Internetwork Magnetic Fields Seen in Fe I 1564.8 nm
aa r X i v : . [ a s t r o - ph . S R ] S e p Draft version September 29, 2020
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Internetwork Magnetic Fields Seen in Fe I Yoichiro Hanaoka and Takashi Sakurai National Astronomical Observatory of Japan2-21-1 Osawa, MitakaTokyo 181-8588, Japan (Received June 1, 2019; Revised January 10, 2019; Accepted September 29, 2020)
Submitted to ApJABSTRACTWe studied the properties of internetwork magnetic fields in the solar photosphere taking advantageof full-disk Stokes
V /I maps of the Fe I I I Keywords:
Sun: photosphere — Sun: magnetic fields — Sun: infrared — techniques: polarimetric INTRODUCTIONThe solar surface is filled with magnetic field. Moreover, active regions and supergranulation network boundariesin the quiet Sun, which have strong magnetic fields, are its dominant components. However, magnetic fields ininternetwork regions inside the networks are also an important component of the solar magnetic field in spite of theirweak field strength. According to one recent estimation, the strength of the internetwork fields is approximately 130 Gat the optical depth of unity (Danilovic et al. 2016), and it is presumed that their flux occupies 14 % of that of the quietSun magnetic field (Goˇsi´c et al. 2014). The fact that the weak magnetic field spreads over the internetwork regionswas noted by Livingston & Harvey (1971) and many studies have been conducted. In particular, both ground-basedand space-borne sophisticated instruments have enabled advanced observations of the weak and small-scale elementsof internetwork fields, which have contributed to the progress of the research of internetwork fields.Bellot Rubio & Orozco Su´arez (2019) summarized such studies in their review paper. They presented widely ac-cepted properties of internetwork fields as some “agreed facts”; namely, internetwork fields are mostly weak (the orderof hectogauss (hG)), highly inclined, and have large filling factors. However, the interpretations derived from variousobservations still show discrepancies. Internetwork fields are challenging targets because of their small scale and weakfield strength, and the observations carried out with different techniques have not necessarily reached consistent con-clusions. For instance, the results based on absorption lines in visible wavelengths and those based on infrared linesare sometimes inconsistent with one another.
Corresponding author: Yoichiro [email protected]
Hanaoka & Sakurai
The visible wavelength observations, which realize high spatial resolutions and high polarization sensitivities, aresuitable to measure small-scale weak magnetic fields in internetwork regions. On the other hand, polarization mea-surements of infrared lines at Fe I µ m (1564.8 nm and 1565.3 nm) have also been a powerful tool for the study ofinternetwork fields since Lin (1995). In particular, the Fe I I
630 nm line because of its large Land´e factor g and its long wavelength. The magneticfield of the order of kilogauss (kG) in the active regions or the network boundaries is beyond the “weak-field” regimeof the Fe I I µ m lines, while some results based on the Fe I µ m lines consistent with thosefrom visible lines have also been published (Dom´ınguez Cerde˜na et al. 2006; Mart´ınez Gonz´alez et al. 2008b).In this study, we attempted to shed light on this topic from a different perspective using full disk images of theFe I I I V /I (degree of circularpolarization) profiles, which are reliably measured even in lower-resolution observations but with high polarimetricsensitivities, statistically utilizing large amount of data accumulated for years (as briefly reported by Hanaoka et al.2015).One of the benefits of the data analyzed here is that they are full-disk data showing the Stokes signals from thecenter to the limb simultaneously. This fact is important because the center-to-limb variation of the polarizationsignals provides a hint for the inclination distribution of the internetwork fields. The horizontal magnetic field presentsno line-of-sight magnetic field component at the disk center, but its line-of-sight component increases toward the limbwith the distance from the disk center, and the circular polarization will appear. If the horizontal magnetic fieldhas a randomly distributed azimuth as discussed in the previous studies, it will be observed as fluctuating circularpolarization signals. On the other hand, the vertical magnetic field, which is dominant in the kG components, shows thelimbward decrease of the line-of-sight component. Harvey et al. (2007) demonstrated the manifestation of internetworkfields in the full disk magnetograms taken with the Synoptic Optical Long-term Investigations of the Sun (SOLIS;Keller et al. 2003) and the Global Oscillation Network Group (GONG; Harvey et al. 1988) instruments using visiblewavelength lines. They found a limbward increase of the line-of-sight magnetic field components, which is consistentwith the highly inclined field. This result is also supported by the studies using high resolution observations of thecenter-to-limb variation by Lites et al. (2008) and Stenflo (2013), and particularly by Lites et al. (2017), who analyzeda large amount of linear polarization data (representing the transverse magnetic field component) obtained at variousdistances from the disk center with the Hinode SOT. They found an increase of the linear polarization components nternetwork Magnetic Fields Seen in Fe I µ m observations, Meunier et al. (1998) found thelimbward decrease of the weak magnetic field signal, and Mart´ınez Gonz´alez et al. (2008a) concluded that there is no center-to-limb variation in internetwork fields.Another benefit of the data used here is that they cover most of solar cycle 24; therefore, we can study the solar cycledependence of the polarization signals of the internetwork regions. Some researchers have studied the internetworkfield data obtained at different phases in the solar activity. For example, S´anchez Almeida (2003), Buehler et al.(2013), and Lites et al. (2014) concluded that there is no cycle variation. On the other hand, other studies haveshown the solar cycle dependence of the weak magnetic field on the solar surface (Faurobert et al. 2001; Jin et al.2011; Faurobert & Ricort 2015). However, Faurobert & Ricort (2015) inferred that the variation was caused by noiseor contamination from active regions, and Faurobert et al. (2001) and Jin et al. (2011) presumed that the measuredweak magnetic field was not necessarily the internetwork fields. These previous studies used visible wavelength lines,but this study is the first to approach the solar cycle variation of the internetwork fields observed with the Fe I OBSERVATION AND DATA2.1.
Instrument and Observation
Because the infrared spectropolarimeter of the SFT is described in detail by Sakurai et al. (2018), we briefly explainits specifications in relation to the Fe I I ′′ spatially and 6.3 or 6.7 pm (63 or 67 m˚A) spectrally. In 2015,the optical system was modified to accommodate two cameras for simultaneous observation of the two wavelengthbands around He I I ′′ at first, and it was changed to2 ′′ in 2011 July. Therefore, a scan have been provided a data cube covering the full disk with the spatial sampling of2–3 ′′ and the spectral sampling of 6.3 – 6.7 pm. The true spatial resolution taking the diffraction limit and the seeingdegradation into account is 3–4 ′′ , and the nominal spectral resolution determined by the slit and the grating is 6.7 pm.It takes about 3.5 s to obtain a data set at a single slit position, and to complete a full-Sun scan, approximately twohours is required. Weather permitting, 1–3 full-disk scans are performed daily.Currently we are using a rotating waveplate as the polarization modulator. The typical root-mean-square (RMS)noise level of the Stokes V /I signal in a pixel of the detectors, namely the polarization sensitivity, is about 0.00016.Thanks to the high full-well capacity ( ∼ e − ) of InGaAs detectors of the cameras and the high efficiency of thepolarization modulation ( ∼
70 %), this low noise level is achieved by combining 192 polarization-modulated imagestaken at each slit position, whose total exposure time is 2 s. Until 2013, we used ferroelectric liquid crystals as thepolarization modulator. Due to the low efficiency of the polarization modulation, about one tenth of the current value,the typical noise level was about 0.00046. However, even before the replacement of the polarization modulator, thenoise level is low enough to enable the detection of the weak polarization signals in the internetwork regions.2.2.
Preparation of the Data for the Analysis
We selected 99 sets of full-disk scan data of Fe I V /I signals (degree of circular polarization derived from the Stokes V and I values measured in Hanaoka & Sakurai R e l a t i v e B r i gh t ne ss D i ff e r en c e D eg r ee o f P o l a r i z a t i on ( I n t e r ne t w o r k ) D eg r ee o f P o l a r i z a t i on ( A c t i v e A r ea s ) (a) Stokes I(b) Stokes I(c) Stokes V/I Figure 1.
Average Stokes I and V /I profiles of the Fe I x -axis represents the offset from the line center. Panel (a) shows the Stokes I profiles of the internetwork regions in the four zones shown in Figure 4 from the disk center area (light brown) toward thelimb (dark brown). Panel (b) shows the relative Stokes I profiles with respect to the disk center area. In panel (c), the Stokes V /I profiles that have their peak positions at various offsets from the line center are shown. Solid lines represent the averageprofiles of the internetwork pixels (plotted using the scale on the left), and dotted lines represent those of the active regions ornetwork boundaries (plotted using the scale on the right). The magnetic field strengths, which give the Zeeman splitting of thecorresponding wavelength offsets, are also provided in the x -axis. The gray stripes in the background represent the wavelengthsampling steps. the same pixel) and examined the peak(s) in each Stokes V /I profile. The positive and negative peaks in a Stokes
V /I profile represent the σ -components shifted from the line center by the Zeeman effect.The Stokes data used in the analysis were prepared as follows. Raw observed spectra were discretely sampled bythe detector pixels; the sampling was not necessarily symmetric to the center of the Fe I ± ± ± ± ± ± ± ±
10, and ±
12 steps from the line center. Thiswavelength range covers approximately ± ± nternetwork Magnetic Fields Seen in Fe I λ = 4 . × − gBλ nm, where g = 3 is the Land´e factor, B is the magneticfield strength in gauss, and λ is the wavelength of the line in nm. As mentioned above, the wavelength sampling stepchanged from 6.3 to 6.7 pm in 2015. Hence, hereafter we express the distance from the line center with the numberof offset steps as well as the offset wavelengths. For all spectral data, we fitted the bottom position of the Fe I V /I profiles,which are antisymmetric to the line center, the signals in the red wing were subtracted from those in the blue wing.Figures 1(a) and 1(c) show some folded Stokes I and V /I wing profiles (discussion for Figure 1 will be presentedlater). In Figure 1(c), the average Stokes
V /I profiles in the internetwork regions and the active regions / networkboundaries, which peaked at each of 1, 2, 3, 4, 5, 6, 8, 10, and 12-steps offsets from the line center, are plotted.We defined the peak position of each folded Stokes
V /I profile as the wavelength offset where the absolute degree ofcircular polarization becomes the maximum. These profiles, which have different peak positions, are considered torepresent different magnetic field strengths as shown in the x -axis of Figure 1(c). However, in the weak-field regime,the peak Stokes V /I is always located where the gradient of the Stokes I profile takes the largest value. If we take theeffective line width in our observation degraded by the spectral resolution into account and ignore the Doppler shiftof the line, the wavelength of the maximum gradient is located at 2-steps from the line center, which approximatelycorresponds to 300–400 G in terms of the Zeeman splitting. Therefore, if the peak position of a Stokes V /I profile islocated at the offset of 2-steps, it is presumed that the magnetic field strength is 300–400 G or less. In this way, thepeak position of each Stokes
V /I profile can be connected to a magnetic field strength without an inversion.Thus, the prepared data were statistically analyzed mainly using the peak positions as an indicator of the magneticfield strength; the results are presented in the next section. RESULTS3.1.
Properties of the Stokes
V /I
Signals Seen in Full-Disk Fe I Here, we present the properties of the Stokes
V /I signals in the full-disk maps of the Fe I Appearance of the Full-Disk Stokes
V /I
Maps of Fe I Figure 2 shows the Stokes I and Stokes V /I maps of Fe I I maps shown in Figures 2(a) and 2(b) show that the number of sunspots on these two dates were quite different.As the spatial sampling was approximately 2 ′′ , each map is comprised of approximately 1000 × V /I signals at the offsets of 1 –12 steps from the line center were calculated as mentioned above. InFigures 2(c)– 2(f), the Stokes
V /I maps at the wavelength offsets of 6 and 2 steps are shown. The actual wavelengthoffsets are 37.7 and 12.6 pm for 2014 May 10 and 40.3 and 13.4 pm for 2019 August 10.The maps at 6-steps offset, shown in Figures 2(c) and 2(d), seem to be ordinary longitudinal magnetograms showingremarkable polarization signals in and around the active regions or network boundaries in the quiet Sun. On theother hand, the maps at 2-steps offset from the line center, Figures 2(e) and 2(f), are much different from those at6-steps offset. The whole disk shows grainy appearance; it is filled with small scale positive and negative signals. Theylook as if they are random noise, but actually they fluctuate about five times more than the noise level mentioned inSection 2.1. Taking a closer look, we can find that the polarization signals are rather weak around the disk center,similarly to the finding by Harvey et al. (2007) in full-disk magnetograms. This is an opposite tendency to the signalsseen in Figures 2(c) and 2(d), which show apparent limbward weakening. Contrary to the active regions and networkboundaries seen in Figures 2(c) and 2(d), where the vertical magnetic fields dominate, the grainy signals are consideredto represent highly inclined magnetic fields, as explained in Section 1.From Figure 2, it is expected that the peak in Stokes
V /I profiles of the active regions or network boundaries arelocated far from the line center, while those of the grainy signals are located around the wavelength offset of 2 steps.The wavelength offset of 6 steps or 37.7 / 40.3 pm corresponds to the Zeeman splitting of the magnetic field of 1.1 k/ 1.2 kG as shown in Figure 1(c), and therefore, it is natural that the so-called kG components, active regions andnetwork boundaries in the quiet Sun, show remarkable polarizations in Figures 2(c) and 2(d). On the other hand,
Hanaoka & Sakurai -0.0030.+0.003 D eg r ee o f P o l a r i z a t i on (a) (b) Stokes I (c) (d)
Stokes V/I 6 Steps from Line Center (e) (f)
Stokes V/I 2 Steps from Line Center
Figure 2.
Sample Stokes I (panels (a) and (b)) and V /I maps ((c)–(f)) of the Fe I V /I signals at the 6-steps offset ((c)37.7 / (d) 40.3 pm) from the line center; panels (e) and (f) are those at the 2-steps offset ((e) 12.6 / (f) 13.4 pm). Regardlessof the solar activity, the Stokes
V /I maps at the 6-steps offset and the 2-steps offset show remarkable differences, particularlyin the internetwork regions. nternetwork Magnetic Fields Seen in Fe I (a) 2014 May 10 (b) 2019 Aug 10 Figure 3.
Distribution of the offsets of the peak positions in the Stokes
V /I profiles from the center of the Fe I the wavelength offset of 2 steps (12.6 / 13.4 pm) corresponds to the Zeeman splitting of 365 / 392 G. Therefore, thepolarization signals, which are more remarkable in the Stokes V /I maps at 2-steps offset than in those at 6 steps offset,are presumed to correspond to the magnetic field of the hG order.From the above, the grainy signals seen in Figures 2(e) and 2(f), which, in principle, would correspond to highlyinclined hG magnetic fields and are spread over the entire solar disk, can be interpreted as the internetwork magneticfields. This means that the internetwork magnetic fields were captured in the full-disk Stokes
V /I maps of the Fe I ′′ . Next, we present thebehavior of the peak positions of the Stokes V /I signals.3.1.2.
Distribution of the Offsets of the Peak Positions in the Stokes
V /I
Profiles
Figure 3 shows the spatial distribution of the offsets from the line center of the peak positions in the Stokes
V /I profiles of individual pixels in the full-disk maps. Panels (a) and (b) show the data for 2014 May 10 and 2019 August10; the corresponding Stokes
V /I maps are presented in Figures 2. The wavelength offsets of the peak positions areclassified by color in Figure 3. Pixels with the offsets of 10 or 12 steps are shown in red, and those with the offsets of6 or 8 steps are shown in yellow. The offsets of 6–12 steps roughly correspond to the Zeeman splitting of the magneticfield strength of 1–2 kG. They are particularly conspicuous in Figure 3(a), and they approximately coincide with theactive regions or network boundaries in the quiet Sun, as shown in Figure 2(c). In Figure 3(b), we see yellow patchescorresponding to the network boundaries, which are seen as white and black spots sparsely scattered over the Stokes
V /I map in Figure 2(d). By contrast, the pixels in blue, which have the peak offset within 3 steps from the line center,dominate the rest of the pixels and are spread over the entire disk both in Figures 3(a) and 3(b). The pixels withpeak offsets of 4 or 5 steps are shown in green and are also scattered over the entire disk. Particularly in Figure 3(b),which presents the data taken during low solar activity, most of the pixels are painted in blue or green. It is evidentthat the distribution of the offsets of the peak positions is generally consistent with that of the kG fields in the activeregions or network boundaries and hG fields in the remaining areas shown in Figure 2.On the other hand, we can find a certain number of red or yellow pixels scattered over the entire disk. Since theyare mostly isolated with a low degree of polarization, they are considered to be the result of noise. If there is nomagnetic field, or if mixed polarities of positive and negative components of the magnetic field cancel each other outin a pixel, the Stokes
V /I signal remains at zero or is very weak for an entire profile. In such cases, noise dominatesthe measured polarization signals, and the Stokes
V /I peak appears at any wavelength and is offset randomly. Suchpixels may appear anywhere outside of the active regions or the network boundaries. There have been some reportsin which the kG components are present in the internetwork magnetic fields; however, these have been hypothesized
Hanaoka & Sakurai AA ≥ ≤ ≥ ≤ (a) 2014 May 10 (b) 2019 Aug 10 Color Code for CLV
Figure 4.
Distribution of the active region and network boundary pixels (AA [active area]; red and light red, degrees ofpolarization exceeding three times the noise level) and that in the internetwork (IN) pixels (blue and light blue, degrees ofpolarization less than three times the noise level) for the data on 2014 May 10 and those on 2019 August 10. Red represents theactive region and network boundary pixels with the offsets of the peak positions of the Stokes
V /I signals at 6 steps or morefrom the line center; light red represents those offset 5 steps or less. Blue stands for the internetwork pixels with the offsets ofthe peak positions at 3 steps or less from the line center, and light blue stands for those offset at 4 steps or more. The disk wasdivided into four zones having different distances from the disk center for the analysis of the center-to-limb variation (CLV).Their borders are indicated by dashed lines. The inset shows the colors representing the four zones, which are used in lateranalysis. to be spurious signals produced through incomplete measurements or inversions (see Bellot Rubio & Orozco Su´arez2019, and references therein). With regard to our data, the majority of such pixels have noise-dominant profiles.3.1.3.
Defining the Internetwork Pixels in the Stokes
V /I
Maps
To study the properties of the Stokes
V /I signals of the internetwork fields more quantitatively, we isolated theinternetwork pixels in the Stokes
V /I maps, excluding the pixels in the active regions or the network boundaries inthe quiet Sun.Figure 3 suggests that the offsets of the peak positions of the Stokes
V /I signals could be a criterion to distinguishthe active regions / network boundaries and the internetwork regions. However, as seen above, some of the pixelswith large offsets do not correspond to the kG magnetic field. Furthermore, it is presumed that there are pixelsincluding both the kG component and the hG one and they show high degrees of polarization in a wide range ofthe wavelengths. The peak wavelength offsets of such pixels are not necessarily as large as the ordinary kG pixels.Therefore, it is important to take the degree of polarization into account. The wavelength offsets of 6–12 steps (37.7/ 40.3 – 75.5 / 80.5 pm) correspond to the Zeeman splitting of the magnetic field of about 1–2 kG, and the pixelsshowing substantial polarization in this range can be considered to include kG magnetic elements inside. Then wedefined the pixels containing the kG magnetic field by using the maximum absolute (unsigned) degree of polarizationof Stokes
V /I signals at the offsets from the line center of 6, 8, 10, and 12 steps. If the polarization at any of theseoffsets exceeds three times of theRMS noise level, the corresponding pixel is presumed to include the kG magnetic field.The influence of the choice of the threshold to the results is mentioned in the next subsection. With this criterion, weclassified the pixels containing kG field as active regions or network boundaries in the quiet Sun, and the pixels freefrom kG components as internetwork pixels.In Figure 4, the defined active region and network boundary pixels are shown in red and light red for the data on2014 May 10 and 2019 August 10. The threshold of the degree of polarization of the Stokes
V /I signals used to definethe active region and network boundary pixels was 0.00048 (3 × RMS noise level mentioned in Section 2.1) after thereplacement of the polarization modulator in 2014. The pixels in red show the offsets of the peak positions of theStokes
V /I signals at 6 steps or more from the line center, and the maximum degree of polarization of the Stokes nternetwork Magnetic Fields Seen in Fe I F r a c t i on F r a c t i on (a) 2014 May 10 (b) 2019 Aug 10 Figure 5.
Histogram of the wavelength offsets from the line center of the peak positions of the Stokes
V /I signals for the dataon 2014 May 10 and those on 2019 August 10. The fractions of the internetwork regions (blue and light blue) and those of theactive regions or network boundaries (red and light red) are stacked, and are shown with the same colors as those in Figure 4.
V /I signals exceeds 0.00048. On the other hand, the pixels in light red show the offset of the peak positions within 5steps from the line center, yet they show the Stokes
V /I signals exceeding 0.00048 at the wavelength offset of 6 stepsor more. The pixels in light red are distributed in the vicinity of the red pixels, and they correspond to the activeregions and the network boundaries seen in Figures 2(c) and 2(d) as well as the pixels in red. This fact supports theclassification of the pixels in light red as the active region or network boundary pixels.The remainder of the pixels (with degrees of polarization less than three times the noise level at the offset of 6 stepsor further) are considered to be the internetwork pixels. We particularly focus on those which have the offsets of thepeak positions of the Stokes
V /I signals within 3 steps from the line center; they are analyzed later and shown in bluein Figure 4. The pixels in light blue have the peak offset at 4 steps or more from the line center and they scatterwidely over the entire disk. As discussed above, the pixels with large peak offsets in the internetwork pixels are, inmany cases, non-magnetic or cancelled-magnetic pixels.Figures 5(a) and 5(b) show histograms of the wavelength offsets of the peak positions in the Stokes
V /I profilesfor the data on 2014 May 10 and those on 2019 August 10. The fractions of the active region / network boundarypixels and those of the internetwork pixels are shown in the same colors as in Figure 4. The fractions of the pixels inthe active regions or network boundaries stand out at the offset of 6 or 8 steps in panel (a) owing to the high solaractivity. On the other hand, a conspicuous peak formed by the internetwork pixels can be found around the offsetof 2 steps, namely 12.6 / 13.4 pm from the line center in both panels. These wavelength offsets correspond to theZeeman splitting of 365 / 392 G. However, as stated in Section 2.2, the offset of 2 steps corresponds to the Stokes
V /I peak position of the weak field regime. Thus, while the definite magnetic field strength of the internetwork pixels isunknown, it is presumed to be 300–400 G or less. Though the results of the former measurements of the internetworkmagnetic field showed some scattering, they were mostly up to 350 G (Khomenko et al. 2003). Therefore, the result300–400 G or less does not contradict the previous results.The internetwork pixels and the active region or network boundary pixels in the quiet Sun are defined for all the 99sets of data selected for this study. Owing to the replacement of the polarization modulator, the RMS noise level wasmuch reduced after 2014, as mentioned in Section 2.1. The 3 × RMS noise level changed from 0.00138 to 0.00048. It ispresumed that the higher threshold applied to data until 2013 to define the active region or network boundary pixels0
Hanaoka & Sakurai F r a c t i on F r a c t i on D eg r ee o f P o l a r i z a t i on D eg r ee o f P o l a r i z a t i on (a) (b) Figure 6. (a) Histograms of the peak position offsets from the line center of the Stokes
V /I signals of the Fe I θ < ◦ ), and the colored lines indicate those for four zones with different distancesfrom the disk center, as shown in Figure 4. The plausible maximum positions of the fraction distributions are indicated withthe vertical bars. The gray stripes in the background show the wavelength sampling steps. The numbers of offset steps fromthe line center are labelled at the bottom of each panel. The results shown here are based on the kG-field threshold of 3 × RMSnoise level, but in addition, the range of the histograms for the entire disk based on the thresholds of 2 × and 4 × RMS noiselevel are also shown with white strips around the dotted lines. (b) Average degrees of the polarization for each peak positionoffset of the Stokes
V /I signals. The values for the entire disk and the four zones are shown in the same way as in panel (a). Inaddition, the standard deviations of the degrees of the polarization of the entire disk are shown with dashed lines. As in panel(a), the range of the average degrees of the polarization for the entire disk based on the thresholds of 2 × and 4 × RMS noiselevel are shown with white strips around the dotted lines. might have missed some of them, and it should be noted that in such cases some of them are mistakenly classified asthe internetwork pixels. 3.1.4.
Center-to-Limb Variation of the Internetwork Stokes
V /I
Signals
Next, we describe the detailed properties of the Stokes
V /I signals of the internetwork magnetic fields, and focus ontheir center-to-limb variation. We divided the solar disk into four zones, one circle and three annuli, which correspondto the angular distances θ from the disk center of θ < ◦ , 20 ◦ < θ < ◦ , 40 ◦ < θ < ◦ , and 60 ◦ < θ < ◦ , as shownin Figure 4. The area beyond 75 ◦ was not used for the analysis because the position of the limb of the Stokes imagescould not be accurately determined. In Figure 6 (and in later presentations), these zones are represented by differentcolors, from light brown to dark brown, as shown in the inset of Figure 4.Figure 6(a) shows the histograms of the offsets from the line center of the peak position of the Stokes V /I signals inthe internetwork regions for the data from 2014 May 10 and 2019 August 10. The dotted lines indicate the histogramsfor the entire disk ( θ < ◦ ). As already shown in Figure 5, the pixels with the offset of 2 steps were most numerouson both dates. This histograms are based on the threshold of 3 × RMS noise level to remove the kG-field as describedin the previous subsection. To check the influence of the choice of the threshold, we calculated the histograms based nternetwork Magnetic Fields Seen in Fe I × to 4 × RMS noise level, and their ranges are also shown with white strips around the dottedlines. The fact that the strips are very narrow means that the results hardly depend on the choice of the threshold.Although the maximum of the fraction in each histogram falls at the offset of 2 steps, taking the fractions at theoffsets of 1 and 3 steps into the consideration, we defined the plausible position of the “true” maximum as the vertexof a parabola fitted to the fractions at the offsets of 1, 2, and 3 steps. The offsets of the maximum positions thusderived for the whole disk for the two dates are indicated by vertical bars drawn with dotted lines. The offset ofthe maximum position on 2019 August 10 was somewhat larger than that on 2014 May 10. However, the wavelengthsampling widths on these dates were not identical, and the difference between these offsets was much smaller than thesampling widths. Therefore, it cannot be concluded that there was a difference between the distributions of the offsetson two observing dates near the most active period and the most quiet period of the Sun.In Figure 6(a), the light-brown lines indicate the histograms of the zone of θ < ◦ , and the lines get darker towardthe limb. All the zones also show that the fraction at the 2-steps offset was most numerous. However, the plausible“true” offsets from the line center of the maximum fraction indicated in Figure 6(a) with colored vertical bars show aslight increase with the increase of the distance from the disk center. This is the common property for the data fromboth 2014 May 10 and 2019 August 10.This does not necessarily mean that the magnetic field strength increases toward the limb and it leaves the weakfield regime. The average Stokes I profiles of the internetwork region in the four zones normalized at the far wing areshown in Figure 1(a). The brightness in the wing decreases from the disk center to the limb. Figure 1(b) shows therelative difference of the profiles with respect to that around the disk center. The brightness around 20 pm from theline center showed the most notable decrease, which means that the width of the Fe I V /I profile in the weak-field regime is located where thegradient of the Stokes I profile takes the largest value. The profiles shown in Figure 1 indicate that the maximumStokes V /I signal in the weak-field regime moved further from the line center with the increase of the angular distancefrom the disk center. Therefore, the systematic difference in the peak position seen in Figure 6(a) was produced bythe broadening of the Fe I V /I signals. While the offsets of the peak positions of the Stokes
V /I signals discussed above justgive the upper limit of the absolute magnetic field strengths in the weak field regime, the degree of polarization isproportional to the longitudinal (line-of-sight) component of the internetwork magnetic fields. The dotted lines showthe average degrees of polarization for the entire disk ( θ < ◦ ), and the dashed lines show their standard deviations.The average degree reached its maximum at 2-steps offset from the line center, where the fractions shown in Figure6(a) also reached their maximum. The range of the average degrees of polarization based on the thresholds from 2 × to 4 × RMS noise level are also shown with white strips. The behavior of the average degrees of polarization does notdepend on the choice of the threshold.The degrees of polarization of the zone including the disk center ( θ < ◦ ) shown with the light-brown lines inFigure 6(b) were clearly lower than that of the other zones except at the offset of 1 step. Furthermore, there wasa tendency for the degree of polarization to increase with the angular distance from the disk center. This can beconsidered an indication that the horizontal component dominates in the internetwork field and longitudinal magneticfield components observed as the Stokes V /I signals were small near the disk center.At the offset of 2 steps in Figure 6(b), it is evident that the degree of polarization increased until the 40 ◦ < θ < ◦ zone. However, the zone of 60 ◦ < θ < ◦ shows a lowering of the polarization. This does not necessarily contradictthe assumption that the horizontal component dominates in the internetwork magnetic fields; we must consider thedegradation of the spatial resolution toward the limb due to the foreshortening effect. In the internetwork region smallpositive and negative polarities are mixed within a small spatial scale. The foreshortening increases the chance thatboth the polarities are included within a pixel and cancel each other out. Thus, we can presume that the degree ofpolarization near the limb, particularly that in the 60 ◦ < θ < ◦ zone, suffers a lowering of the polarization.The dominance of the horizontal field is sometimes considered to be an apparent effect appearing in isotrop-ically distributed orientations, where the frequency of the inclination is proportional to sine of the inclination(S´anchez Almeida & Mart´ınez Gonz´alez 2011). If this is true, the Stokes V /I signals seen in the full-disk maps showthe same properties regardless of the angular distance from the disk center. However, the results show an increase ofthe degree of polarization toward the limb. In principle, this indicates the true dominance of the horizontal orientationin the internetwork magnetic field. Nevertheless, we should note the possibility that the degree of dominance of the2
Hanaoka & Sakurai S un s po t N u m be r O ff s e t o f M a x i m u m ( p m ) D i ff e r en c e ( p m ) (a)(b)(c) Figure 7.
Time variation of the sunspot number and those of the offsets of the fitted maxima of the distributions of the peakpositions of the Stokes
V /I signals of the internetwork pixels from 2010–2019. Panel (a) shows the sunspot relative numbersobtained by the NAOJ. Panel (b) shows the variation of the offset of the fitted maximum calculated from the internetwork pixelsin the entire disk ( θ < ◦ ). The position of the offsets of 2 steps from the center of the Fe I horizontal field depends on the angular distance from the disk center, because the formation height of the Fe I Stability of Various Properties of the Stokes
V /I
Signals
Next, we investigated the stability of the various properties of the Stokes
V /I signals of the internetwork magneticfields seen in Section 3.1 during the years of 2010–2019. This period covers most of solar cycle 24, as seen in Figure7(a), where the sunspot number obtained by the NAOJ is presented. As mentioned in Section 2, we picked 99 datafrom the Fe 1564.8 nm line during this period.3.2.1.
Wavelength Offsets of the Peak Absolute Polarization nternetwork Magnetic Fields Seen in Fe I V /I signals, which are shown with vertical bars in Figure 6(a) for the data from 2014 May 10and 2019 August 10. The variation of the sunspot number is also presented. The fitted offsets for all internetworkpixels ( θ < ◦ ) are shown with plus signs in Figure 7(b). The long gap extending over 2013–2014 was due to afailure caused by a lightning strike. After this gap, the polarization modulator was changed and the noise level wasgreatly improved. The long gap from 2016–2017 was due to a repair work. During the short gap in 2015, the opticalsystem was replaced and the wavelength sampling step was changed from 6.3 pm to 6.7 pm. In Figure 7(b), the offsetwavelengths corresponding to 2 steps from the line center are shown with dotted lines. As mentioned in Section 3.1,the offsets of the fitted maxima of the distributions of the Stokes V /I peak positions show a systematic differenceacross the gap in 2015 due to the change of the spectral sampling, which is a kind of quantization error due to coarsesampling. The offset remained slightly above the offset of 2 steps through the entire period. Therefore, in spite of thesystematic differences, we can conclude that there was no detectable variation tracking the solar activity in the offsetsof the fitted maxima, while the sunspot number showed conspicuous variation.Figure 7(c) shows the variations of the relative wavelength offsets of the four zones with different distances from thedisk center with respect to the offset of all pixels shown in Figure 7(b). The colors of the lines correspond to the zonesshown in the inset of Figure 4. Figure 6 shows that the wavelength offset increased from the disk center to the limb.We can confirm that this tendency was consistent from 2010–2019. Until 2013, the differences among the four areaswere larger than those after 2014, likely due to the differences in the noise level. Therefore, we can conclude that therewas no detectable cycle variation in the wavelength offset of the maximum of the distribution of the peak positionsof the Stokes
V /I signals in the internetwork fields. This means that the strength of the internetwork magnetic fieldsremain within the weak field regime throughout a solar cycle.3.2.2.
Average Degrees of Polarization
Figure 8 shows the time variations of the average degrees of polarization of the pixels where the offset of the peakpositions of the Stokes
V /I signals is 2 or 3 steps from the line center. The degrees of polarization calculated fromthe entire internetwork pixels ( θ < ◦ ) for the offset of 2 steps are shown in Figure 8(a) and those for 3 steps areshown in Figure 8(c). The values for the data from 2014 May 10 and 2019 August 10 have already shown in Figure6(b). The 1 × and 3 × RMS noise levels in the Stokes
V /I signals both before and after the gap of 2013–2014 areshown with dotted lines and dashed lines. The average degree of polarization until 2013 and that from 2014 did notshow systematic change during their respective periods. Figures 8(b) and 8(d) show the relative variations of theaverage degrees of polarization in the four zones with different angular distances from the disk center with respect tothe degrees of polarization of all pixels shown in Figures 8(a) and 8(c). The colors of the lines correspond to the zonesshown in the inset of Figure 4. As mentioned in Section 3.1.4, Figure 6 shows that the average degree of polarizationat the offset of 2 steps increases until the 40 ◦ < θ < ◦ zone toward the limb and that the lowering of the polarizationis seen in the 60 ◦ < θ < ◦ zone. Figure 8(b) shows a similar tendency throughout the measured period. The relativedifferences until 2013 and those after 2014 showed a systematic difference, which was likely due to the changes in noiselevels. However, the general characteristics were maintained during the respective periods. The relative differences ofthe average degrees of polarization at the offset of 3 steps in Figure 8(d) show increases of the polarization toward thelimb throughout the measured period; this tendency became clearer after 2014 than it was before 2013. Nonetheless,these relative differences show no detectable systematic variations until 2013 or after 2014.However, it should be noted that a possible slight increase in the average polarization until 2013 was evident inFigures 8(a) and 8(c). This is presumably due to contamination from the increasing non-internetwork pixels, becausethe thresholds for determining the active region or network boundary pixels were high until 2013. Some of the pixelswith a polarization slightly below the threshold, which would be classified as active regions or network pixels with thethreshold after 2014, were classified as internetwork pixels until 2013.From the above, we can confirm that there was no obvious change in the properties of the degree of polarizationthroughout the period of 2010–2019. As a whole, there is no result showing changes correlating with the solar activityvariation. Thus, we can conclude that the magnetic field properties of the internetwork regions show no detectablecycle variation. SUMMARYIn this paper, we presented the properties of the internetwork magnetic field in the solar photosphere derived fromthe full-disk Fe I Hanaoka & Sakurai D eg r ee o f P o l a r i z a t i on D i ff e r en c e D eg r ee o f P o l a r i z a t i on D i ff e r en c e (a) Peak Offset = 2 Steps(b) Peak Offset = 2 Steps(c) Peak Offset = 3 Steps(d) Peak Offset = 3 Steps Figure 8.
Time variations of the average degrees of the polarization of the internetwork pixels. Panels (a) and (b) show thepolarization of the pixels where the offset of the peak positions of the Stokes
V /I signals is 2 steps from the line center, andpanels (c) and (d) show that for 3 steps. Panels (a) and (c) show the polarizations calculated from the internetwork pixels inthe entire disk ( θ < ◦ ). The degrees of polarization corresponding to 1 × and 3 × RMS noise level are shown with dashed anddotted lines, respectively; these levels jumped due to the replacement of the polarization modulator in 2014. Panels (b) and (d)show the differences in the degree of polarization from the values in panels (a) and (c) at various distances from the disk center.The colors of the lines correspond to the zones shown in the inset of Figure 4. nternetwork Magnetic Fields Seen in Fe I ad hoc observations with a small field of view used inprevious studies. In those studies, the method of analysis was mostly based on the inversions of the polarizationsignals; however, we statistically analyzed large amount of polarization data without inversions. Although we werenot able to obtain various parameter values from the Stokes inversions, the results were robust and free from thecomplexities arising from inversions. Thus, the analysis here shows the properties of the internetwork field derivedfrom a quite different point of view from those in the previous studies.Nevertheless, the obtained results showed that the internetwork is filled with small-scale magnetic fields, the strengthsof which are within the weak-field regime of the Fe I I µ m lines often show inconsistencies; however, our results based on the analysis of the data taken withthe Fe I V /I signals here, but the data contain full-Stokes information. Therefore, to expandthe study using the linear polarization signals is the next step. Furthermore, in addition to the Fe I I I I V /I