Comparison of polarization at two lunar eclipse events

Abstract

We present the results of imaging polarimetry of the eclipsed Moon on 2014 October 8. The observed polarization degree was less than 1% in both the V and R bands, which contrasts with the reported 2%–3% polarization during the eclipse of 2015 April 4. We examined the Earth’s atmospheric data on the two dates and found that the high (≥7 km) cloud distribution was more inhomogeneous for the 2015 eclipse than it was for the 2014 eclipse. The polarization position angle observed during the 2015 eclipse can be explained if the major polarizing source was high clouds with an intermediate horizontal optical thickness. We suggest a possible dependence of the lunar eclipse polarization on the Earth’s high cloud distribution.

1 Introduction

Lunar eclipses provide ground-based observers with opportunities to investigate the Earth’s transmitted light, because sunlight that is refracted and scattered in the Earth’s atmosphere illuminates the Moon in the Earth’s umbra. Regular-intensity spectroscopy of the eclipsed Moon has been conducted to obtain the Earth’s transmission spectrum (e.g., Pallé et al. 2009; Vidal-Madjar et al. 2010; Yan et al. 2015) with the intention of testing spectroscopic observations of transiting Earth-like exoplanets.

There have only been a few polarimetric studies of the eclipsed Moon, probably because astronomers seldom expect detectable polarization during lunar eclipses, which occur with a fairly straightforward Sun–Earth–Moon geometry. One of the few polarimetric observations was performed by Coyne and Pellicori (1970), who reported the detection of significant polarization (∼2%) during the lunar eclipse of 1968 April 13. They speculated that the polarization might have been caused by multiple scattering in the Earth’s atmosphere on the path from the Sun to the Moon, though no further investigation was made. To date, the cause of polarization during a lunar eclipse remains uncertain.

Takahashi et al. (2017) revisited this issue both as a physical question and in anticipation of possible application to transit polarimetry of planets in the Solar System and beyond. The first ever spectropolarimetry of the eclipsed Moon was conducted on 2015 April 4. A polarization of up to 2%–3% was observed at wavelengths of 500–600 nm, which was consistent with the report by Coyne and Pellicori (1970). Using simple calculations, Takahashi et al. (2017) showed that light transmitted through Earth’s atmosphere should be vertically polarized as a result of a flux excess of horizontal double scattering over vertical double scattering.

When a lunar eclipse occurs, the Earth appears from a viewpoint on the Moon as a bright ring of light from the limb, as photographed by Surveyor III (Shoemaker et al. 1968) and KAGUYA (SELENE).¹ Because observation of the eclipsed Moon corresponds to that of the limb-integrated Earth light, Takahashi et al. (2017) pointed out that some type of atmospheric inhomogeneity along the limb is necessary to explain the observed polarization. Otherwise, the net transmitted Earth light (and thus in-umbra Moon) would be unpolarized even though the light transmitted at every position on the limb is vertically polarized. However, with only the data from a single event, it was not possible to identify the atmospheric inhomogeneity which (in combination with the asymmetry in double scattering) causes polarization of the net transmitted Earth light.

To help identify the atmospheric inhomogeneity, we present the results of our analysis of the polarimetric data obtained during the lunar eclipse of 2014 October 8. The brightness of the eclipsed Moon is known to vary from event to event owing to the effects of stratospheric aerosols (Sekiguchi 1983; Keen 1983). Similarly, its intensity spectra also vary, especially at wavelengths shorter than 600 nm (García Muñoz & Pallé 2011; García Muñoz et al. 2011). It is natural to speculate on event-to-event variation in the polarization of the eclipsed Moon. A comparison of the polarimetric results from the two eclipse events of 2014 October 8 and 2015 April 4, along with an examination of the similarities and differences in the Earth’s atmospheric conditions, will narrow down the factor(s) that control the net polarization of the Earth’s transmitted light.

2 Observations

The timeline of the lunar eclipse of 2014 October 8 is summarized in table 1. Imaging polarimetry of the eclipsed Moon in the V and R bands was conducted at three sites in Japan within the framework of an Optical and Infrared Synergetic Telescopes for Education and Research (OISTER) campaign program. At the Nayoro Observatory, Hokkaido University, we used the Multi-Spectral Imager (MSI; Watanabe et al. 2012) at the Cassegrain focus of the Pirka 1.6 m altazimuth telescope; at the Nishi-Harima Astronomical Observatory (NHAO), University of Hyogo, we employed a simultaneous imaging/spectroscopic polarimeter called POL (Nishida 2008; Fujita et al. 2009) mounted at the Cassegrain focus of the 60 cm equatorial telescope; and at the Higashi-Hiroshima Observatory, Hiroshima University, our observations were conducted using the Hiroshima One-shot Wide-field Polarimeter (HOWPol; Kawabata et al. 2008) at the Nasmyth focus of the Kanata 1.5 m altazimuth telescope.

Table 1.

Timetable of the lunar eclipse of 2014 October 8.*

Time (UT)	Event
9:15	Beginning of partial eclipse
∼9:40	Ingress of target region into umbra
10:25	Beginning of total eclipse
10:55	Greatest eclipse
11:25	End of total eclipse
∼12:10	Egress of target region out of umbra
12:35	End of partial eclipse

*The times with respect to the partial/total eclipse are the predictions by the Ephemeris Computation Office of the National Astronomical Observatory of Japan.

The observations are summarized in table 2. The MSI observes two separate fields with sizes of |$\sim \! {40^{\prime \prime }} \times {3{^{\prime}_{.}}3}$|⁠. The image of each of these fields is split into a pair of ordinary and extraordinary images by a Wollaston prism. The POL views a single field with a size of |$\sim \! {2{^{\prime}_{.}}2} \times {7^{\prime }}$|⁠, which is divided into four channels corresponding to linearly polarized light with four polarization position angles: 0°, 45°, 90°, and 135°. The HOWPol has a single |$\sim \! {1^{\prime }} \times {15^{\prime }}$| field of view (FOV), though the effective field for polarimetry is limited to |$\sim \! {1^{\prime }} \times {8^{\prime }}$| owing to the size of the half-wave plate (HWP). Similar to the POL, in a single exposure the HOWPol provides four images corresponding to four different polarization position angles. Both the MSI and HOWPol are equipped with a rotatable HWP in front of or inside the instrument. For every exposure, the HWP was sequentially switched to orientation angles of 0°, 45°, |${22{^{\circ}_{.}}5}$|⁠, and |${67{^{\circ}_{.}}5}$|⁠.

Table 2.

Summary of lunar eclipse polarimetry on 2014 October 8.*

Telescope (diameter)	Instrument	FOV	Time (UT)
Pirka (1.6 m)	MSI	|$\sim \! 40^{\prime \prime } \times {3{^{\prime}_{.}}3} \times 2$|	9:50–12:10
NHAO 60 cm tel.	POL	|$\sim \! {2{^{\prime}_{.}}2} \times 7^{\prime }$|	9:40–12:40
Kanata (1.5 m)	HOWPol	∼1″ × 15″	9:50–12:10

Because the Moon passed through the northern part of the Earth’s umbra during the eclipse, all the telescopes were pointed toward a region near the southern edge of the Moon, which stayed within the umbra for the longest duration. The selected target region was the crater Boussingault at lunar coordinates of 54° E, 70° S. The Moon was imaged on one half of the instrument’s view, whereas the sky was located in the other half of the view (figures 1–3). The use of a relatively large sky area was intended to properly measure and remove the sky intensity. The position angle of the view was fixed in the equatorial system on the sky, and thus an instrumental rotator was used for the altazimuth telescopes (the Pirka and Kanata telescopes). We monitored the target region over almost an entire period when the target was within the umbra (from UT ∼9:40 to ∼12:10).

Fig. 1.

Image frame obtained by Pirka/MSI. The four subframes consist of two pairs of ordinary and extraordinary images. The two pairs correspond to two separate fields. In each pair, the left subframe was regarded as an extraordinary image. The FOV of a single subframe is |$\sim \! 40^{\prime \prime } \times {3{^{\prime}_{.}}3}$|⁠. North is up, and east is left.

Fig. 2.

Image frame obtained by NHAO 60 cm telescope/POL. The four subframes correspond to images at four polarization position angles: from the left subframe to the right, 45°, 135°, 90°, and 0° as measured counterclockwise with respect to equatorial north. The POL views a single field. The FOV of a single subframe is |$\sim \! {2{^{\prime}_{.}}2} \times 7^{\prime }$|⁠. North is up, and east is left.

Fig. 3.

Image frame obtained by Kanata/HOWPol. The four subframes consist of two pairs of ordinary and extraordinary images of a single field. The left and right pairs are denoted as pairs #1 and #2, respectively. The origin orientations of Wollaston prisms corresponding to the two pairs are offset by 45° (table 5). In each pair, the left subframe is regarded as an ordinary image. The FOV of a single subframe is ∼1″ × 15″, but the effective FOV for polarimetry is limited to ∼1″ × 8″. North is up, and east is left.

3 Data reduction

Removal of the sky background intensity from lunar eclipse data is an important and laborious task. The intensity of the sky around the Moon originates mainly from moonlight scattered in the local atmosphere on the path from the Moon to the ground-based observer and thus is stronger in regions closer to the Moon. We need to properly estimate the background values on the Moon based on those in its vicinity. After bias/dark subtraction and flat-fielding, the two-dimensional images—collection of I(x_i, y_i) or pixel values at a chip position (x_i, y_i)—were reduced to one-dimensional data [I(y_i)] by averaging the values along the x axis (or the east–west direction) for ease of later calculation. We applied the two-step sky subtraction method, which was developed by Takahashi et al. (2017) for spectropolarimetry of the lunar eclipse of 2015 April 4. First, the sky intensity was linearly extrapolated from a sky area toward the Moon and subtracted from the original values at the lunar positions; second, residual subtraction was conducted, where we fitted a linear or quadratic function to the sky-subtracted values at the lunar positions and evaluated the residuals of the first procedure. The second procedure considers a case in which the positional dependence of the sky background is not exactly linear, and some residuals remain after the first subtraction, especially for positions farther from the sky region. After these steps, we obtained one-dimensional sky-subtracted intensity values.

We derived the normalized Stokes parameters (q = Q/I and u = U/I) and polarization degree (P) from the intensity values. The derivation is described in detail in the Appendix. As stated in the Appendix, there is general agreement between the results from the MSI and those from the HOWPol, though a significant discrepancy is recognized between the results from the POL and those from the other two instruments. After careful reexamination of our reduction procedures, we concluded that the instrumental polarization was not completely corrected by our methods for the POL data and excluded it from the discussion below (see appendix 1.3).

4 Results and discussion

Time series of q, u, and P are presented in figures 4–6. We see overall agreement between the results from the MSI and HOWPol. However, there are some discrepancies of up to 0.5% (e.g., in P at UT ∼10.5 hr, figure 6), which suggests that our results may include a systematic error of 0.5%. As the maximum measured value for P is ∼0.5% (figure 6), we cannot claim detection of significant polarization. Note that we do not discuss the polarization position angle because we did not detect significant polarization. A major conclusion we can draw from our observations is that the polarization degree of the in-umbra Moon during the eclipse of 2014 October 8 was less than 1%.

Fig. 4.

Time series of q and u in the V band for the Moon during the eclipse of 2014 October 8. The filled/red and open/blue points represent q and u, respectively. The MSI data are displayed as circles, whereas the HOWPol data are shown as triangles. The vertical dashed lines indicate approximate times of the target’s ingress to and egress from the Earth’s umbra. The vertical solid lines represent the beginning and end times of totality. (Color online)

Fig. 5.

Time series of q and u in the R band for the Moon during the eclipse of 2014 October 8. The same styles as in figure 4 are used. (Color online)

Fig. 6.

Time series of P in the V and R bands for the Moon during the eclipse of 2014 October 8. The filled/green and open/orange symbols represent data in the V and R bands, respectively. The other styles are the same as in figure 4. (Color online)

This contrasts strikingly with the polarization degree of the in-umbra Moon during the eclipse of 2015 April 4, which was measured to be 2%–3% at its maximum at wavelengths of 500–600 nm (Takahashi et al. 2017). Below we discuss the question of what caused the difference in polarization between the two events.

Takahashi et al. (2017) suggested that polarization of the in-umbra Moon was caused by a combination of asymmetric double scattering in the Earth’s atmosphere (on the light path from the Sun to the Moon) and atmospheric inhomogeneity along the limb. Their model predicted that when sunlight is transmitted through Earth’s atmosphere, flux excess of horizontal double scattering over vertical double scattering can be expected, which means that transmitted light should be vertically polarized. During a lunar eclipse, Earth’s atmosphere is seen from the Moon as a bright ring of light from the limb, as described in section 1. The limb light, or the transmitted light, should have a radial polarization pattern. If some type of atmospheric inhomogeneity along the limb exists, and thus the polarized flux along the limb is not point-symmetric, then the limb light will have a net polarization. In contrast, if there is no inhomogeneity, the light will not have a net polarization.

The photographs of the illuminated limb of Earth taken by Surveyor III and KAGUYA show a clear inhomogeneity in the brightness along the limb, which was attributed to the cloud distribution at the time (Shoemaker et al. 1968). By analogy with this, we focus on the inhomogeneous cloud distribution as a possible cause of polarization of the limb-integrated transmitted light. We retrieved the Earth’s cloud fraction data on 2014 October 8 and 2015 April 4 acquired by the Moderate Resolution Imaging Spectroradiometer (MODIS) instrument aboard the Terra satellite.² Note that global data from the exact times of the eclipses are not available, because MODIS sweeps an entire hemisphere in 1–2 days with a 2330 km-wide viewing swath.³ The acquisition times of our data range ±12 hr around mid-eclipse. We examined the distribution of the cloud fraction near the day/night terminator, namely, regions whose angular separation from the sublunar point about the Earth’s center ranges from 80° to 100°.

In preparation for the following discussion, we introduce the position angle of a point on Earth’s limb as viewed from the Moon. The position angle θ of point X is defined as the angle of the geodesic line LX with respect to LN as measured clockwise, where points L and N represent the sublunar point and the north pole on Earth, respectively. Because the polarization position angles of the light transmitted above points at θ and θ + 180° should be the same (perpendicular to the horizon), θ is folded by 180°. We obtained the cloud fraction f over every 30° bin of θ. The blue bars in figure 7 (upper line) represent f for six θ bins on 2014 October 8 and 2015 April 4. It appears from the figure that f is almost equally distributed for the entire θ range on both dates.

Fig. 7.

Distribution of cloud fractions. Each blue line is a visualization of a cloud fraction f for a specific position angle θ. The length of each blue line represents the f value. The angle between a blue line and the positive x axis corresponds to 2θ. The radius of a black circle indicates |$\bar{f}$|⁠, or the averaged cloud fraction over all the θ bins. The red lines (which are often invisible) with a circle at the end show |$\boldsymbol{\bar{f}}$|⁠, or the averaged “cloud fraction vectors” (where the blue lines are regarded as vectors). The length of each red line (absolute value of |$\boldsymbol{\bar{f}}$|⁠) and the corresponding θ value are given in table 3. (Color online)

We conducted the same analysis for clouds whose top height (⁠|$z$|_c) is greater than 7 km (approximate scale height of the Earth’s atmosphere) by referring to the cloud top height values embedded in the MODIS data. The lower line in figure 7 displays the results. The cloud distribution on 2014 October 8 seems to be fairly homogeneous. In contrast, for 2015 April 4, the cloud fraction for θ ∼ 75° appears to be greater than that for other position angles.

To support a more quantitative discussion, we calculated some indices. The average of the f values in all the θ bins, denoted as |$\bar{f}$|⁠, is shown in table 3 and corresponds to the radius of each black circle in figure 7. For the “cloud fraction vectors” f, which are visualized as blue lines in figure 7, with a length of f and an orientation of 2θ with respect to the positive x axis, we derived the average, |$\boldsymbol{\bar{f}}$|⁠. The averages are displayed as red lines in figure 7 (though the lines are often too short to see), and their absolute values and position angles are given in table 3. If the cloud distribution has a large inhomogeneity, |$\boldsymbol{\bar{f}}$| has a relatively large absolute value. We define the inhomogeneity index (IHI) as ||$\boldsymbol{\bar{f}}$||$|/\bar{f}$| and give its values in table 3 as well. The results show that the IHIs for clouds of all heights are small (on the order of several percent) for both dates. When we focus on high clouds, however, the IHI on 2015 April 4 is as large as 0.3, and is 2.5 times greater than that on 2014 October 8.

Table 3.

Comparison of cloud distributions on 2014 October 8 and 2015 April 4.*

Date	|$\bar{f}$|	Abs. |$\boldsymbol{\bar{f}}$|	θ for |$\boldsymbol{\bar{f}}$|	IHI
Cloud top height: |$z$|c ≥ 0 km
2014 October 8	73.1%	4.2%	175°	0.057
2015 April 4	61.8%	0.7%	107°	0.012
Cloud top height: |$z$|c ≥ 7 km
2014 October 8	16.7%	2.1%	85°	0.13
2015 April 4	16.4%	5.3%	80°	0.32

*See section 4 and the caption of figure 7 for explanations of the indices.

The greater inhomogeneity in the high cloud distribution on 2015 April 4 compared to that on 2014 October 8 seems consistent with our observed contrast in polarization for the two lunar eclipse events: low polarization in 2014 and high polarization in 2015.

Takahashi et al. (2017) showed that light transmitted at lower altitudes, namely, below the scale height (∼7 km), is weak owing to a large horizontal optical thickness (even for a clear atmosphere), and thus makes an insignificant contribution to the flux and polarization of the total transmitted light. Therefore, it is reasonable that the cloud distribution at high altitudes plays a major role.

For the high clouds on 2015 April 4, the θ value for |$\boldsymbol{\bar{f}}$| is 80° (table 3), which is close to the observed polarization position angle (Θ ∼ 90°) during the lunar eclipse on that date (Takahashi et al. 2017).⁴ We discuss this fact considering two cases: (1) high clouds have a large horizontal optical thickness (τ_H ≫ 1) and (2) high clouds have an intermediate horizontal optical thickness (τ_H ∼ 1).

For case (1), high clouds block the transmitted light. Hence, the polarized flux is reduced if there are clouds on the light path. If a point on the limb with a certain θ (denoted as θ_c) is especially cloudy, the Θ value of the limb-integrated transmitted light is expected to be θ_c ± 90°, because the polarized flux with Θ = θ_c is diminished. As θ_c on 2015 April 4 can be regarded as ∼80°, the expected Θ value is ∼170°, which is in disagreement with the observed result (Θ ∼ 90°). Therefore, case (1) is unlikely.

For case (2), a high cloud can act as a polarizing source instead of a light-blocking curtain. Although Takahashi et al. (2017) considered mainly Rayleigh scattering by atmospheric molecules as the polarizing source, they did not exclude other polarizing scatterers, as noted in the paper. As in the case of Rayleigh scattering, net vertical polarization is expected owing to a flux excess of horizontal scattering over vertical scattering, given that (a) the scatterer is positively polarizing (polarization is vertical to the scattering plane), and (b) the scatterer’s horizontal optical thickness (τ_H) is greater than the vertical optical thickness (τ_V). Condition (a) is true for a cirrus cloud in a phase angle range of 30°–130° (Coffeen 1979). Condition (b) also seems true if we recall that cirrus and cirrostratus clouds at a 7–15 km level are more extended horizontally than vertically (Lamb & Verlinde 2011). Takahashi et al. (2017) predicted that the polarized flux is most significant for an altitude at which τ_H is approximately unity. If clouds with τ_H ∼ 1 exist somewhere at high altitudes, the vertically polarized flux may be enhanced. As θ_c on 2015 April 4 was ∼80° (for |$z$|_c ≥ 7 km), the expected Θ for the limb-integrated transmitted light is ∼80°, which is in approximate agreement with the observed Θ of ∼90°. Therefore, case (2) is more consistent with the observations than case (1).

Takahashi et al. (2017) listed the following two types of inhomogeneities along the limb as possible causes of the net polarization observed during the lunar eclipse in 2015: a greater scale height of the atmosphere and/or a smaller cloud coverage above the equatorial region compared with the polar regions. The latter was based on the idea discussed above as case (1), which proved to be unlikely.

The former was proposed because the polarized flux emerging from a point on the Earth’s limb was expected to be proportional to the scale height of the atmosphere—which is proportional to the atmospheric temperature—at the limb point (Takahashi et al. 2017, appendix C.3). According to the COSPAR International Reference Atmosphere (CIRA-86; Fleming et al. 1988), the pressure-weighted vertically averaged temperature in April is ∼265 K near the equator (0° latitude) and ∼240 K near the poles (80° N and 80° S latitudes). Supported by this fact, Takahashi et al. (2017) pointed out that inhomogeneous scale heights (temperatures) may have contributed to the net polarization observed in 2015 April. However, the temperature of CIRCA-86 in October is very close to that in April (∼265 K near the equator and 230–240 K near the poles) which is natural because both months are around the equinox. Therefore, it seems to be difficult to explain the contrast in the net polarization between 2014 October and 2015 April. Nonetheless, it should be noted that this discussion is based on the data of a reference atmosphere, not on actual measurements observed during the lunar eclipse events.

5 Concluding remarks

We analyzed polarimetric data obtained during the lunar eclipse of 2014 October 8. The polarization degree was less than 1% in both the V and R bands, which differs from the reported 2%–3% polarization for the eclipse of 2015 April 4 (Takahashi et al. 2017). Examination of data obtained from an Earth-observing satellite revealed greater inhomogeneity in the high (≥7 km) cloud distribution for the 2015 eclipse than for the 2014 eclipse. The polarization position angle observed during the 2015 eclipse can be explained if the major polarizing source was optically intermediate clouds at high altitudes. We suggest that the net polarization of the Earth’s transmitted light may be controlled by inhomogeneity in the high cloud distribution, coupled with asymmetric double scattering. However, this should be regarded as a hypothesis at this stage. It is necessary to collect more observational data to confirm the hypothesis.

Funding

This work was supported by Japan Society for the Promotion of Science (JSPS) KAKENHI Grant Number 15K21296, and the Optical and Near-Infrared Astronomy Inter-University Cooperation Program, funded by Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan.

Acknowledgment

We thank Michiko Ishibashi and Hiroyuki Maehara for their backup observations at Saitama University and The University of Tokyo (Kiso Observatory), respectively.

Appendix 1.

Derivation of polarization degrees

A.1.1 Pirka/MSI

We followed the procedures adopted by Ishiguro et al. (2017) and Ito et al. (2018) for the MSI data to derive the polarization degrees and polarization position angles (except for their error estimates). The raw normalized Stokes parameters (q₀ and u₀) were derived using the following equations:

\begin{eqnarray} R_q & = & \sqrt{\frac{I_{{\rm e},0^\circ }}{I_{{\rm o},0^\circ }} \frac{I_{{\rm o},45^\circ }}{I_{{\rm e},45^\circ }}}, \\ \nonumber \end{eqnarray}

(A1)

\begin{eqnarray} R_u & = & \sqrt{\frac{I_{{\rm e},{22{^{\circ}_{.}}5}}}{I_{{\rm o},{22{^{\circ}_{.}}5}}} \frac{I_{{\rm o},{67{^{\circ}_{.}}5}}}{I_{{\rm e},{67{^{\circ}_{.}}5}}}}, \\ \nonumber \end{eqnarray}

(A2)

\begin{eqnarray} q_0 & = & \frac{R_q-1}{R_q+1},\\ \nonumber \end{eqnarray}

(A3)

\begin{eqnarray} u_0 & = & \frac{R_u-1}{R_u+1}, \end{eqnarray}

(A4)

where I_{x, φ} denotes the sky-subtracted intensity. The first subscript, o or e, stands for ordinary or extraordinary rays, respectively. The second subscript, φ, represents the orientation angle of the HWP. Taking a double ratio |$(\frac{I_{{\rm e},\phi }}{I_{{\rm o},\phi }}) / ( \frac{I_{{\rm e},\phi ^{\prime }}}{I_{{\rm o},\phi ^{\prime }}})$| in equations (A1) and (A2) cancels the effects on measurement of both time-varying telluric extinction and the static contrast between the instrumental (behind the HWP) throughput for an ordinary beam and that for an extraordinary beam. A single pair of (q₀, u₀) is derived from four pairs of (I_o, I_e) corresponding to four different φ values.

Then, depolarization in the instrument was corrected by

\begin{eqnarray} \left( \begin{array}{c}q_1\\ u_1 \end{array}\right) = \frac{1}{p_{\rm eff}} \left( \begin{array}{c}q_0\\ u_0 \end{array}\right), \end{eqnarray}

(A5)

where p_eff represents the instrumental polarization efficiency. The value of p_eff is 99.68 ± 0.02% for the V band and 99.71 ± 0.01% for the R band, which were measured by observing dome flat light through a polarizer.

Next, we removed artificial polarization within the instrument (including the telescope). The position angle of the instrumental polarization rotates in accordance with the instrumental rotator angle, ψ. Hence, the instrumental polarization is removed by

\begin{eqnarray} \left( \begin{array}{c}q_2\\ u_2 \end{array}\right) = \left( \begin{array}{c}q_1\\ u_1 \end{array}\right) - \left[ \begin{array}{cc}\cos (2\psi _q) &\quad -\sin (2\psi _q)\\ \sin (2\psi _u) &\quad \cos (2\psi _u) \end{array}\right] \left[ \begin{array}{c}q_{\rm inst}\\ u_{\rm inst} \end{array}\right], \end{eqnarray}

(A6)

where ψ_q(ψ_u) represents the averaged ψ during observations of |$I_{0^\circ }$| and |$I_{90^\circ }$| (⁠|$I_{{22{^{\circ}_{.}}5}}$| and |$I_{{67{^{\circ}_{.}}5}}$|⁠), and q_inst (u_inst) is the instrumental q (u) for ψ = 0°. The values of (q_inst, u_inst) were measured by observing unpolarized standard stars and are listed in table 4.

Table 4.

Correction parameters for the MSI data.*

Parameter	V	R
p  eff	99.68 ± 0.02%	99.71 ± 0.01%
q  inst	0.613 ± 0.006%	0.478 ± 0.006%
u  inst	0.365 ± 0.006%	0.283 ± 0.006%
t  off	|${4{^{\circ}_{.}}15} \pm {0{^{\circ}_{.}}13}$|	|${3{^{\circ}_{.}}81} \pm {0{^{\circ}_{.}}12}$|

*See the text for explanations of the parameters.

Because (q₂, u₂) are defined in the instrumental coordinate system, they were converted into values in the equatorial system by

\begin{eqnarray} \left( \begin{array}{c}q_3 \\ u_3 \end{array}\right) = \left[ \begin{array}{cc}\cos (2t) &\quad \sin (2t) \\ -\sin (2t) &\quad \cos (2t) \end{array}\right] \left( \begin{array}{c}q_2\\ u_2 \end{array}\right), \end{eqnarray}

(A7)

where t = t_off − (position angle of the instrument). The values of the offset t_off are given in table 4.

At this stage, we obtained (q₃, u₃) at every y position on the Moon, as we derived them from one-dimensional intensity data. The (q₃, u₃) data were sampled over 150 pixels (∼|${1{^{\prime}_{.}}0}$|⁠), avoiding the edge of the Moon (with a 15 pix or 6″ margin from the end). The extracted (q₃, u₃) values were averaged into a (⁠|$\bar{q_3}$|⁠, |$\bar{u_3}$|⁠) pair. We also calculated (σ_q3, σ_u3), the standard deviation of (q₃, u₃) over the sampling region, for use in error estimates.

Because the MSI observes two separate fields simultaneously, we obtained two pairs of (⁠|$\bar{q_3}$|⁠, |$\bar{u_3}$|⁠) out of a sequence of four exposures. The difference in q or u values between the two pairs is small (generally less than 0.2%). The two pairs were averaged again to acquire the final single pair of (q, u) for a single sequence. The standard deviations between the two pairs were also calculated and were denoted as (σ_q4, σ_u4). The total errors in q and u were evaluated by

\begin{eqnarray} \sigma _q & = & \sqrt{\sigma _{q3}^2 + \sigma _{q4}^2}, \end{eqnarray}

(A8)

\begin{eqnarray} \sigma _u & = & \sqrt{\sigma _{u3}^2 + \sigma _{u4}^2}. \end{eqnarray}

(A9)

The derived q and u for the MSI are shown as circles in figures 4 and 5 for the V and R bands, respectively.

The polarization degree (P₀) and the position angle of polarization (Θ) were derived by

\begin{eqnarray} P_0 = \sqrt{q^2 + u^2}, \end{eqnarray}

(A10)

\begin{eqnarray} \tan (2 \Theta ) = u/q, \end{eqnarray}

(A11)

where Θ is defined in the range 0°–180° and is selected in such a way that cos (2Θ) has the same sign as q. The errors in P₀ and Θ were determined by

\begin{eqnarray} \sigma _P & = & \frac{\sqrt{q^2\sigma _q^2 + u^2\sigma _u^2}}{P_0}, \end{eqnarray}

(A12)

\begin{eqnarray} \sigma _\Theta & = & {28{^{\circ}_{.}}65} \frac{\sigma _P}{P_0}. \end{eqnarray}

(A13)

Finally, to correct the possible bias arising from noise, we adopted the modified asymptotic estimator developed by Plaszczynski et al. (2014), as follows:

\begin{eqnarray} P = P_0 - \sigma _P^2 \frac{1 - e^{-P_0^2/\sigma _P^2}}{2 P_0}. \end{eqnarray}

(A14)

The acquired P is plotted as circles in figure 6.

A.1.2 Kanata/HOWPol

The derivation of P for the HOWPol data is essentially identical to that for the MSI data, though there are some minor differences according to the user’s convention. We did not apply the correction of instrumental depolarization corresponding to equation (A5) because p_eff for the HOWPol is sufficiently high. The rotating instrumental polarization was corrected by

\begin{eqnarray} \left( \begin{array}{c}q_2\\ u_2 \end{array}\right) = \left( \begin{array}{c}q_0\\ u_0 \end{array}\right) - P_{\rm inst} \left\lbrace \begin{array}{c}\cos [2(-\alpha -C)]\\ \sin [2(-\alpha -C)] \end{array}\right\rbrace , \end{eqnarray}

(A15)

where P_inst is the degree of instrumental polarization; α is defined as

\begin{eqnarray} \alpha = - 90^\circ - \arctan \left(\frac{\sin H}{\tan \phi _{\rm obs} \cos \delta - \sin \delta \cos H} \right) \end{eqnarray}

(A16)

using the hour angle (H) and declination (δ) of the target and the latitude of the observatory |$\phi _{\rm obs}\ (= {31{^{\circ}_{.}}38}\, {\rm N})$|⁠; C is a constant parameter. The position angles of the derived polarization were converted to those in the equatorial system by adding the offset value of the prism origin, Θ_prism—this procedure corresponds to equation (A7). The values of the correction parameters are shown in table 5. The one-dimensional (q₃, u₃) were sampled over 200 pixels (⁠|$\sim {2{^{\prime}_{.}}0}$|⁠) with a margin of 30 pixels (∼18″) from the edge.

Table 5.

Correction parameters for the HOWPol data.*

Parameter	Pair #1	Pair #2
P  inst (R band)	3.94%	3.65%
P  inst (V band)	3.11%	2.74%
C	54°	8°
Θprism	145°	100°

*See the text for explanations of the parameters.

The HOWPol observes a single field and provides two pairs of ordinary and extraordinary images in a single exposure. Thus, we obtain two pairs of (⁠|$\bar{q_3}$|⁠, |$\bar{u_3}$|⁠) out of a sequence of four exposures. The final q, u, P, Θ, and their errors for the HOWPol were derived in the same manner as those for the MSI [equations (A8)–(A14)]. The acquired q, u, and P are plotted as triangles in figures 4–6.

A.1.3 NHAO 60 cm telescope/POL

Because the POL on the NHAO 60 cm telescope does not have a rotatable HWP, the derivation of P differs from that for the MSI and HOWPol. A single set of (q, u) or (P, Θ) is derived from a set of (⁠|$I_{0^\circ }$|⁠, |$I_{45^\circ }$|⁠, |$I_{90^\circ }$|⁠, |$I_{135^\circ }$|⁠), where I is the sky-subtracted intensity, and the subscript denotes the oscillation position angle of the light, which are acquired simultaneously with a single exposure.

The raw q and u (denoted as q″ and u″) were calculated as

\begin{eqnarray} q^{\prime } & = & \frac{I_{0^\circ }-I_{90^\circ }}{I_{0^\circ }+I_{90^\circ }}, \end{eqnarray}

(A17)

\begin{eqnarray} u^{\prime } & = & \frac{I_{45^\circ }-I_{135^\circ }}{I_{45^\circ }+I_{135^\circ }}. \end{eqnarray}

(A18)

Alternation of a polarization state in the instrument (including the telescope) was corrected by assuming a linear relationship between the true values (q, u) and the measured values (q″, u″), as follows:

\begin{eqnarray} \left( \begin{array}{c}q^{\prime }\\ u^{\prime } \end{array}\right) = \left( \begin{array}{c}b_1\\ c_1 \end{array}\right) + \left( \begin{array}{cc}b_2 &\quad b_3\\ c_2 &\quad c_3 \end{array}\right) \left( \begin{array}{c}q\\ u \end{array}\right). \end{eqnarray}

(A19)

The correction parameters b_i and c_i were determined by observing unpolarized and strongly polarized standard stars. Their values are summarized in table 6. The one-dimensional (q, u) were sampled and averaged over 150 pixels (⁠|$\sim {1{^{\prime}_{.}}6}$|⁠) with a 30 pix (∼20″) margin from the edge, and then P and Θ were calculated.

Table 6.

Correction parameters for the POL data.*

Parameter	V	R
b  1	0.005	0.002
b  2	1.05	1.10
b  3	|$- $|0.07	|$- $|0.06
c  1	|$- $|0.007	|$- $|0.004
c  2	0.18	0.18
c  3	1.05	0.99

*See the text for explanations of the parameters.

The derived P was ∼1%–1.5% during the eclipse. This disagrees with the results from the MSI and HOWPol, which showed polarization degrees of 0%–0.5% (figure 6). We examined our reduction procedures by considering pixel-scale differences among the four channels and trying other sky-subtraction methods (e.g., dealing with two-dimensional images instead of stacking them into one-dimensional data), none of which resolved the discrepancy. We concluded that the instrumental polarization was not fully removed by our methods, probably because there might be additional/different instrumental polarization for the Moon owing to the strong stray light. If the POL was equipped with a rotatable HWP, we might have been able to cancel out the effect by taking a double ratio, as we did for the MSI and HOWPol [equations (A1)–(A2)]. Because the results from the POL are doubtful, we excluded them from the discussion.

Footnotes

1

〈http://global.jaxa.jp/press/2009/02/20090218_kaguya_e.html〉.

2

Platnick, S., et al. 2015. MODIS Atmosphere L2 Cloud Product (06_L2). NASA MODIS Adaptive Processing System, Goddard Space Flight Center, USA 〈http://dx.doi.org/10.5067/MODIS/MOD06_L2.061〉. The data were retrieved via NASA’s EOSDIS Worldview web application at 〈https://worldview.earthdata.nasa.gov〉.

3

〈https://terra.nasa.gov/about/terra-instruments/modis〉.

4

Note that both the position angle of a point on Earth’s surface as viewed from the Moon, θ, and the observed polarization position angle, Θ, are measured with respect to (Earth’s) equatorial north, and thus their values are directly comparable. θ is measured clockwise whereas Θ is measured counterclockwise. This is because the views are in the opposite directions.