Polarization of the Corona Observed During the 2017 and 2019 Total Solar Eclipses

subsec:instobs

We conducted polarization measurements of white-light corona at four sites during the 2017 eclipse and at three sites during the 2019 eclipse. We successfully obtained data under good weather and instrument conditions at two of the sites for each eclipse. The sites, where the data used for the analysis were obtained, are presented in Table \ireftbl:table1, including the eclipse circumstances at these sites and instrumental parameters.

We prepared several, mostly identical instruments for the multi-site polarimetry observations, which include a small telescope, filter wheel, and digital single-lens-reflex (DSLR) camera. The telescope adopted was Takahashi FS-60Q, with an objective lens diameter and focal length of 60 and 600 mm, respectively. The field of view spans an area of $3^{\circ}.4\times 2^{\circ}.3$ ($13\times 9$ $R_{\odot}$, covering the elongation of $>$ 4 $R_{\odot}$). A DSLR camera has a detector with Bayer filters for color imaging, and can be used to obtain broadband images of the white-light corona in red (R), green (G), and blue (B) color-channels. Their central wavelengths are approximately 620, 530, and 470 nm, respectively, for Nikon D810a, and 600, 530, and 470 nm, respectively, for Canon EOS6D. As described in the following sections, the polarization information, as well as multi-color data, enabled us to separate the observed brightness and polarization into K-corona, F-corona, and sky background.

The filter wheel (fabricated by Koheisha, Kawagoe, Japan) was attached between the telescope tube and the camera. In the filter wheel, we installed three linear polarizers, which have transmission axes of $0^{\circ}$, $60^{\circ}$, and $-60^{\circ}$, respectively, relative to the horizontal axis of the camera frame. In addition, a neutral density filter, which has approximately the same thickness and transmission as the polarizers, was also installed in the filter wheel to obtain ordinary white-light coronal images. In this conventional modulator, which is the same type as that of LASCO and SECCHI, the polarization state is altered with the rotation of the filter wheel. In our eclipse observations, we took five images with different exposure times, 1, 1/4, 1/16, 1/64, 1/256, and 1/1024 s, for each filter/polarizer. It takes approximately 30 s to obtain a set of data with the neutral density filter and three polarizers. During the totality period, 3–4 sets of the polarization data were taken at each site, as presented in Table \ireftbl:table1. The images taken with five different exposures for each filter or polarizer were stacked to obtain a single image with a high dynamic range and less noise.

As mentioned above, we carried out observations at multiple sites. One of the objectives of multi-site observations is to detect the evolution of the corona during several tens of minutes, which cannot be detected at a single site owing to the short duration of totality (see Hanaoka et al., 2018; Peticolas et al., 2019; Boe et al., 2020, for successful examples during the 2017 eclipse). In fact, time differences between the two sites were negligible in both eclipses, as observed in Table \ireftbl:table1. However, the obtained data are appropriate in verifying the consistency between the results obtained from different sites.

subsec:dataproc

To obtain polarization maps, first, we stacked images of each filter/polarizer taken with five different exposures during 3–4 polarization modulations. To successfully stack the images, it is necessary to apply some corrections to the images, in addition to dark and flat corrections.

Digital data recorded by the camera, Canon EOS6D, indicated a slightly non-linear response to the input light intensity (the deviation from the linear response is up to 6 %), which we corrected. In addition, systematic errors of the exposure times were also corrected. The nominal exposure times were 1, 1/4, 1/16, … s; however, actual ratios between data numbers in the images taken with adjacent exposure times possibly deviated from 1:4 owing to the systematic errors in exposure times. We estimated the true exposure times based on the ratio of the data numbers in the images (the deviation from the nominal values were at most 1 %), and the data numbers were normalized to the values per second using these true exposure times. The non-linear response and the error in the exposure time are closely related to each other, and therefore, we applied these corrections together so that the images taken with different exposure times become consistent to each other.

The position of the Sun in the images taken during totality swayed owing to the tracking errors. We derived relative displacement among the images via the correlation of the coronal fine structures. By applying the correction for this image displacement, we stacked images of each of the filter and polarizers. The overexposed and underexposed parts in the images were omitted for the stacking, and only the data within 4–97 % of the full pixel-response range were used.

Next, we derived linear polarization signals from the stacked images of the three polarizers. The derivation of the polarization signals as a form of Stokes parameters is explained in, for example, Inhester et al. (2021).

Raw polarization signals converted from the intensities of three polarization states may contain systematic errors. A reason for these errors could be attributed to the variation in the observation target during the polarization modulation. The polarization modulation with a filter wheel is sequential and relatively slow, and polarization modulated images are taken at different time moments. In such a case, spurious polarization signals can be produced. The coronal brightness was verified to exhibit a negligible change during totality in all the observations, and this type of error can be ignored. However, the brightness of the background sky slightly changed during the totality period; even so, the polarization of the sky is eliminated in our analysis, as described in Section \irefsubsec:removalsky.

Nevertheless, we observed systematic errors; the raw linear polarization signals of the corona, which were expected to be tangential to the Sun, exhibited a few kinds of systematic deviations from the tangential direction. The polarization of the sky modifies the orientation of the polarization signals, but the contribution of the sky can be ignored at a small elongation from the solar disk. However, a deviation was found even at a small elongation. This suggests that there were some sort of errors in the data used for the calculation of the polarization signals. Therefore, we applied three types of corrections explained below to minimize the deviation from the tangential direction at 1.1 $R_{\odot}$, where the coronal brightness is more than a hundred times larger than that of the sky.

The first correction is for the throughput difference between the polarizers. In the 2019 observations, we used a light diffuser to obtain flat-field images, and unpolarized light input from the diffuser was effective in correcting the throughput difference. However, in 2017, we obtained flat-field data with the blue sky. To eliminate the influence of the polarization and brightness gradient of the sky, we took images of the sky by rotating the entire optics (telescope, filter wheel, and camera) every $90^{\circ}$ around the telescope optical axis. However, the throughput difference could not be eliminated completely. Such a throughput error triggers a systematic deviation in the orientation of the polarization signals. We estimated the throughput difference by assuming that the correct throughputs provide the minimum deviation of the polarization signals from the tangential direction. For the 2017 data, we applied up to 1 % corrections for the throughputs of the polarizers. Regarding the 2019 data, we confirmed that a minimum deviation is realized without throughput corrections.

The second correction is the positional displacement among the images taken with different polarizers. The images of each polarizer, which were stacked, exhibit almost the same coronal structure; therefore, they can be aligned via the correlation between the coronal fine structures. In contrast, the alignment between the images of different polarization states has an ambiguity, because the images exhibit different brightness distributions owing to the polarization. Therefore, we shifted the polarization images individually to minimize the deviation of the polarization signals from the tangential orientation. The images were shifted approximately $5^{\prime\prime}$ at the maximum.

The third correction is the orientation error of the filter wheel. Although the relative angle differences between the transmission axes of the polarizers are fixed to be $60^{\circ}$, the rotation angle of the wheel may deviate from the camera frame (the reference axis) owing to the tolerance of assembling the instruments at the observation sites. Again, we estimated the roll angle of the wheel by assuming that the correct roll angle provides the minimum deviation of the polarization signals from the tangential orientation, and this angle was $2^{\circ}.5$ at the maximum.

After applying these three corrections, we obtained Stokes $I$, $Q$, and $U$ data for the R, G, and B channels, respectively, which contain information on the polarization of the corona, as well as the sky background. The obtained Stokes $I$ maps of the corona are presented in Figure \ireffig:fig1 in the form of the composite of the RGB images. In the later quantitative analysis, we used the data averaged over $11^{\prime\prime}.8\times 11^{\prime\prime}.8$ (See Section \irefsubsec:bpb); the root-mean-square (RMS) random noise level in the these “macropixels” in areas of the lowest brightness of these Stokes $I$ images was approximately 1.5 %.

To analyze the coronal data quantitatively, the brightness of the corona is required to be expressed in the unit of the solar disk’s average brightness, $B_{\odot}$. We took several images of the solar disk for the reference before the first contact, after the fourth contact (only for 2017) and during the partial eclipse phase, with the same instruments used for coronal observations, but with neutral density filters. We took solar disk images using a stack of two neutral density filters with a total density of approximately 5.6 (transmittance of $10^{-5.6}$). Although it is difficult to measure the spectral transmittance of the filters together (too dense for a measuring instrument), that of each of the filters can be measured with high accuracy. Because the transmittance of the neutral density filters depends on the wavelength, we calculated the effective transmittance for each of the R, G, and B channels using the spectral characteristics of the sensitivity of each color channel of the cameras and that of the solar light (see Hanaoka et al., 2012). The brightness values of the disk derived from these images were compensated for these effective transmittances. Although the transmittance of the neutral density filters and the spectral distribution of the solar light were well defined, the spectral response of the RGB channels of the cameras is expected to include a certain amount of error. This error resulted in the uncertainty of the effective transmittance of 2–20 % depending on the channels.

The disk-brightness data before and after the totality were interpolated to estimate the brightness at the moment of totality, which was used to calibrate the brightness of the corona. However, there is an uncertainty because of the possible change of the sky condition around the totality. Hanaoka et al. (2012) estimated the uncertainty was $\pm 5$ % for the 2008 and 2009 eclipses. We should expect that there is the same level of uncertainty in the brightness calibration here.

The brightness calibration was carried out for data obtained at one of the observing sites for each eclipse, where we took disk images with the above-mentioned filters. The brightness scale of the data taken at the other sites were adjusted to the calibrated data. The error of the effective transmittance of the neutral density filters and the possible change of the sky condition around the totality were the largest sources of the uncertainty in the brightness calibration in our analysis.

Finally, we obtained calibrated polarization maps of the RGB channels taken during the two eclipses observed at the two sites. Representative polarization maps obtained at Salem (for the 2017 eclipse) and at CTIO (for the 2019 eclipse) are presented in Figure \ireffig:fig2. In addition to the tangential linear polarization around the solar disk, we can find a background polarization component originating from the sky. In Figure \ireffig:fig2, the direction of the zenith is indicated. The polarization of the sky occurs approximately along the vertical direction. It is well-known that the orientation of the sky’s polarization in the vicinity of the Sun during totality is almost vertical (e.g., Ney et al., 1961; for actual measurements of the sky’s polarization during the 2017 eclipse, see Shaw et al., 2019, Eshelman et al., 2020, Snik et al., 2020, and Vorobiev et al., 2020). In our observations, the polarization of the sky during the 2019 eclipse is more significant than that of the 2017 one; as observed in Table 1, the altitude of the Sun during the 2019 eclipse was much lower than that during the 2017 eclipse.

subsec:removalsky

As seen in Figure \ireffig:fig2, the polarization of the sky extends over the corona. To obtain the brightness and polarization of the corona, it is necessary to eliminate sky background, particularly in the part distant from the solar disk. We estimated the polarization and brightness of the sky separately to remove the sky background. Actually, stray light from the instrument and atmosphere also overlaps with the image of the corona; however, it is a minor component of the eclipse observations. For this first-order analysis, we assumed that the stray light is negligibly small.

The linearly polarized component of the sky can be estimated based on the fact that it has a completely different orientation distribution from that of the corona. In Figure \ireffig:fig2, the sky appears to have constant polarization; in fact, small variations in the field of view can also be observed. Considering the variation in the field of view, we estimated the polarization of the sky based on two assumptions. The first one is that the Stokes $Q$ and $U$ components of the sky polarization vary linearly in the field of view (in other words, the Stokes $Q$ and $U$ are depicted by a tilted plane). The second assumption is that the orientation of the linear polarization of the corona after the elimination of the sky background is tangential to the Sun. We estimated the $Q$ and $U$ components of the sky, such that the polarization of the K+F corona at 4 $R_{\odot}$ remaining after the removal of the sky background becomes tangential. As illustrated in Figure \ireffig:fig2, up to approximately 4 $R_{\odot}$, the entire circumference is within the field of view, and at this elongation, the sky is the dominant component in the polarization brightness.

Next, we estimated the brightness of the sky background based on two other assumptions. The first assumption is that the sky’s degree of polarization is constant throughout the field of view, and it is the same value in all of the RGB channels. Formerly derived spectral dependence of polarization on the mid-eclipse sky relatively diverges, and no definite tendency can be determined (refer to the summary by Eshelman et al., 2020). Therefore, we assumed a constant degree of polarization regardless of the RGB channels. The second assumption is that the spectral distribution of the brightness of the K+F corona can be approximated by that of the F-corona at far from the solar disk, because the F-corona becomes dominant beyond 2–3 $R_{\odot}$. We adopted the spectral distribution of the F-corona presented in Boe et al. (2021), namely $\lambda^{0.91}$, where $\lambda$ represents the wavelength. We determined the average sky brightness at 4 $R_{\odot}$, such that the coronal brightness of the RGB channels remaining after the elimination the sky background follows this relationship. The atmospheric absorption, which affects shorter wavelengths more significantly, may slightly alter the apparent color of the F-corona; however, it was ignored for simplicity. These two assumptions or constraints cannot be necessarily satisfied simultaneously for three color-channels. Therefore, we adopted a combination of the approximately same degree of polarization for three color-channels, which precisely provides the assumed spectral distribution of the F-corona. The brightness of the sky in the entire field-of-view was calculated from the derived degree of polarization and polarization brightness of the sky.

The estimated sky brightness ranged approximately 8 (R) – 24 (B) $\times 10^{-10}$ $B_{\odot}$ during the 2017 eclipse and approximately 14 (R) – 35 (B) $\times 10^{-10}$ $B_{\odot}$ during the 2019 eclipse, and the typical RMS fitting error of the sky brightness was 3 %. In addition, the degrees of polarization were approximately 0.32 and 0.61 during the 2017 and 2019 eclipses, respectively.

After eliminating the sky background, we obtained the Stokes $I$, $Q$, and $U$ components of the K+F corona; its brightness, polarization brightness, and degree of polarization can be derived from the Stokes components. Figure \ireffig:fig3 presents the polarization maps of the K+F corona. It can be confirmed that the polarization of the K+F corona is mostly tangential. A deviation remains unvanished from the tangential orientation in the polarization of the corona, as presented in Figure \ireffig:fig4. To reduce the random errors owing to shot noise, and to exhibit the large scale pattern, the deviation is averaged in areas with a position-angle width of $3^{\circ}$ $\times$ a radial step of 0.1 $R_{\odot}$ in Figure \ireffig:fig4. Except in the large elongation areas beyond 4 $R_{\odot}$, the magnitude of the deviation is not much different from the results by Vorobiev et al. (2020), who derived the polarization of the inner corona during the 2017 eclipse. In the large elongation area, a small error in the estimated polarization of the sky significantly affects the residual polarization. Actually, at 4 $R_{\odot}$, the RMS error of the polarization brightness after the removal of the sky is as large as 40 % in the $11^{\prime\prime}.8\times 11^{\prime\prime}.8$ macropixels except in the streamer regions, because the polarized component of the K+F corona comes from only the K-corona, which is much fainter than both the F-corona and the sky at this elongation. Nevertheless, the large scale deviation at 4 $R_{\odot}$ remains within a few degrees.

subsec:bpb

The brightness and polarization brightness of the K+F corona, $B_{K+F}$ and $pB$, at six elongations ranging 1.2–4.0 $R_{\odot}$, which were derived from the eclipse observations, are presented in Figures \ireffig:fig5 and \ireffig:fig6. The plotted values were averaged over $11^{\prime\prime}.8\times 11^{\prime\prime}.8$, which is similar to the pixel size of the LASCO C2 full-resolution data, for comparison. The typical uncertainty ranges of $B_{K+F}$ mentioned in Section \irefsubsec:dataproc are presented by color bars in Figures \ireffig:fig5 and \ireffig:fig6. Although the three color-channel data were independently calibrated, the results agree well at the low elongations. This fact suggests that the brightness calibration for the three color channels were consistently performed well within the uncertainty ranges. Furthermore, the data of the two sites are basically consistent with each other. The fact that the $pB$ values taken at two sites agree well indicates that the operation of the filter wheels did not experience failure. In addition, the results of the two eclipses, both of which occurred under low solar activity, are similar to each other. With a closer look, it can be determined that the brightness of the 2019 eclipse, which occurred closer to the solar minimum, is relatively smaller than that of the 2017 eclipse. A coronal mass ejection was observed above the east limb during the 2017 eclipse (Boe et al., 2020), and it contributed to the noticeable brightness excess of the corona around the eastern equator. Throughout the height range, the $B_{K+F}$ data are consistent with the brightness of the typical minimum corona previously published by, for example, van de Hulst (1950).

In Figures \ireffig:fig5 and \ireffig:fig6, we overplotted the $pB$ obtained by K-Cor of MLSO (at 17:19:39 for the 2017 data and 20:39:34 for the 2019 data; taken from the MLSO’s web page, https://www2.hao.ucar.edu/mlso/mlso-home-page) for the inner corona up to 2.0 $R_{\odot}$ in panels (a)–(c). K-Cor measures the polarization brightness of the corona at the height range of 1.05–3.0 $R_{\odot}$. We binned the K-Cor data with $2\times 2$ pixels, corresponding to $11^{\prime\prime}.3\times 11^{\prime\prime}.3$, for comparison. A dip extending around the position angle 10–60^∘ (from the north to the northeast) observed in the K-Cor data of Figure \ireffig:fig5 occurs because of the Moon. The observation wavelength range of K-Cor is 720–750 nm and it is beyond our observation wavelengths; however, the $pB$ does not significantly change in these wavelength ranges. The eclipse observations and K-Cor data indicate a difference of approximately 30 % in some parts. However, even if we take the uncertainty of the brightness calibration of the eclipse data into consideration, as a whole, we can find a relatively good agreement at 1.2 and 1.5 $R_{\odot}$, where the noise owing to the sky background in the K-Cor data is sufficiently small. Structural coincidence between the coronal data obtained at eclipse observations and the K-Cor data was also confirmed by Judge et al. (2019) for the 2017 eclipse and by Boe et al. (2021) for the 2019 eclipse.

In the panels (d)–(f) showing $B_{K+F}$ and $pB$ at elongations of 2.5, 3.0, and 4.0 $R_{\odot}$ of Figures \ireffig:fig5 and \ireffig:fig6, we plotted LASCO C2 data (obtained from the LASCO-C2 Legacy Archive, http://idoc-lasco.ias.u-psud.fr/sitools/client-portal/doc/; refer to Lamy et al., 2020). LASCO C2 takes the $B_{K+F}$ and $pB$ data between 2.2 and 6.5 $R_{\odot}$. The data at approximately 2.5 $R_{\odot}$ and farther, which are free from interference fringes owing to the occulting disk, can be adopted for quantitative comparison. The LASCO C2 data used here were obtained with the orange filter (bandpass: 540–640 nm); this wavelength range extends over the G and R channels of our observations. For the 2019 eclipse, the LASCO data obtained at the time close to our eclipse observations are available (20:38:07 for $B_{K+F}$ and 21:13:30 for $pB$). In contrast, for the 2017 eclipse, although the $B_{K+F}$ data close to our observations are available (17:12:07), the closest $pB$ data were obtained at distant times (02:54:10 and 20:59:55). Therefore, for 2017, we plotted both the 02:54:10 and 20:59:55 $pB$ data with solid and dashed lines. The available $pB$ data of LASCO C2 has a pixel size of $23^{\prime\prime}.8\times 23^{\prime\prime}.8$ instead of the original pixel size, $11^{\prime\prime}.9\times 11^{\prime\prime}.9$, which was used in the $B_{K+F}$ data.

Figures \ireffig:fig5 and \ireffig:fig6 demonstrate that the eclipse observations and LASCO C2 data exhibit good agreement for $B_{K+F}$ in both eclipses. However, the $pB$ data of LASCO C2 are systematically smaller than those of the two eclipses, even if we take the error of the brightness calibration into account.

subsec:degpol

Figures \ireffig:fig7 and \ireffig:fig8 represent the degree of polarization of the K+F corona, $p_{K+F}=pB/B_{K+F}$. For the 2017 eclipse, the polarization data presented by Vorobiev et al. (2020) are also depicted for elongations up to 2 $R_{\odot}$, and for both eclipses, the LASCO C2 data are presented for elongations beyond 2 $R_{\odot}$. For the 2017 eclipse when the LASCO $pB$ data are obtained separately from the eclipse time, the LASCO $pB$ data at 02:54:10 and 20:59:55 were divided by the LASCO $B_{K+F}$ data, taken closely to the $pB$ data, at 02:48:08 and 20:48:07, respectively.

Vorobiev et al. (2020) observed the 2017 eclipse with Rochester Institute of Technology Polarization Imaging Camera (RITPIC), a special polarization imaging camera with pixel-based polarizers on the detector, at Madras, Oregon, USA, where one of our observations was carried out. They took coronal images with approximately $4\times 4$ $R_{\odot}$ field of view in the Bessell B and R wavelength bands (approximately correspond to the B and R channels of DSLR cameras), and derived the polarization parameters. Their $p_{K+F}$ data were binned into a $11^{\prime\prime}.5\times 11^{\prime\prime}.5$ pixel in Figure \ireffig:fig7. Their data agree with ours very well.

However, reflecting the discrepancy observed in the $pB$ data of Figures \ireffig:fig5 and \ireffig:fig6, the $p_{K+F}$ data of LASCO C2 exhibits a systematic difference from the eclipse data. The $p_{K+F}$ values of LASCO C2 are 0.68 times those of the eclipses in average.

Unlike $B_{K+F}$ and $pB$, which need to be calibrated using the average brightness of the solar disk, $p_{K+F}$ can be determined free from the brightness calibration, because its derivation can be done based on the relative variation of the brightness on images taken with linear polarizers with different polarization orientations. Furthermore, linear polarizers have a high polarimetric performance. Ordinary plastic film linear polarizers show a high extinction ratio ($10^{3}$ – $10^{5}$) over a wide wavelength range, and their polarization properties are little affected by incident angle of light or temperature. Therefore, the uncertainty of $<$ 2 % in $p_{K+F}$ derived by Vorobiev et al. (2020) can be considered as a typical error in the measurements using ordinary linear polarizers.

Besides the uncertainty of the brightness calibration, the eclipse $B_{K+F}$ data have an ambiguity triggered by the error of the sky brightness. Here we present the uncertainty range of $p_{K+F}$ determined by taking the extreme error of the sky brightness into account. The highest possible sky brightness is the measured total brightness near the outer edge of the field of view, approximately at 6.5 $R_{\odot}$; even at this elongation, $B_{K+F}$ is not zero, and therefore, the true sky brightness must be lower than the total brightness at this elongation. The lowest possible sky brightness is the polarization brightness of the sky; this means that the sky’s degree of polarization is 100 %. The ranges of the $p_{K+F}$ between those calculated based on the highest and lowest sky brightness are presented in Figures \ireffig:fig7 and \ireffig:fig8 by color bars.

If the true sky brightness is lower than the estimated one, the eclipse $B_{K+F}$ should be larger, and it leads to a smaller $p_{K+F}$. To verify whether the discrepancy between the eclipse observations and LASCO data was triggered by the ambiguity of the sky’s brightness, we presented $p_{K+F}$ calculated assuming the lowest possible sky brightness in the panels (d)–(f) in Figures \ireffig:fig7 and \ireffig:fig8 with gray bands. The range between the maximum and minimum of the calculated $p_{K+F}$ is depicted by the bands. In the case of the 2.5 and 3.0 $R_{\odot}$, the degree of polarization based on the low sky-brightness assumption is still higher than the LASCO values. Finally at 4.0 $R_{\odot}$, it approaches the LASCO values. Therefore, even an unrealistic assumption that the degree of polarization of the sky is 100 % cannot explain the discrepancy except at the 4.0 $R_{\odot}$.

Lamy et al. (2020) discussed the consistency between LASCO C2 data and the eclipse data presented by Vorobiev et al. (2020). However, their eclipse data cover up to 2 $R_{\odot}$, and the LASCO C2 data provide reliable results beyond 2.5 $R_{\odot}$. There is no overlap, and it appears difficult to reach a definite conclusion. However, our wide-field data suggest that a systematic discrepancy exists between the eclipse data and the LASCO data in terms of the polarization, in both the 2017 and 2019 eclipses.

subsec:kandf

Next, we consider the isolation of the K- and F-corona from the observed K+F corona based on the $B_{K+F}$ and $pB$ data.

The $B_{K+F}$ is the sum of the brightness of the K-corona, $B_{K}$, and that of the F-corona, $B_{F}$, such as, $B_{K+F}=B_{K}+B_{F}$. The $pB(=p_{K+F}B_{K+F})$ can be written as $pB=p_{K}B_{K}+p_{F}B_{F}$ using the degree of polarization of the K- and F-corona, $p_{K}$ and $p_{F}$. In the range of elongations discussed here, $p_{F}$ can be assumed to be zero. Therefore, $pB=p_{K}B_{K}$, and $B_{K+F}=pB/p_{K}+B_{F}$. To isolate the K- and F-corona, it is necessary to know $p_{K}$ and $B_{F}$.

We carried out the isolation of the K- and F-corona on the assumption that both the $p_{K}$ and $B_{F}$ solely depend on the elongation from the disk center. Actually, a calculation of $p_{K}$ by Lamy et al. (2020), based on the electron density model by Baumbach (1938), indicates that the $p_{K}$ solely depends on the elongation (refer to their Figure \ireffig:fig2). They indicate that the $p_{K}$ asymptotically reaches approximately 0.64 at large elongations. Furthermore, as demonstrated by Boe et al. (2021), a realistic electron-density distribution prediction of the corona for the 2019 eclipse (PSI-MHD model; magnetohydrodynamic model by Predictive Science Inc.) provides an approximately constant $p_{K}$ regardless of the position angle at each elongation near the limb ($<$1.5 $R_{\odot}$). Beyond this elongation, $p_{K}$ exhibits fluctuation; there is a tendency that the bright coronal streamers show a high degree of polarization. The PSI’s MHD modeling (http://www.predsci.com/corona/coronal_modeling.html) provides the $B_{K}$ and $pB$ distributions for other eclipses as well, and we can confirm that they exhibit a similar tendency to that of the 2019 eclipse. Regarding the F-corona, although the F-corona is known to extend along the ecliptic, in the inner corona, the ellipticity of the F-corona is negligible (e.g., Saito, Poland, and Munro, 1977; see also Koutchmy and Lamy, 1985).

If we adopt a constant $p_{K}$ regardless of the position angle at an elongation $r$, $B_{F}$ can be expressed using the observed values of $B_{K+F}$ and $pB$ as

where $\theta$ represents the position angle. The assumption mentioned above requires $B_{F}(\theta,r)$ to be independent of $\theta$. Therefore, we determined $p_{K}(r)$, such that the fluctuation (standard deviation) of $B_{F}(\theta,r)$ calculated by the above equation becomes minimum. The isolated $B_{K}=pB/p_{K}$ and $B_{F}$ are presented in Figures \ireffig:fig9 and \ireffig:fig10. In panels (a)–(c), $B_{K}$ and $B_{F}$ derived independently for the RGB channels are presented for the elongations at 1.2, 1.5, and 2.0 $R_{\odot}$. For larger elongations, we calculated $B_{K}$ and $B_{F}$ on the assumption that $p_{K}=0.64$ (Lamy et al., 2020) for all the color channels, and they are depicted in panels (d)–(f). The uncertainty ranges of the $B_{K+F}$ presented in Figures \ireffig:fig5 and \ireffig:fig6 are displayed again for the comparison.

The color of the F-corona presumed from the $B_{F}$ curves of the three color channels in Figures \ireffig:fig9 and \ireffig:fig10 appears approximately uniform at each elongation; the rather rough assumptions explained above successfully reproduced the fixed color of the F-corona. Looking at the details, we can find that the $B_{F}$ data from the eclipses at 2.5 $R_{\odot}$ or farther exhibit two broad increases around the equator (position angle P.A.$=90^{\circ}$ and $270^{\circ}$); these are caused by the elliptical distribution of the F-corona at large elongations. In addition, dips in the eclipse $B_{F}$ data can be found at the peaks of the K-corona streamers. This fact probably indicates that the true degree of polarization around the peaks of streamers is relatively higher than the surroundings, as presented by the PSI-MHD model (Boe et al., 2021).

There are some other measurements of $B_{K}$ and $B_{F}$. The data from the LASCO C2 catalog are presented in Figures \ireffig:fig9 and \ireffig:fig10 (panels (d)–(f) in both the figures) with thick and thin black lines. The LASCO $B_{F}$ data for the 2017 presents fairly good coincidence to our results ($B_{F}$ data for the 2019 eclipse are unavailable). The LASCO $B_{K}$ data were estimated based on the polarization. They basically show similar behavior to our results, except that they are systematically lower than our results.

In Figure \ireffig:fig10(a)–(d), we presented $B_{K}$ and $B_{F}$ estimated by Boe et al. (2021; numerical data are from https://www.ifa.hawaii.edu/SolarEclipseData/NarrowCont.html) for the 2019 eclipse with thick and thin gray lines. Their $B_{K}$ and $B_{F}$, binned with a position-angle width of $3^{\circ}$ and a radial step of 0.02 $R_{\odot}$, are expressed in unit of the brightness of the solar disk center, and therefore, those in unit of the average brightness of the disk actually about 20 % higher than the presented values. They obtained narrow-band coronal images at several wavelengths, and calibrated them using the MLSO K-Cor measurements of $pB$ and the relation between $pB$ and $B_{K}$ derived from the PSI-MHD model. Furthermore, they estimated $B_{F}$ based on the spectral characteristics of the narrow-band coronal images. Even if we take the uncertainty of the data into account, their $B_{K}$ data are larger than our results and the LASCO data, while their $B_{K}$ data show a similar position-angle dependence to that of the other observations. The F-corona estimated by Boe et al. (2021) are not only significantly higher than our estimation, but also shows a notable dependence on the position angle. In addition, Boe et al. (2021) suggested that the F-corona is polarized. The polarization of the F-corona was also pointed out by Morgan and Cook (2020); they estimated the degree of polarization of $6.6\pm 0.9$ % between 4 and 5.5 $R_{\odot}$. This means that $pB$ at 4 $R_{\odot}$ shown in Figures \ireffig:fig5 and \ireffig:fig6 is mostly coming from the F-corona, and $B_{K}$ at 4 $R_{\odot}$ is much smaller than that presented in Figures \ireffig:fig9 and \ireffig:fig10. Polarization of the F-corona potentially gives significant impact on the estimations of the K-corona brightness particularly at the large elongations.

In Figure \ireffig:fig11, the estimated $B_{K}$ averaged around the streamers and poles, $B_{F}$ averaged over all position angles, and $p_{K}$ at several elongations up to 2 $R_{\odot}$ are presented. The data for the two sites, two eclipses and three color-channels are plotted together; they mostly overlap each other except the $B_{K}$ at streamers, and this indicates the consistency among the independently derived values. The systematic difference at the streamers in 2017 and 2019 is approximately 20 %. As mentioned in Section \irefsubsec:bpb, this corresponds to the higher activity level of the corona during the 2017 eclipse than that during the 2019 eclipse. In spite of this difference, the $B_{K}$ values are basically similar to those for the equator and poles by Koutchmy and Lamy (1985; presented with plus symbols in panel (a)).

The $B_{F}$ data in Figure \ireffig:fig11(b) are also similar to the previous measurements by Koutchmy and Lamy (1985) and Duerst (1982) (shown with plus symbols in panel (b)) except at 1.1 $R_{\odot}$. The fitting at 1.1 $R_{\odot}$ failed because the brightness of the F-corona was far smaller than that of the K-corona.

In the $B_{F}$ data of the three color-channels shown in Figure \ireffig:fig11(b), the brightest was the R channel and followed in order by G and B. It is natural because the F-corona is red; however, it is not as red as expected from the relation $\lambda^{0.91}$ (Boe et al., 2021), which we adopted in Section \irefsubsec:removalsky for the spectral distribution of the F-corona at 4 $R_{\odot}$. On the contrary, the RGB brightness of the K+F corona beyond 4 $R_{\odot}$ (virtually pure F-corona) inferred from the sky brightness derived in Section \irefsubsec:removalsky is redder than that expected from the relation $\lambda^{0.91}$. These facts suggest the possibility that the assumption of the spectral distribution of the F-corona is too red (namely the eliminated sky is too blue). Actually, adopting the relation $\lambda^{0.48}$ for the F-corona and applying it to estimate the sky brightness at 4 $R_{\odot}$, we can obtain more consistent spectral distribution of the F-corona from 2 $R_{\odot}$ to beyond 4 $R_{\odot}$ (however, within 2 $R_{\odot}$, an even smaller exponent is required). However, the estimation of the F-corona brightness is based on various assumptions as mentioned above, and the results from the two sites of the two eclipses show considerable dispersion regarding the color of the F-corona; therefore, further investigations are needed to judge whether the spectral distribution of the F-corona should be updated or not.

The $p_{K}$ values presented in Figure \ireffig:fig11(c) are not significantly different from the average of the $p_{K}$ predicted by the PSI-MHD model (Boe et al., 2021, presented with stars in panel (c)). They are also similar to the $p_{K}$ derived from a simple model calculation by Lamy et al. (2020).

Consequently, we can conclude that the plausible $B_{K}$, $B_{F}$, and $p_{K}$ values were successfully derived from the observed $B_{F+K}$ and $pB$ self-consistently without external calibration data, only assuming that the $p_{K}$ and $B_{F}$ solely depend on the elongation from the disk center. The $p_{K}$ and $B_{F}$ derived here should be considered a type of average, because the assumption for the $p_{K}$ and $B_{F}$ is relatively rough. If the $B_{F}$ has significant dependence on the position angle or the F-corona is substantially polarized as pointed by Boe et al. (2021) and Morgan and Cook (2020), the results need to be reexamined. Nevertheless, the results support the reliability of the $B_{F+K}$ and $pB$ measured in the two eclipses. Generally, to verify whether the measured $B_{K+F}$ and $pB$ values are correlated with the plausible $B_{F}$, $B_{K}$, and $p_{K}$ values will be a suitable test for the consistency between $B_{K+F}$ and $pB$ values.