of different properties of the two Fe i photospheric lines. For instance, a decrease over the magnetic cycle of EQW together with an increase of FWHM would indicate variations of quiet-Sun temperature induced by the presence of unresolved magnetic features. On the other hand, a decrease of the EQW together with a decrease of the FWHM would indicate an increase of the effective temperature of the plasma.

Finally, it is worth noting that Martínez González et al. (2006) showed that the same profiles of Fe i 630.1 and 630.2 nm lines can be produced by different thermodynamic stratifications, magnetic field, and microturbulence values, so that results obtained from spectro-polarimetric inversions should be treated with caution. However, the method that we propose is less affected by this degeneracy. In fact, we have shown that the EQW parameter of the two iron lines is almost insensitive to magnetic field effects, while the FWHM is mostly sensitive to magnetic field effects. The EQW variations can therefore be utilized to derive information on temperature variations, while the FWHM can be utilized to derive information on magnetic flux variations. Note, however, that a proper modeling of the observed data (e.g., Holzreuter & Solanki 2012) is required to quantitatively estimate these variations. In particular, modifications of the velocity distributions, which can result from either variations of the magnetic flux or of the effective temperature, can also affect the shapes of lines. Nevertheless, since velocity distributions are known to narrow in the case of increase of magnetic flux (e.g., Ishikawa et al. 2007; Morinaga et al. 2008), while they broaden and skew toward upflow velocities in the case of an increase of the effective temperature (e.g., Gray & Toner 1985; Nordlund & Dravins 1990), we expect velocity effects not to hamper the possibility of disentangling magnetic flux and effective temperature variations from observations of properties of the two lines. We plan to investigate these effects in the future by analyzing results from numerical simulations of the solar photosphere.

This study was supported by the Istituto Nazionale di Astrofisica (PRIN-INAF-2010) and the Agenzia Spaziale Italiana (ASI/ESS/I/915/01510710). The NSO is operated by the Association of Universities for Research in Astronomy (AURA), Inc., for the National Science Foundation. IBIS was built by INAF/Osservatorio Astrofisico di Arcetri with contributions from the Universities of Firenze and Roma "Tor Vergata," the National Solar Observatory, and the Italian Ministries of Research (MUR) and Foreign Affairs (MAE). We are grateful to A. Tritschler for providing the IBIS data reduction code.

Here we investigate the dependence of the EQW of a magnetically sensitive spectral line on the magnetic field. It is common practice to express the emergent line profile I(λ) in the presence of Zeeman splitting as the sum of three Gaussians (e.g., Borrero 2008):

$$\begin{eqnarray} I(\lambda) &=& 1-\frac{(1-I_0)\sin ^2\gamma }{2}e^{-\frac{(\lambda -\lambda _0)^2}{\Delta \lambda ^2}} - \frac{(1-I_0)(\cos ^2\gamma +1)}{4} \nonumber\\ &&\times \left(e^{-\frac{(\lambda -\lambda _0-\lambda _B)^2}{\Delta \lambda ^2}} + e^{-\frac{(\lambda -\lambda _0+\lambda _B)^2}{\Delta \lambda ^2}} \right). \end{eqnarray} \tag{ A1 }$$

The two Gaussian functions shifted by the amount λ₀ ± λ_B represent the left- and right-hand circularly polarized σ profiles, while the Gaussian function centered at λ₀ represents the profile of the linearly polarized π component. I₀ and Δλ are the IC and Doppler width of the line, in the absence of the magnetic field; γ is the inclination angle with respect to the line of sight; λ_B = Cg_effλ²₀B, where C is a constant, g_eff is the effective Landé factor, λ₀ is the central wavelength of the line, and B is the magnetic field strength.

In the case of magnetic field oriented along the line of sight (γ = 0) the previous formula simplifies to

$$\begin{equation} I(\lambda)=1 - \frac{(1-I_0)}{2}\times \left(e^{-\frac{(\lambda -\lambda _0-\lambda _B)^2}{\Delta \lambda ^2}} + e^{-\frac{(\lambda -\lambda _0+\lambda _B)^2}{\Delta \lambda ^2}} \right). \end{equation} \tag{ A2 }$$

The EQW of the line in that case is, by definition,

$$\begin{eqnarray} &&{\rm EQW}:=\frac{1-I_0}{2}\left [\int _{0}^{\infty }e^{-\frac{(\lambda -\lambda _0-\lambda _B)^2}{\Delta \lambda ^2}} d\lambda + \int _{0}^{\infty } e^{-\frac{(\lambda -\lambda _0+\lambda _B)^2}{\Delta \lambda ^2}} d\lambda \right ]. \nonumber\\ \end{eqnarray} \tag{ A3 }$$

The two terms in the sum are equal as they differ only by a shift within the infinite domain. Since the two terms represent the EQWs of the left (EQW⁻) and right (EQW⁺) circular polarized signal, then EQW = EQW⁻ = EQW⁺ (note that it is easy to verify that these equalities hold for any shape of the profiles, as far as the two circular polarized signals are symmetric).

We now to study the variation of the EQW with the magnetic field. To this end we differentiate Equation (A3) with respect to B:

$$\begin{eqnarray} \frac{d {\rm EQW}}{\it dB} &=& \frac{1-I_0}{2} \frac{d}{\it dB}\left [\int _{0}^{\infty }e^{-\frac{(\lambda -\lambda _0-\lambda _B)^2}{\Delta \lambda ^2}} d\lambda \right. \nonumber\\ &&+ \left. \int _{0}^{\infty } e^{-\frac{(\lambda -\lambda _0+\lambda _B)^2}{\Delta \lambda ^2}} d\lambda \right ]. \nonumber\\ \end{eqnarray} \tag{ A4 }$$

Making the variable change $\tilde{\lambda } = \lambda - \lambda _0$, and changing the order of integration and differentiation, we obtain

$$\begin{eqnarray} \frac{d {\rm EQW}}{\it dB} &=& (1-I_0) Cg_{{\rm eff}}\lambda _0^2 \left [ \int _{-\infty }^{\infty }\frac{(\tilde{\lambda }-\lambda _B)}{\Delta \lambda ^2} e^{-\frac{(\tilde{\lambda }-\lambda _B)^2}{\Delta \lambda ^2}} d\tilde{\lambda } \right. \nonumber\\ &&+ \left. \int _{-\infty }^{\infty }\frac{(\tilde{\lambda }+\lambda _B)}{\Delta \lambda ^2} e^{-\frac{(\tilde{\lambda }+\lambda _B)^2}{\Delta \lambda ^2}} d\tilde{\lambda } \right], \end{eqnarray} \tag{ A5 }$$

where we have assumed that the rest wavelength is much larger than the Zeeman splitting. It is easy to see that the right-hand side of Equation (A5) is null, as the terms in the integrand that are proportional to $\tilde{\lambda }$ are uneven, and the terms that are proportional to λ_B cancel. Thus, under the assumption that I₀ and Δλ do not depend on the field, the EQW of a Zeeman split line is independent of the field strength. However, this assumption does not hold in a stratified atmosphere because the Zeeman splitting changes the opacity at a given wavelength, and therefore alters the height and physical parameters the line samples. Indeed, as shown by previous studies (e.g., Fabbian et al. 2010), the EQW of a magnetic sensitive line is sensitive to the strength of the magnetic field. These variations are small in the weak field case (i.e., λ₀ ± λ_B < Δλ), and are non-negligible only for large fields. Moreover, the sensitivity of the EQW to the field depends on the atomic parameters of the line, as these determine its opacity.