1 Introduction

The white-light emission of the solar K-corona originates in Thomson scattering of photospheric light by free electrons in the corona. Detailed descriptions of the Thomson-scattering theory can be found in various articles (see, e.g., Minnaert, 1930; Baumbach, 1938; van de Hulst, 1950; Shklovskii, 1965; Billings, 1966; Inhester, 2015).

It is known that the degree of polarisation of Thomson-scattered light by coronal electrons is a sensitive function of the scattering angle between the incident-light direction and the direction towards the observer (Billings, 1966). This effect allows us to estimate an effective distance of the coronal-scattering volume off the plane of the sky (POS), and it has been employed to obtain 3D positions of coronal mass ejections (CMEs) from coronagraph images obtained with different polariser orientations (see, e.g., Moran and Davila, 2004; Dere, Wang, and Howard, 2005; Vourlidas and Howard, 2006; Mierla et al., 2009).

In a recent study, Floyd and Lamy (2019), taking advantage of the much improved determination of the polarisation of the corona (see, e.g., Lamy et al., 2014) from the LASCO-C2 images (Brueckner et al., 1995), applied the polarised-ratio technique to a set of 15 CMEs observed in the time interval from 1998 to 2014. The authors showed that despite the unfavourable conditions affecting the LASCO-C2 polarisation measurements, the polarimetric reconstruction works quite well and is able to produce meaningful 3D CME morphology and direction of propagation. Even in the case of the very fast CME of 18 July 2002 (speed around 2000 km s−1), its localisation could be obtained for the bulk of it (Floyd and Lamy, 2019).

As the observed polarised brightness is much less sensitive to stray light, it has been also widely used to estimate the electron density of the average corona and of the total mass of CMEs (see, e.g., Quémerais and Lamy, 2002; Feng et al., 2015; Decraemer, Zhukov, and Van Doorsselaere, 2019). As the estimation of the CME mass and propagation direction has important implications for space-weather forecasts, it is important to have accurate measurements of the polarisation parameters.

The accurate measurements of the polarisation parameters can be checked by the tangential orientation of the polarisation direction. There are limits to this check, both experimental (mirrors may rotate the polarisation (Floyd and Lamy, 2019), atmospheric effects for ground-based measurements (see Koutchmy et al., 1999), limited precision, etc.), and theoretical (non-random orientation of dust, relativistic electrons (Molodensky, 1973; Kishonkov and Molodenskii, 1975), and magnetically sensitive resonant-line scattering (see, e.g. Mierla et al., 2011; Qu et al., 2013; Heinzel et al., 2020). In this article we present the observation of a systematic deviation of the polarisation angle. This deviation occurs near the propagation front of a fast CME. This led us to the conclusion that we have to account for yet another effect that may lead to rotated polarisation angles. This effect has so far not been considered in the literature. We discuss this effect quantitatively and compare it with possible alternative reasons in the discussion section. The article is structured as follows: In Section 2 we introduce the data, and in Sections 3 and 4 we describe how they are processed and analysed. Discussions and conclusions are presented in Sections 5 and 6, respectively.

2 Observations

In this work, we use COR1 and COR2 images taken on 12 July 2012. COR1 is the coronagraph with the innermost field of view of the Sun Earth Connection Coronal and Heliospheric Investigation (SECCHI) instrument suite (Howard et al., 2008) onboard the twin Solar TErrestrial RElations Observatory spacecraft (STEREO: see Kaiser et al., 2008). Each of the STEREO/COR1 telescopes has a field of view from 1.4 to 4 solar radii [R⊙] and observes in a white-light waveband 22.5 nm wide centred at the H\(\alpha \) line at 656 nm (Thompson and Reginald, 2008).

Like the COR1, the SECCHI outer field-of-view coronagraph, COR2, observes the coronal signal in visible light from 2 to 15 R⊙, with a passband from 650 to 750 nm. Both COR1 and COR2 coronagraphs take polarised images at three different polarisation angles at 0∘, 120∘, and 240∘.

The COR1 images have dimensions of \(512\times 512\) pixels and a pixel size of 15′′. The exposure time is one second and the image cadence is five minutes. Each set of three polarised images (\(I_{0}\), \(I_{120}\), and \(I_{240}\)) is taken in approximately 18 seconds (COR1-A) and 24 seconds (COR1-B), with 9 seconds between the frames in the case of COR1-A data, and 12 seconds between the frames for COR1-B.

The COR2 images have dimensions of \(2048\times 2048\) pixels and a pixel size of 15′′. The exposure time is six seconds and the image cadence is one hour. Each set of three polarised images (\(I_{0}\), \(I_{120}\), and \(I_{240}\)) is taken in approximately 60 seconds with 30 seconds between the frames.

On 12 July 2012, a fast and wide CME (associated with an X1.4 flare) was observed at the east limb on COR-A images, at the west limb in COR-B images, and coming from the central disk in LASCO images. The approximate location of the CME source region (the X1.4 flare) was S15W01 as observed from the Earth’s perspective (see, e.g., cdaw.gsfc.nasa.gov/CME_list/halo/halo.html and solarmonitor.org/index.php?date=20120712&region=11520).

The CME was observed for the first time by COR1 at around 16:00 UT, by COR2 at around 16:40 UT, and by LASCO-C2 at around 16:50 UT. COR1 and COR2 coronagraphs observed a structured, three-part CME consisting of a bright circular leading edge, followed by a dark cavity and a bright compact core (see, e.g., Illing and Hundhausen, 1986), while LASCO-C2 and -C3 coronagraphs observed a full-halo CME with apparent angular width of \(360^{\circ}\). A diffuse shock (see, e.g., Vourlidas et al., 2003) was also observed ahead of the leading edge.

3 Data Processing

Data processing consists of three steps:

  1. i)

    Pre-process the data using the IDL SolarSoft secchi_prep.pro routine.

  2. ii)

    Extract the K-corona signal by subtraction of the base-difference image.

  3. iii)

    Extract the polarised parameters – polarised brightness, total brightness, degree of polarisation, and the deviation angle from the tangential – from three K-corona images with different polarisations \(I_{0}\), \(I_{120}\), and \(I_{240}\).

3.1 Data Pre-Processing

We run secchi_prep on each polarised COR1 and COR2 image. By default, secchi_prep applies the following calibration to COR images: division by exposure duration; correction for onboard image processing; subtraction of the CCD bias; multiplication by the calibration factor to convert the image values from digital number (DN) to mean solar brightness (MSB); vignetting and flat-field correction; and optical distortion correction (COR2 only).

The COR1 and COR2 images were brought to the same dimensions: \(1024\times 1024\) pixels with the pixel size of 7.5′′ for COR1 and 29.4′′ for COR2.

We divided each image by the calibration factor \(6.780 \times 10^{-11}\) for COR1 and \(1.350 \times 10^{-12}\) for COR2 to have units of DN s−1. We preferred DN s−1 as it offers a more straightforward way to display the data. The choice of the units does not influence our results.

3.2 Extracting K-Corona Signal

In general, one raw white-light coronagraph image contains signal from the solar corona (K-corona: electron or continuum corona; F-corona: Fraunhofer or dust corona; E-corona: emission lines like H\(\alpha \), Fe x, Fe xiv, Ca xv, He i D3 line, etc.), from the cosmic rays, planets, comets, stars, as well as from the instrument-related contribution: bias, dark current, stray light, etc. Some of this signal is polarised, and some is unpolarised.

The K-corona signal is due to the scattering of the sunlight (photospheric light) by the free electrons of the corona, also known as the Thomson-scattered light. The F-corona is the scattering of the light on the dust particles around the Sun (contains Fraunhofer absorption lines). The E-corona is formed from the emission of highly ionised atoms of the solar corona. One more component, the thermal (T-corona) is produced by thermal emission of dust particles heated by the Sun (Shopov et al., 2008).

The K-corona contains features such as streamers, CMEs, outflows, inflows, etc. In the ideal case we would like to eliminate all the unwanted contributions and keep the signal only from the K-corona, or in our case only from the CME Thomson-scattered light. In order to eliminate the unwanted signal, one can employ different procedures: i) subtraction of the monthly minimum background from a raw white-light coronagraph image; ii) subtraction of a pre-event image from a raw white-light coronagraph image (the so-called base-difference image technique); iii) subtraction of a hourly minimum background from a raw white-light coronagraph image.

These procedures are described in detail in Appendix A.

For this study we used base-difference images by subtracting the frame before the CME was observed, at 15:50 UT in COR1 and at 16:09 UT in the COR2 field of view (FOV). Three different pre-event images are used, one each for the three polarisers. Henceforth, if not stated otherwise we will always be using base-difference images.

The influence of different backgrounds on our results is discussed in Appendix B.

Figure 1 shows an example of one set of three polarised COR1-A images taken on 12 July 2012, 16:35 UT.

Figure 1
figure 1

COR1 images taken on 12 July 2012, with orientation of the polariser of \(0^{\circ}\) taken at 16:35:00 UT (left panel), \(120^{\circ}\) at 16:35:09 UT (middle panel), and \(240^{\circ}\) at 16:35:18 UT (right panel). Images are displayed on a logarithmic scale. Base-difference images were used. The radius of the occulter is 1.47 R⊙ and the circle inside the occulter outlines the solar disk.

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3.3 Extraction of the Polarisation Parameters

From COR1 and COR2 polarised images processed as explained above, we derived polarised quantities: total brightness (B – Equation 1), polarised brightness (pB – Equation 2), the degree of polarisation (p – Equation 3), and the polarisation-angle orientation (\(\alpha \) – Equation 4) of the K-corona signal. The deviation angle from the tangential [\(\Delta \alpha \)] is calculated as the difference between the tangent to the solar limb and the polarisation angle. Positive values of the deviation angle correspond to the clockwise rotation and the negative values to the counter-clockwise rotation.

$$\begin{aligned} B& = \frac{2}{3} \,(I_{0} + I_{120} + I_{240}) \end{aligned}$$
(1)
$$\begin{aligned} pB& = \frac{4}{3} \,\sqrt{(I_{0} + I_{120} + I_{240})^{2} - 3\,(I_{0}I_{120} + I_{0}I_{240} + I_{120}I_{240})} \end{aligned}$$
(2)
$$\begin{aligned} p& = \frac{pB}{B} \end{aligned}$$
(3)
$$\begin{aligned} \alpha &= \frac{1}{2} \,\arctan{ \frac{\sqrt{3}\,(I_{240} - I_{120})}{2I_{0} - I_{120} - I_{240}}}. \end{aligned}$$
(4)

These formulae correspond to frequently used ones that utilise Stokes parameters I, Q, and U. Here, we use explicit formulae to simplify the analysis presented in Section 4.2. In Equation 1 and Equation 2 we used the directions of \(0^{\circ}\), \(120^{\circ}\), and \(240^{\circ}\) of the polarisers, instead of the usual directions of \(0^{\circ}\), \(60^{\circ}\), and \(120^{\circ}\) because these are the angles at which the COR1 polarisers are reported to be oriented, (see, e.g., Thompson and Reginald, 2008). Note that \(60^{\circ}\) and \(240^{\circ}\) polariser orientations are identical.

\(I_{0}\), \(I_{120}\), and \(I_{240}\) represent the images recorded by the COR1 instrument at three polarisation angles, from which a pre-event image was subtracted as mentioned in the previous section. In this way, the three polarised images \(I_{0}\), \(I_{120}\), and \(I_{240}\) contain largely only the component of the CME K-corona.

The polarisation angles for COR2 images were corrected with the values provided by the SolarSoft routine cor1_fitpol.pro (Thompson, 2015). This was necessary because the COR2 polarisers do not have a specific alignment.

4 Data Analysis and Interpretation

4.1 CME Polarisation Parameters

In Figure 2 we show the extracted polarisation parameters from COR1-A images taken at 16:35 UT (left column) and at 16:50 UT (right column), as follows: polarised brightness \(pB\) (upper panels), total brightness \(B\) (middle-upper panels), degree of polarisation \(p\) (middle-lower panels), and deviation angle from the tangential (lower panels).

Figure 2
figure 2

COR1-A polarisation parameters derived from polarised images on 12 July 2012, 16:35 UT (left panels) and 16:50 UT (right panels) showing: pB-images (upper panels), B-images (middle-upper panels), degree of polarisation (middle-lower panels) and deviation from the tangential angle (lower panels). pB and B images are displayed on a logarithmic scale. The parameters are derived from base-difference images. The circle inside the occulter outlines the solar disk.

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We observe deviations from the tangential greater than \(10^{\circ}\), in regions associated with the excess brightness of the CME (see also Figure 3). The degree of polarisation is around 70% in regions associated with the CME signal (see also Figure 3).

Figure 3
figure 3

Radial profiles of the polarisation parameters along the dashed line in Figure 2.

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We apply the same technique to COR1-B and COR2-A and -B images, to see if the same behaviour is observed in different instruments.

Figure 4 displays the total brightness calculated from COR1-A (upper-left panel), COR1-B (upper-right panel), COR2-A (lower-left panel) and COR2-B (lower-right panel) images. The images are shown on a logarithmic scale. The two COR-B images show a rather consistent structure of the CME, whereas the COR2-A structure is much more complex than the COR1-A one.

Figure 4
figure 4

Total brightness from COR1-A (upper-left panel), COR1-B (upper-right panel), COR2-A (lower-left panel), and COR2-B (lower-right panel) observations. The images are displayed on a logarithmic scale. Base-difference images were used. The circle inside the occulter outlines the solar disk.

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Figure 5 shows the degree of polarisation from COR1-A (upper-left panel), COR1-B (upper-right panel), COR2-A (lower-left panel) and COR2-B (lower-right panel).

Figure 5
figure 5

Degree of polarisation from COR1-A (upper-left panel), COR1-B (upper-right panel), COR2-A (lower-left panel), and COR2-B (lower-right panel) observations. Base-difference images were used. The circle inside the occulter outlines the solar disk.

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The degree of polarisation is around 65% but it can go as high as 80% in some CME regions from COR1-A. These high values of the degree of polarisation are more observed in COR2 images.

In Figure 6 is shown the deviation angle from the tangential using COR1-A (upper-left panel), COR1-B (upper-right panel), COR2-A (lower-left panel), and COR2-B (lower-right panel) data.

Figure 6
figure 6

Deviation angle from the tangential from COR1-A (upper-left panel), COR1-B (upper-right panel), COR2-A (lower-left panel), and COR2-B (lower-right panel) observations. Base-difference images were used. The circle inside the occulter outlines the solar disk.

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Deviations from the tangential up to 10 – \(15^{\circ}\) are seen in COR1-A images (upper-left panel of Figure 6), in all pixels where a significant CME excess brightness is observed, i.e. in both the front and the core of the CME (see also Figure 2). It is more visible on the front part of the CME (the red and blue areas in the front and back part of the CME frontal structure). The deviations are smaller in COR1-B images. The same deviations are observed also in monthly minimum and hourly minimum background subtracted images, as discussed in Appendix B. For COR1, a systematic constant deviation of the polarisation angle by \(-1^{\circ}\) was found (Inhester et al., 2021).

For COR2-A data, the deviation from the tangential increases to \(30^{\circ}\) at the front of the CME (blue area in the lower-left panel of Figure 6).

4.2 Tests on Deviation from the Tangential

To further investigate the deviations from the tangential described above, we perform a few tests, as displayed in Figure 7. We used base-difference COR1-A images, at 16:35 UT for these tests.

  1. i)

    In the first test (upper-left panel in Figure 6), we derive the polarisation angle from the three polarised images (\(I_{0}\), \(I_{120}\), and \(I_{240}\)) recorded by COR1-A at 16:35 UT.

  2. ii)

    For the second test (left image of the upper row in Figure 7), we kept \(I_{0}\) as observed and assigned \(I_{120}\) and \(I_{240}\) a zero value in all pixels.

    Figure 7
    figure 7

    Deviation angle from the tangential calculated in six different scenarios: Upper row, from left to right: \(I_{120}\) and \(I_{240}\) set to zero, \(I_{0}\) as observed; \(I_{0}\) and \(I_{240}\) set to zero, \(I_{120}\) as observed; \(I_{0}\) and \(I_{120}\) set to zero and \(I_{240}\) as observed. Lower row, from left to right: \(I_{0}\) set to zero and \(I_{120}\) and \(I_{240}\) as observed; \(I_{120}\) set to zero and \(I_{0}\) and \(I_{240}\) as observed; \(I_{240}\) set to zero and \(I_{0}\) and \(I_{120}\) as observed. The circle inside the occulter outlines the solar disk.

    Full size image
  3. iii)

    For the third test (middle image of the upper row in Figure 7), we kept \(I_{120}\) as observed and assigned \(I_{0}\) and \(I_{240}\) a zero value in all pixels.

  4. iv)

    For the fourth test (right image of the upper row in Figure 7), we kept \(I_{240}\) as observed and assigned \(I_{0}\) and \(I_{120}\) a zero value in all pixels.

  5. v)

    For the fifth test (left image of the lower row in Figure 7), we kept \(I_{120}\) and \(I_{240}\) as observed and assigned \(I_{0}\) a zero value in all pixels.

  6. vi)

    For the sixth test (middle image of the lower row in Figure 7), we kept \(I_{0}\) and \(I_{240}\) as observed and assigned \(I_{120}\) a zero value in all pixels.

  7. vii)

    For the seventh test (right image of the lower row in Figure 7), we kept \(I_{0}\) and \(I_{120}\) as observed and assigned \(I_{240}\) a zero value in all pixels.

We see that the observed deviation angles are almost entirely due only to the combination of \(I_{0}\) and \(I_{240}\) images, which are taken 18 seconds apart. In this time interval, a CME with a speed of 600 km s−1 will move about 10,800 km, i.e. more than two pixels (= 10,489.8 km) in COR1-A FOV, which may justify the deviations observed. The speed of 600 km s−1 was the measured speed of the CME at 2.41 R⊙ (16:35 UT) in the COR1-A FOV.

In the future, this interpretation will be further tested based on the analysis of synthetic CME observations built starting from a full MHD 3D simulation, similar to what was recently done by Bemporad, Pagano, and Giordano (2018).

The values of the displacement of the CME during a polarisation sequence are summarised in Table 1.

Table 1 The displacement of the CME during a polarisation sequence. The first column shows the instrument taking the images, the second column displays the time at which a sequence of three polarised images was taken, the third column shows the speed of the CME in km s−1, the fourth column shows the distance the CME travelled during the time interval the sequence was taken, which is displayed in column 5. The last column shows the displacement of the CME, in pixels, during the time interval that a sequence of three polarised images was taken.
Full size table

4.3 CME Kinematics

We calculated the projected speeds of the CME’s leading edge observed by COR1 and COR2 at its central position angle (PA). The PA is measured from the North Pole, counter-clockwise. For LASCO-C2, where the CME was observed as a full halo, we take the values from the LASCO CME catalogue: cdaw.gsfc.nasa.gov/CME_list/. The speed was measured at the PA corresponding to the fastest-moving segment of the CME leading edge.

The values of the speeds as derived from different datasets are as follows:

  • LASCO: speed of 885 km s−1 (linear fit), at a PA of 158∘. From a second-order polynomial fit, the speed goes up to 1100 km s−1 at 5.5 R⊙.

  • COR1-A: speed from 200 km s−1 at around 2 R⊙ (16:20 UT) to 1100 km s−1 at around 3.9 R⊙ (16:55 UT), PA = 107∘.

  • COR1-B: speed from 500 km s−1 at around 2 R⊙ (16:25 UT) to 900 km s−1 at around 3.2 R⊙ (16:45 UT), PA = 254∘.

  • COR2-A: speed of 1000 km s−1, at PA = 107∘ and height of 14 R⊙ (18:39 UT) (decelerating from 1200 km s−1).

  • COR2-B: speed of 1250 km s−1, at PA = 254∘ and height of 14 R⊙ (18:24 UT) (slightly decelerating).

These values are shown also in Figure 8.

Figure 8
figure 8

Projected speeds versus height as measured in COR1-A at PA \(107^{\circ}\) (black triangles), COR1-B at PA 254∘ (red squares), COR2-A at 107∘ (green triangles), COR2-B at PA 254∘ (orange squares), and LASCO-C2 at 158∘ (blue crosses). The errors are derived by assuming an error in height of five pixels.

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It is observed that the CME accelerates quite a lot in the regions of the corona below 4 R⊙ and keeps approximately a constant speed above 4 R⊙.

For a COR1 pixel size of 7.5′′, i.e. 5245 km pixel−1 (COR1-A) and 5527 km pixel−1 (COR1-B), a CME moving at 1000 km s−1 will cross the single pixel in 5.24 seconds (COR1-A) and 5.53 seconds (COR1-B). This is larger than the nominal exposure time of one second. For a COR2 pixel size of 29.4′′, i.e. 20,549 km pixel−1 (COR2-A) and 21,662 km pixel−1 (COR2-B), a CME moving at 1000 km s−1 will cross the single pixel in 20.55 seconds (COR2-A) and 21.66 seconds (COR2-B), which is again larger than the nominal exposure time of six seconds. Hence, plasma displacements during the exposure times can be neglected.

4.4 CME 3D Reconstruction

We applied three different reconstruction techniques in order to derive the 3D position of the CME, namely: triangulation model (Inhester, 2006), polarisation-ratio (PR) model (see, e.g., Moran and Davila, 2004; Dere, Wang, and Howard, 2005; Vourlidas and Howard, 2006; Floyd and Lamy, 2019), and a graduated cylindrical shell (GCS) model (see Thernisien, 2011).

The polarisation-ratio technique was applied on a small region at the central part of the CME leading edge and on the core of the CME, as seen in COR1-A at 16:35 UT. Triangulation was applied on the same regions. GCS was applied on the CME observed by COR1 at 16:35 and by COR2 at 17:08 UT. The results of GCS are displayed in Figure 9.

Figure 9
figure 9

GCS reconstruction for the CME observed by STEREO/COR-A (left panels), LASCO-C2 (middle panels), and STEREO/COR-B (right panels) at around 16:35 UT (upper panels) and 17:08 (lower panels) on 12 July 2012.

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The results from different reconstruction techniques are summarised in Table 2 together with the location of STEREO-A, STEREO-B, and Earth on 12 July 2012 16:00 (also shown in Figure 10).

Figure 10
figure 10

Position of the three spacecraft (STEREO-A (red), STEREO-B (blue), and SOHO - Earth perspective (green) on 12 July 2012, 16:00). The red arrow indicates the direction of propagation of the CME as derived from GCS reconstruction. The view is from above, in the Heliocentric Earth Ecliptic (HEE) coordinate system, with the \(x\)-axis towards the Earth and its \(z\)-axis towards the Ecliptic North Pole (Thompson, 2006).

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Table 2 3D position of the CME as derived from different reconstruction techniques (in HEEQ coordinates; see Thompson, 2006). The last three entries show the positions of the spacecraft, also in HEEQ.
Full size table

It is observed that the three methods, GCS, triangulation, and the polarised-ratio method, yield the same values for the position of the scattering material at distances around 2 R⊙. At higher distances (around 6 R⊙) the CME is slightly deflecting towards the Equator.

Given the position of the spacecraft and the direction of propagation of the CME, this was observed at i) the central meridian as seen from the Earth; ii) \(30^{\circ}\) behind the East limb as observed by STEREO-A and iii) \(25^{\circ}\) behind the West limb as observed by STEREO-B.

This is also visible in the STEREO/Extreme Ultraviolet Imager (EUVI) 195-A and -B data where dimmings appear as being located behind the limb (see, e.g., cdaw.gsfc.nasa.gov/stereo/daily_movies/2012/07/12/index.html).

5 Discussion

5.1 Discussion on the Degree of Polarisation

The degree of polarisation of the K-corona, depends on the distance from the solar surface and on the scattering angle to the observer. Besides the position in space of the structure (streamer, coronal mass ejection, etc.), the effective line-of-sight integration length is also important when deducing the degree of polarisation (Koutchmy, 1972).

We have found that the degree of polarisation is around 65% but it can go as high as 80% in some CME regions from COR1-A. Floyd and Lamy (2019) showed that for an electron in the POS at 2 R⊙, the polarisation is about 0.8 and increases to nearly 1 at 3 R⊙. These values decrease with increasing angle from the POS.

Similar or lower values of the degree of polarisation were observed for coronal streamers. Lamy et al. (2020) by studying polarised LASCO-C2 images showed that the brightest streamers culminate at a peak degree of polarisation of around 0.4 at 2.2 R⊙, at the maximum of solar activity. Dürst (1982), by studying the solar eclipse of 16 February 1980, found that the degree of polarisation of the K-corona is strongly increasing with height in the inner corona, and it is almost constant for heights above 2 R⊙. Several streamers located in or near the POS have maximum polarisation near 80%. Koutchmy, Picat, and Dantel (1977), by studying the eclipse of 30 June 1973, found polarisations as high as 60% in regions associated with coronal streamers. In another solar-eclipse study, Vorobiev et al. (2020) found that the degree of polarisation of the F+K corona (including also the sky) peaks at around 47% in the equatorial features, at a distance of around 1.5 R⊙.

An unexpectedly large degree of polarisation (exceeding the maximum non-relativistic Thomson-scatter value) was observed by a number of authors (Koutchmy and Schatten, 1971; Pepin, 1970; Skomorovsky et al., 2012; Qu et al., 2013). This was attributed by Qu et al. (2013) to the contamination of the scattered white-light signal by coronal ion emission lines. As the majority of these studies were made during solar eclipses one extra source of contamination may come from the sky polarisation (see Koutchmy et al., 1999). Obviously this is not the case for space data.

5.2 Discussion on the Polarisation Angle

The anisotropy of the incident light causes the observed scattered radiation to exhibit a polarisation parallel to the visible solar limb (tangential to the solar limb).

In our case we observe deviations from the tangential greater than \(10^{\circ}\), in regions associated with the CME brightness excess. Inhester et al. (2021) analysed COR1 observations of the quiescent corona and found a systematic constant deviation of the polarisation angle by \(-1^{\circ}\), which exceeded the calculated variance of the polarisation angle. As the most simple and most probable explanation, the authors proposed not-precise-enough knowledge of the orientation of the polarisers and thus an incorrect demodulation matrix. The offset of \(2.5^{\circ}\) in mechanical mounting of the polarisers in COR1 would provide a systematic deviation of order of \(1^{\circ}\). Such an explanation is not applicable in our case, since the polarisation angle varies across the CME and takes larger values. We look for other possible explanation in the following sections.

5.3 Possible Interpretation of the Observed Deviation Angles: Relativistic Electrons?

It has been shown that deviation from the tangential may occur, and among possible physical reasons of such deviations we mention the Thomson scattering by the fast moving (relativistic) electrons in the solar corona and the scattering by the supra-thermal electrons (Molodensky, 1973; Kishonkov and Molodenskii, 1975). These kinds of deviations have been reported only seldom in the literature (Pepin, 1970; Badalyan, 1988; Skomorovsky et al., 2012; Qu et al., 2013) and some results deduced from photographic films/plates are of limited precision and significance (Park et al., 2001; Kim and Popov, 2015), especially when radial neutral filters were not used. Other studies explicitly found no deviation from the tangential polarisation within the measurement error (Koutchmy et al., 1993; Filippov, Molodensky, and Koutchmy, 1994; Kim et al., 1996; Kulijanishvili and Kapanadze, 2005; Vorobiev et al., 2020).

In the case of a CME-driven shock wave the semi-relativistic electrons can be expected at the front or the flanks of the CME inducing radio emission (Type-II and Type-III radio bursts) (see, e.g., Claßen and Aurass, 2002; Magdalenić et al., 2014). The energies of impulsive electron events near the Sun could be in the range of 1 – 40 keV (sometimes going down to 0.1 keV) (Reid and Ratcliffe, 2014). A small, e.g. order of 10%, population of high-energy electrons is sufficient to produce bursts. For a net rotation of the polarisation angle, the contribution of all electrons adds up incoherently. So if there are 10% beamed high-energy electrons their scatter with rotated polarisation adds to the unrotated Thomson background of the other 90%. This may strongly reduce the observed net rotation. If the electrons are not beamed but just isotropically heated, then each energetic electron in the tail of the distribution may rotate its scatter but if all contributions are added up this results not in a net observable rotation of the polarisation direction but just in a random broadening of the polarisation directions that probably is observed as a slightly reduced polarisation degree.

Small relativistic effects may be observed even for an electron energy of 0.03 keV with deviations relative to the non-relativistic limit of a few % for the degree of polarisation and two degrees away from the tangential direction for the polarisation angle (Inhester, 2015).

5.4 Possible Interpretation of the Observed Deviation Angles: Line Contamination?

The visible emission lines (H\(\alpha \), Helium D3-line, Fe xiv, Fe x, etc.) are part of the signal of a coronagraph image together with the K-corona, F-corona, and the stray light.

As described in Section 3, COR1 observes in a white-light waveband 22.5 nm wide centred at the H\(\alpha \) line at 656 nm and COR2 observes in visible light with a passband from 650 to 750 nm. This implies that the CME signal observed by COR1 may be “contaminated” with H\(\alpha \) emission.

To investigate the potential contribution from H\(\alpha \) emission, we applied the polarisation-ratio technique on COR1-A data at 16:35. If the H\(\alpha \) contribution is significant we should observe low polarisation in the region of our CME (see, e.g., Poland and Munro, 1976; Mierla et al., 2011). Instead, from Figure 3 it is clear that a high degree of polarisation of order of 0.7 is observed in the CME region. Note also that the H\(\alpha \) emission is supposed to be observed in the core of the CME, but the deviation from the tangential was most visible in the leading edge.

Let us compare our reasoning with other events. As was pointed out by Floyd and Lamy (2019), the Helium D3 line at 587.7 nm, falls right at the centre of the LASCO-C2 orange bandpass and it was seen in the core of many three-part CMEs imaged by C2. The authors concluded that the prominence material embedded in LASCO CMEs is expected to have a maximum polarisation of the order of 17% and a direction somewhat deviating from the tangential. The same conclusion was derived from another study by Heinzel et al. (2020). In quiescent prominences, the polarisation of the He i D3 emission is low, typically 2% and its direction deviates from the tangential by approximately \(\pm 15^{\circ}\) (Leroy, Ratier, and Bommier, 1977).

Dolei, Bemporad, and Spadaro (2014), by studying three blobs observed in COR1 images, demonstrated that the H\(\alpha \) contribution to the total white-light intensity observed at the location of the blobs is between 95 and 98%. They also showed that the H\(\alpha \) polarised component is of the order of a few percent of the H\(\alpha \) total emission from the blobs.

Mierla et al. (2011) demonstrated that the major part of a CME core emission (more than 85%) is H\(\alpha \) radiation, and only a small fraction is Thomson-scattered light. This results in the reduction of the CME polarisation signal. The CME was associated with a filament eruption, and it was observed by the COR1 instrument.

The same was deduced from the eclipse of 2019 with the CATE experiment (D. Elmore, private communication, 2021). They observed a prominence at the NNE limb with polarisation close to zero over the polarised coronal background.

For this study, EUV images at 304 and 195 Å taken by EUVI and the Solar Dynamics Observatory/Atmospheric Imaging Assembly (AIA) were inspected and no filament/prominence was found. Nevertheless, the CME had a three-part structure with a bright leading edge, followed by a dark cavity and a bright core, meaning it was a flux-rope-like ejecta. Such CMEs without any associated filaments are often reported in the literature, e.g. Howard et al. (2017) and Vourlidas et al. (2013). From the 3D reconstruction described above we saw that the two methods, triangulation and the polarisation-ratio, yield the same values for the position of the scattering material.

Thus, we can conclude that there is no H\(\alpha \) emission associated with our CME, i.e. the signal of the CME is due entirely to the Thomson-scattered light.

5.5 Other Effects That Can Modify the Polarisation Angle and the Degree of Polarisation

Other phenomena that can create low polarisation in coronagraph images are F-corona emission (e.g. Morgan and Habbal, 2007) and Thomson scattering from enhanced plasma density far away from the POS (e.g. Billings, 1966). The F-corona contribution is ruled out because it forms a diffuse background and does not vary rapidly in time. The subtraction of the image taken 45 minutes before our image at 16:35 should have removed any F-corona contribution.

As shown above, the CME appears at different locations in each polarised image resulting in an error in the measured polarisation. This effect may be taken care of by binning the images. Thernisien, Vourlidas, and Howard (2011) showed that substantial binning of the LASCO-C2 images would be required to suppress the effect, for instance by a factor of about 36 (from the original format of \(1024\times 1024\) pixels) for a CME travelling at 500 km s−1.

We performed a similar test with the COR1-A image at 16:35 UT. By binning the image to a size of \(64\times 64\) (i.e. 16 times less than the original image) we noted that the deviation angle from the tangential is around \(0 \pm 3^{\circ}\), while in the binned \(128\times 128\) images one can still see small deviations (around \(7^{\circ}\)), especially in the front part of the CME. In these regions, the original image can have a deviation as high as \(15^{\circ}\) or more (see Figure 3). The binned images of \(32\times 32\) were too noisy to conclude anything.

The degree of polarisation for the binned images behaves in a similar way to the deviation angle images, i.e. the high values (above 0.65) decrease or disappear with the size of the image.

5.6 Discussion on Different Effects Influencing the Polarisation Analysis

In a recent study by Floyd and Lamy (2019), the main limitations the authors encountered when deriving the polarisation parameters of 15 CMEs observed by LASCO-C2 were: i) the contamination of the LASCO-C2 passband filter with the He i D3 line at 587.7 nm. ii) the duration of ten minutes of a LASCO-C2 polarisation sequence resulting in the blurring of the polarisation in the case of the fast CMEs. The front of fast CMEs appeared at different locations in each polarised image resulting in an error in the measured polarisation. iii) using a pre-event image resulted in artifacts affecting the pB-images of the LASCO-C2 CMEs because traces of underlying coronal structures are interfering with the CME signal.

In another study, Lamy et al. (2020), using the newest calibrated LASCO-C2 polarised images, analysed the polarised K-corona over the two complete Solar Cycles 23 and 24, and they compare it with a variety of eclipse data taken during the two solar cycles. The agreement with the eclipse data is in general good except for slight discrepancies affecting the innermost part of the C2 field of view.

To do this analysis, the authors needed to introduce further corrections to the polarised data: i) the correction for the global transmission of the \(0^{\circ}\) polariser; ii) global correction function applied to polarised images; iii) exposure-time equalisation of the three polarised images. They showed that errors on the relative global transmission between polarisers and on the individual exposure times of the triplet forming a polarisation sequence have similar effects, which were corrected by an optimisation procedure based on imposing a tangential direction to the solar limb of polarisation and minimal dispersion around this direction. They concluded that the capability of LASCO-C2 to perform accurate polarimetric measurements is affected by several adverse conditions, prominently the very basic polarisation analyser system (three Polaroid foils), the optical design with two folding mirrors, and the fluctuations in exposure times.

A similar study was performed by Lamy et al. (2021), this time with the newest calibrated LASCO-C3 polarised data. They only used two polarisers in this case, because of the degradation of the \(0^{\circ}\) polariser.

In the case of internally occulted coronagraphs (like COR1) multiple reflections in the instrument from the direct photospheric light are difficult to avoid. When using an externally occulted instrument simulating a total eclipse (LASCO-C2, COR2, etc.), the photospheric light does not reach the instrument, but several new problems appear, as described by Lamy et al. (2020). Some significant improvement was made in the case of the COR1 and COR2 coronagraphs by using a rotating single polariser instead of comparing images taken using different polarisers, as was done with LASCO-C2. However, in the case of COR1 and COR2, although polarisation images are taken through the same optics, the rotation of the polariser introduces some wobble between the images due to the fact that the front and rear surfaces of the polariser are not exactly parallel. This effect is smaller than a pixel, but the slight misregistration between the three images can produce some spurious polarisation signals at sharp transitions in brightness (see, e.g., the analysis of the Comet Lovejoy by Thompson (2015)). This is definitely an issue when analysing comet data, but is not generally a problem for the solar corona, where the variations in brightness tend to be more gradual. In our case the effect is negligible compared to other sources of uncertainty.

To summarise, the causes affecting the polarimetric measurements may be related to i) the instrument design: e.g. the optical design with two folding mirrors, as in the case of LASCO-C2; the basic polarisation analyser system (three Polaroid foils as in LASCO-C2, and one polariser that rotates as in COR). ii) the instrumental operation mode: e.g. the fluctuations in exposure times; the time difference in taking a polarisation sequence (ten minutes for LASCO-C2, a few seconds for COR1 and COR2). iii) the correct extraction of the K-corona signal from other signals (different background subtractions are involved). iv) contamination of the bandpass with emission lines (He i D3 for LASCO-C2 and H\(\alpha \) for COR1).

From the effects specified above, the one affecting most of our measurements is related to the time span in taking a polarisation sequence. The other effects are minimised by our data pre-processing (with the secchi_prep routine), by background subtraction (to keep only the CME K-corona signal – see the discussion in Appendix B) and by the type of the observation (CME without any prominence associated, i.e. no emission-line contamination).

5.7 Improving the Polarisation Analysis

The question arises of how we can improve polarisation measurements of CMEs. Of course, one aspect is low scattered light, but this is not the key aspect since scattered light is subtracted by the pre-event image. Key areas of improvement are the accuracy, hence the technique (three polaroids being the worst) and the speed of the measurements (to “freeze” the motion of the CMEs).

The Solar Orbiter visible light and ultraviolet coronal imager (Metis: Antonucci et al., 2020) has the technical capability of obtaining polarisation measurements of quality superior to those of LASCO-C2 and COR1 thanks to its electro-optical device, which does not require any mechanical motion. The resulting much faster mode of operation will contribute to the reduction of the time delays between the individual images required to obtain the deviation-angle image.

One step forward to improve the derivation of the polarisation parameters was proposed by Vorobiev et al. (2020). The authors used a new polarimetric imaging camera and method: a set of four linear micro-polarisers at 0, 45, 90, and \(135^{\circ}\) put over each “sub-matrix” made of “super pixels” of 2×2 assembled on the CCD chip. They could detect polarisation signals as small as 0.3% and found no strong deviations in the angle of linear polarisation from the tangential direction.

Koutchmy et al. (2019a,b) also performed such analyses by using a rotating linear Polaroid placed in front of the entrance lens of F300 mm at F/5.6 and a Canon D60 color CMOS camera of 24 Mpx. Twelve images were recorded at positions consecutively shifted by 15∘, with a precision better than one degree and a cadence of four seconds using the same exposure time of 1/8 second. No deviation from the tangential direction was found (at the 1∘ level of precision) after subtracting the sky background, as measured over the image of the Moon.

Among the next space missions with new polarisation technical developments is BITSE (Balloon-borne Investigation of Temperature and Speed of Electrons) in the corona (see Gong, Gopalswamy, and Newmark, 2019). BITSE’s detector will be able to directly collect polarised light at four different wavelengths: 385.0, 398.7, 410.0, and 423.3 nm. It will measure the coronal average electron temperature using the Grotrian method.

PUNCH (Polarimeter to UNify the Corona and Heliosphere) will also have polarisation capability. It comprises four matched cameras all operating as a “virtual instrument” to image Thomson-scattered light, from the vantage of four separate spacecraft in Sun-synchronous LEO. PUNCH is the first coronal and solar-wind imager designed specifically to produce three-dimensional images from a single vantage point (DeForest et al., 2019).

The future externally occulted coronagraph ASPIICS (Association of Spacecraft for Polarimetric and Imaging Investigation of the Corona of the Sun), onboard PROBA-3 (Project for On-Board Autonomy), will be the first space-borne instrument to image the polarised corona at distances as low as 1.1 R⊙, with low stray light (Lamy et al., 2010; Renotte et al., 2015; Galano et al., 2018). The coronagraph will be equipped with a filter wheel with six positions to choose between three spectral filters and three linear polarisers. ASPIICS will not suffer from radial vignetting due to its remote external occulter and it will preserve its image quality throughout its FOV. Nevertheless, due to the usage of a filter wheel there will be limitations because of the non-simultaneity in taking the set of three polarised images.

6 Conclusions

In this work we have used SECCHI/COR1 and COR2 polarised white-light data to study a large and fast CME on 12 July 2012. We used the degree of polarisation to calculate the position of the CME in 3D. We compared it with the position derived from triangulation and forward-modelling techniques, and we found very good agreement among them. From the study of the CME kinematics we derived projected speeds up to around 1000 km s−1. The CME studied here showed deviation of the polarisation angle from the tangential in the range of \(10-15^{\circ}\) (or more). Our analysis showed that this is mostly due to the fact that the sequence of three polarised images from which the polarised parameters are derived is not taken simultaneously, but at a difference of few seconds in time. In this interval of time, the CME moves by at least two pixels in the FOV of the instruments, and this displacement results in uncertainties in the polarisation parameters (degree of polarisation, polarisation angle, etc.).

We conclude that the observational effect of the propagation of the CME front through the image combined with non-simultaneous polarisation observations is the most likely reason for the deviation of the polarisation angle from the tangential direction. To our knowledge this effect has not been considered before.

This study is important also for analysing future data from instruments with polarisation capabilities.