Imaging Polarimetry of the 2017 Solar Eclipse with the RIT Polarization Imaging Camera

To prepare for observations in the field, the instrumental response of each pixel was determined for the entire system (telescope, filters, and RITPIC) (Figure 3), using an integrating sphere and a rotating linear polarizer. The relative throughput, polarization efficiency, and relative orientation of every pixel was determined by moving the rotating polarizer over a range of 200°, in 1° increments, and acquiring several images at each position. For this characterization we used a wire grid polarizer from Moxtek, Inc.; the absolute angular position of the polarizer is known to ∼15 and the contrast ratio in the relevant calibration bands is given in Table 1. This procedure is similar to that presented in Vorobiev et al. (2018), where the response non-uniformity of RITPIC is discussed at length. In this case, the entire system was characterized at once, including the telescope, bandpass filters, and the micropolarizer camera itself.

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Using the characterization obtained in this way, we analyzed a set of unpolarized and polarized flat field images to estimate the instrumental polarization of the entire system. Analysis of unpolarized flats (generated using an integrating sphere only) showed that the minimum polarization estimated by the system varied from 0.61% ± 0.1% in the blue channel, to 0.67% ± 0.1% in the red channel. Analysis of polarized flats (with the polarization indicated in Table 1), reveals some residual instrumental effects in our demodulation process (Figure 4). The estimated degree of linear polarization (DOLP) for the polarized flats varies between 99.5% and 100.5%, depending on the angle of the incident polarization. Additionally, the spatial variability (indicated by the error bars in Figure 4) in each frame is ∼±1%, with better performance in the blue channel, than in the red.

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We attribute the angle-dependence of our instrumental response to the fidelity of our characterization process. A systematic error in the determination of the relative throughput or orientation differences between micropolarizer pixels would lead to a similar response. Overall, we expect this systematic uncertainty to decrease as the degree of polarization of incident light decreases, with the deviations shown here representing the upper limits.

Our observations were made from Madras, OR, on 2017 August 21. We acquired sequential images in the Bessel B and R passbands, using a range of exposure times. The intensity of the corona changes by a factor of 200 from the edge of the solar disk to a distance of 1 R_⊙ away (Figure 5). Our sensor's limited dynamic range required the use of exposure bracketing (a.k.a. high dynamic range imaging) to image the corona with sufficient SNR, while remaining in the linear operational regime of the CCD. A total of 42 frames (in seven sequences of 6 exposures each) were acquired during totality; a summary of target regions, exposures, and passbands is given in Table 2. In this way, we obtained a full set of observations at 7 instances over the duration of totality.

We acquired observations at four different exposure levels (two for the B filter), with seven images per exposure level, during totality (see Table 2). First, maps of the normalized Stokes parameters, q and u, were calculated for each of the seven images and combined using a median. This resulted in four median maps, corresponding to the four exposure levels. Then, regions of high (photometric) signal were identified in each map. Pixels with a photometric SNR < 35 and pixels close to saturation were flagged as "bad." The remaining high SNR pixels in the four median maps were combined using a weighted mean; the SNR of each pixel in each map was used as its weight (see Section 3.5).

In this section, we present the results of our linear polarimetry of the Solar corona. First, we show the maps of the normalized Stokes parameters, q and u, in the instrumental reference frame. These maps were made by obtaining the median of the 7 HDR frames acquired through the duration of totality (each consisting of 2 exposures for the B channel and 4 exposures for the R channel). The resulting composite maps are shown in Figure 6. The distribution of the normalized Stokes parameters in the corona shows the overall tangential polarization structure, as well as some smaller scale features. The peak fractional polarization is ∼47% in both q and u. Overall, the fractional polarization in the B channel appears to be slightly higher than through the R filter.

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Using the normalized Stokes parameters, we calculate the DOLP and the AOLP (Figure 7). Although DOLP is a biased estimator of the fractional polarization, it is useful because it is independent of any reference frame, allowing for straightforward comparison of measurements made by different observers/instruments.

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At a glance, the structure of the DOLP is similar to that of the overall intensity (Figure 5). However, there exist several differences. Most significantly, the intensity of the corona (Stokes I) peaks near the limb and rapidly decreases with increasing distance from the Sun. Meanwhile, the DOLP is not maximal near the limb (∼30%), and peaks at a distance of ∼1.5 R_⊙. Furthermore, the rate at which the intensity and DOLP decrease is dramatically different. Near the edges of our field of view, the intensity has decreased by a factor of ∼500, whereas the DOLP has only decreased by a factor of ∼2.

As part of an extensive observing campaign prepared for the 2017 eclipse, all-sky polarimetry by Eshelman et al. (2018), Shaw et al. (2019) revealed that the sky in the direction of the Sun is polarized in a large band, at the ∼40% level, with lower polarization very near the Sun. This pattern is significantly different than the typical symmetric polarization pattern, which is minimal toward the Sun, and increases to a peak at ∼90° away from the Sun. Our observations have the sufficient spatial resolution and dynamic range to measure the foreground sky polarization across the Solar/Lunar disk. The degree of polarization for the Bessel B and R channels are shown in Figure 8. The linear polarization of the sky is clearly defined up to a distance of ∼0.5 R_⊙ from the center of the Solar/Lunar disk, where it begins to become comparable in intensity to the scattered light from the solar corona, which maintains its characteristic circular pattern. The regions where the sky and coronal light have parallel and orthogonal polarization are clearly distinguishable by regions of maximal and minimal fractional polarization, respectively.

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The measurements in the blue and red channels exhibit some systematic differences. First, the degree of polarization in the blue channel (as measured in the circled region of Figure 8) is higher than in the red channel, at 0.16 ± 0.013 and 0.08 ± 0.005, respectively. Second, the angle of polarization of the sky foreground differs between the two channels by ∼11 ± 2°. The angle of polarization of the blue light is more closely aligned with the scattering plane, with an average deviation of ∼7° (toward celestial North).

We believe these measurements accurately reflect the polarization state of the sky and are not an instrumental effect. We have been unable to find similar measurements in the literature to perform a direct comparison; most sky polarization studies performed in the past used either all-sky imaging (Pomozi et al. 2001; Eshelman 2018; Eshelman et al. 2018; Shaw et al. 2019) or pointed observations, focused on the solar meridian and the anti-solar point specifically (Dandekar & Turtle 1971). Our measurements in the blue channel roughly agree with those of Eshelman (2018), within the respective measurement uncertainties. However, the degree of polarization in the red channel differs significantly between their measurements and ours, at ∼0.24 and ∼0.08, respectively.

The source of this discrepancy is not clear. However, it may be due to the different local conditions under which the two measurements were taken. The sky over Madras, Oregon had smoke clouds from a nearby wildfire and some thin high altitude cirrus clouds. Conversely, the sky over Rexburg, Idaho, where the Eshelman (2018) observations took place, were relatively clear. In the future, it may be useful to combine all-sky imaging and more pointed narrow-field observations of sky polarization during a total Solar eclipse.

Because we acquired seven complete sets of measurements throughout the duration of totality, we are able to investigate the temporal evolution of the sky foreground (Figure 9). Throughout totality, the degree of polarization in the blue channel increased slightly, but steadily, while the polarization in the red channel remained mostly constant. Conversely, a much larger change in the angle of polarization was measured in the red channel as the eclipse progressed.

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3.4.1. Sky Foreground Subtraction

As part of our polarization analysis, we subtract the sky foreground early in the demodulation process. In each frame, the sky foreground is estimated by the median of the pixels in the region circled in Figure 8. We assume that in this region the polarized brightness is dominated by the sky, with minimal contribution from the scattered coronal light. In the discussion that follows, all maps have been foreground-subtracted. The foreground-subtracted polarization maps are shown in Figure 10.

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Though the coronal polarization does not change dramatically, some key differences can be seen, as compared to the previous maps (Figure 7). The light behind the Lunar disk is now dominated by scattered light from the corona, which maintains its tangential orientation. The DOLP in this region is ∼23% in the B channel and ∼25% in the R channel. The AOLP throughout the field of view is more uniform without the sky foreground, especially at distances greater than ∼2 R_⊙, as can be seen in the bottom left corner of the Bessel B map. In the future, it would be useful to compare the sky polarization measured behind the Lunar disk to that measured at a greater distance from the Sun, for example over the range of ∼5–15 R_⊙.

In this subsection, we outline the procedure we used to obtain an estimate of the uncertainty associated with our measurements of the normalized Stokes parameters and the DOLP. The Stokes parameters are estimated in each RITPIC frame on a per-pixel basis, using a linear least squares fit,

$$\begin{eqnarray}&&\hat{{\boldsymbol{S}}}={\left({{\boldsymbol{X}}}^{\top }{\boldsymbol{X}}\right)}^{-1}{{\boldsymbol{X}}}^{\top }{\boldsymbol{Y}},\end{eqnarray} \tag{ 1 }$$

where $\hat{{\boldsymbol{S}}}$ is the estimated Stokes vector (excluding the 4th element), ${\boldsymbol{X}}$ describes the micropolarizer performance (including contrast and orientation), and ${\boldsymbol{Y}}$ is the vector of corresponding pixel fluxes, in units of electrons. The uncertainties of the estimated Stokes parameters are calculated using analysis of residuals. The variance associated with each measurement, σ², is calculated using the unbiased estimator ${\hat{\sigma }}^{2}$ (Bajorski 2011) as follows,

$$\begin{eqnarray}&&{\hat{\sigma }}^{2}=\displaystyle \frac{1}{n-2}\displaystyle \sum _{i=1}^{n}{\left({y}_{i}-{\hat{y}}_{i}\right)}^{2}=\displaystyle \frac{1}{2}\displaystyle \sum _{i=1}^{n}{\left({y}_{i}-{\hat{y}}_{i}\right)}^{2},\end{eqnarray} \tag{ 2 }$$

where n = 4 (the number of raw pixels used to determine the Stokes parameters), y_i is an intensity measured by the ith pixel in a set of 4, and ${\hat{y}}_{i}$ is the intensity predicted by the model $\hat{{\boldsymbol{Y}}}={\boldsymbol{X}}\hat{{\boldsymbol{S}}}$. Finally, the variance of the estimated Stokes parameters is determined using the measurement variance and the micropolarizer matrix, ${\boldsymbol{X}}$,

$$\begin{eqnarray}&&{{\boldsymbol{\sigma }}}_{\hat{{\boldsymbol{S}}}}^{2}={\hat{\sigma }}^{2}{\left({{\boldsymbol{X}}}^{\top }{\boldsymbol{X}}\right)}^{-1}.\end{eqnarray} \tag{ 3 }$$

This results in a map of variance for each Stokes parameter (${\hat{\sigma }}_{I}$, ${\hat{\sigma }}_{Q}$, ${\hat{\sigma }}_{U}$,), for each data frame (Figure 11, Left). Using formal uncertainty propagation, we determine the variances of the normalized Stokes parameters: σ_q and σ_u. Next, using the set of 7 measurements, we obtain the median values of the normalized Stokes parameters and the average variance: $\bar{q}$ and $\bar{{\sigma }_{q}^{2}}$ (Figure 11, Middle). Finally, we use the map of Stokes I to find regions with photometric SNR > 35 and mask off saturated pixels. The resulting region of "high quality" data for the 10 ms exposures is shown in the right panel of Figure 11.

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This process is repeated for the remaining frames, with exposures of 50 ms, 1 s, and 5 s. The high SNR regions from each exposure set are combined to generate the maps shown in Figures 6 and 7. The uncertainties from high SNR regions of each exposure set are combined using a weighted mean. The weighting is determined by the SNR of each measurement. In this way, the contribution of the noisier measurements is diminished, if better data are available from a different exposure set. As a result, the uncertainties of the final measurements, σ_q and σ_u, are not the true standard deviation; however, they are still a reasonable estimate of the uncertainty of each measurement.

Lastly, we use the uncertainties σ_q and σ_u to find the uncertainty associated with the DOLP and AOLP in the corona, σ_dolp and σ_aolp, using formal (linear) uncertainty propagation. Strictly speaking, this procedure gives a biased estimate of both the fractional polarization, DOLP, and its uncertainty, σ_dolp. However, the corona is very strongly polarized, even in areas of low polarization. This high "polarimetric SNR" (>20 across our field of view) ensures that the relative effect of the bias is small. The maps of uncertainty are shown in Figure 12. In regions where the polarization is large, our combined measurement of the fractional polarization has an uncertainty of ∼1%–3%. At distances >1.5 R_⊙, the lack of signal from the corona and the sky foreground lead to rapidly increasing uncertainty. A similar trend can be seen in the estimate of the AOLP (Figure 12, Right). Where the signal is high, the angle estimation is precise to ∼1°. Further out, the uncertainty increases to ∼2°.

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For a quick comparison of our measurements to the predictions of a Thomson scattering corona, we use the theoretical curves of van de Hulst (1950). The van de Hulst model is azimuthally symmetric, with a separate treatment for the equatorial and polar regions. This model includes both the highly polarized K corona and the weakly polarized F corona. Each parcel of coronal plasma is primarily illuminated by photons from the solar photosphere, such that at a distance R from the Solar surface, the plasma is illuminated by a "cone" that subtends the solid angle ${\rm{\Omega }}=\tfrac{1}{{R}^{2}}$ on the photosphere. Very near the photosphere, the corona is illuminated significantly by photons from a large fraction of the photosphere and from many directions, leading to a fairly low average polarization. As the distance increases to R ≈ 1.5 R_⊙, the majority of the light is arriving from a relatively small region of the photosphere, nearly radially, resulting in the high fractional polarization of ∼60%. This high polarization fraction is predicted by van de Hulst (1950) to extend to distances of R ≈ 6 R_⊙, contrary to most observations. This discrepancy is usually attributed to the dilution of the highly polarized K corona by light scattered by interstellar dust particles which makes up the weakly polarized F corona.

During this eclipse, the corona showed strong polarization features in the equatorial direction, with weaker polarization (and intensity) at the poles. Large scale and small scale "streamers" are clearly visible. The DOLP of the corona peaks at ∼47% in the equatorial features, at a distance of ∼1.5 R_⊙. To perform a preliminary comparison to the van de Hulst (1950) models, four regions were chosen: one each at the north and south poles, and in areas of maximal and minimal polarization near the equatorial regions (indicated by rectangles in Figure 13). The radial profiles (Figure 14) were created by averaging the polarization in these regions, in the azimuthal direction. Four regions were chosen: one each at the north and south poles, and in areas of maximal and minimal polarization in the equatorial regions.

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The maximal polarization we measured agrees well with the maximum predicted values from van de Hulst (1950). However, the polarization measured at the north and south poles is significantly higher than the model prediction. The minimally polarized regions near the equator are also more significantly polarized. The source of the discrepancy is not clear to us at this time. It is likely that the azimuthally symmetric model does not adequately account for the electron density structure in the real corona. Also, the impact of the sky foreground is difficult to rigorously quantify with the data we have available. Indeed, the sky-subtracted maps show more significant deviations from the model than the un-calibrated measurements, especially further from the Solar center. This may indicate over-subtraction of the sky foreground, because the corona intensity (and thus polarized intensity) is fainter in these regions.

3.6.1. Distribution of the Polarization Angle

The angle of polarization is tangential to the solar limb. This is shown in Figures 7 and 10. To identify any underlying structure in the AOLP, we plot the deviations of the AOLP from the purely tangential direction, denoted by the angle χ, in Figure 15. In regions of strong polarization (and high SNR), our estimate of the AOLP is consistent with the tangential direction within ∼05. Where the polarization is low, like near the south east limb, the deviation increases to ∼1°. In regions further than ∼1.5 R_⊙, the deviation increases to >3°.

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