# Derived electron densities from linear polarization observations of the
visible-light corona during the 14 December 2020 total solar eclipse

Liam T. Edwards Department of Physics
Aberystwyth University
Ceredigion, Cymru, SY23 3BZ  Kaine A. Bunting  Department of Physics
Aberystwyth University
Ceredigion, Cymru, SY23 3BZ  Brad Ramsey Department of Physics
Aberystwyth University
Ceredigion, Cymru, SY23 3BZ  Matthew Gunn Department of Physics
Aberystwyth University
Ceredigion, Cymru, SY23 3BZ  Tomos Fearn Department of Computer Science
Aberystwyth University
Ceredigion, Cymru, SY23 3DB  Thomas Knight  Department of Physics
Aberystwyth University
Ceredigion, Cymru, SY23 3BZ  Gabriel Domingo Muro Department of Physics
Aberystwyth University
Ceredigion, Cymru, SY23 3BZ  Space Radiation Lab,
California Institute of Technology,
Pasadena, CA, USA 91125  Huw Morgan Department of Physics
Aberystwyth University
Ceredigion, Cymru, SY23 3BZ

###### Abstract

A new instrument was designed to take visible-light (VL) polarized brightness
($pB$) observations of the solar corona during the 14 December 2020 total
solar eclipse. The instrument, called the Coronal Imaging Polarizer (CIP),
consisted of a 16 MP CMOS detector, a linear polarizer housed within a
piezoelectric rotation mount, and an f-5.6, 200 mm DSLR lens. Observations
were successfully obtained, despite poor weather conditions, for five
different exposure times (0.001 s, 0.01 s, 0.1 s, 1 s, and 3 s) at six
different orientation angles of the linear polarizer (0∘, 30∘, 60∘, 90∘, 120∘,
and 150∘). The images were manually aligned using the drift of background
stars in the sky and images of different exposure times were combined using a
simple signal-to-noise ratio cut. The polarization and brightness of the local
sky is also estimated and the observations were subsequently corrected. The
$pB$ of the K-corona was determined using least squares fitting and
radiometric calibration was done relative to the Mauna Loa Solar Observatory
(MLSO) K-Cor $pB$ observations from the day of the eclipse. The $pB$ data was
then inverted to acquire the coronal electron density, $n_{e}$, for an
equatorial streamer and a polar coronal hole, which agreed very well with
previous studies. The effect of changing the number of polarizer angles used
to compute the $pB$ is also discussed and it is found that the results vary by
up to $\sim$ 13% when using all six polarizer angles versus only a select
three angles.

Eclipse Observations; Polarization, Optical; Instrumentation and Data
Management; Spectrum, Visible

##  1 Introduction

A total solar eclipses (TSE) provides a unique opportunity to observe the
visible-light (VL) corona down to the solar limb for a few minutes during
totality, and allow continuous observations from the limb out to several solar
radii. Not only does the Moon block out the solar disk, but it also lowers the
local sky brightness along the eclipse path which makes it perfect for
observing the significantly fainter corona (Lang, 2010). This lower coronal
region ($<$ 2 R⊙) is the origin of the solar wind and is therefore crucial to
observe in order to better understand how it is formed and accelerates to
supersonic speeds (Habbal, 2020; Strong et al., 2017; McComas et al., 2007).
TSEs have been used to obtain valuable information about a variety of solar
phenomena, including coronal streamers (Pasachoff & Rušin, 2022), coronal mass
ejections (CMEs) (Boe et al. (2021c); Filippov et al. (2020); Koutchmy et al.
(2004)), coronal holes (Pasachoff & Rušin, 2022), plasma flows (Sheeley &
Wang, 2014; De Pontieu et al., 2009), prominences (Jejčič et al., 2014), and
coronal jets (Hanaoka et al., 2018). Physical properties of the coronal plasma
have also been studied extensively during TSEs, typically, temperature,
density, and velocity (Del Zanna et al., 2023; Muro et al., 2023; Bemporad,
2020; Reginald et al., 2014; Habbal et al., 2011, 2010; Reginald et al., 2009;
Habbal et al., 2007).

A coronagraph is required in order to observe the VL corona outside of a total
solar eclipse. First designed by Bernard Lyot in the 1930’s (Lyot & Marshall,
1933), it consists of a telescope with an opaque disk that is positioned to
block out the bright disk of the Sun. This is crucial because the photosphere
is $\sim$ 106 times brighter than the corona itself. In fact, the Earth’s sky
is also much brighter than the corona - of the order $\sim$ 105 times brighter
at 20 R⊙ \- therefore, any ground-based coronagraph, such as the COronal Solar
Magnetism Observatory’s (COSMO) K-Coronagraph (K-Cor; Hou et al. (2013)), is
affected by the Earth’s own atmosphere. To overcome this issue, several space-
based coronagraphs have been launched in recent decades, for example, the
Large Angle Spectrometric Coronagraph (LASCO; Brueckner et al. (1995)) onboard
the Solar and Heliospheric Observatory (SOHO; Domingo et al. (1995)) and the
Sun-Earth Connection Coronal and Heliospheric Investigation’s (SECCHI; Howard
et al. (2008)) COR1/2 instruments onboard the Solar Terrestrial Relations
Observatory (STEREO; Kaiser et al. (2008)). However, despite the unprecedented
access to the corona they have given the field of solar physics, there is
still a fundamental issue for any type of coronagraph to overcome which is the
stray light resulting from the diffraction of incoming light at the occulter
edge. In order to mitigate this, most occulters will block not only the solar
disk but also a portion of the very lower solar corona - out to around 1.5 R⊙
for internally occulted coronagraphs (Verroi et al., 2008). One way to
circumvent this issue is to increase the distance between the detector and the
occulter, which is possible to achieve in space by the use of formation flying
cube satellites (e.g., ASPIICS; Lamy et al. (2010)), however, these type of
instruments are still in their infancy. As a result, observations of the inner
corona during a TSE are unmatched in the VL regime.

The VL corona is composed of light from several different sources, primarily
from the scattering of photospheric light by free electrons in the corona and
dust in the interplanetary plane, termed the K- and F-corona, respectively. In
the case of the K-corona, the scattering process - known as Thomson scattering
- produces a strongly tangentially polarized component to the total VL
brightness (for an in-depth overview of this mechanism see Inhester (2015)),
whereas the F-corona is considered to be unpolarized below $\sim$ 3 R⊙ (Morgan
& Habbal, 2007). As a result, observing the VL corona below this height with a
linear polarizer during a TSE can be considered to be a measurement of the
K-coronal brightness only. There are also other types of brightness
contributions to the total coronal brightness, namely the E-corona (emission)
which consists of spectral line emission from highly ionized atoms, but these
are considered to be negligible for the purposes of this work. The K-coronal
brightness component, $B_{K}$, represents the structure and amount of coronal
plasma irrespective of its temperature, unlike EUV or x-ray observations. The
K-coronal component of the polarized brightness (pBk) gives the electron
density ($n_{e}$) which is calculated using an inversion method first
developed by van de Hulst (1950) and later improved upon by Hayes et al.
(2001) and Quémerais & Lamy (2002). Equation 1 describes the relationship
between the polarized brightness of the K-corona and the electron density
which is used to acquire the electron density from the TSE images:

$pB_{k}\propto\int_{LOS}n_{e}(r)\cdot G(s,\rho)\,ds$ (1)

where $G$ is a geometrical weighting function, $\rho$ is the distance between
the Sun and the intercept between the line-of-sight (LOS) and the plane-of-sky
(POS), and $s$ is the distance between the intercept point on the POS and an
arbitrary point in the corona along the LOS (see Figure 1 in Quémerais & Lamy
(2002) for more detail). This is a well-established technique and several
studies have used this inversion of $pB$ to obtain coronal electron densities
(e.g. Liang et al. (2022); Bemporad (2020); Skomorovsky et al. (2012); Hayes
et al. (2001); Raju & Abhyankar (1986); Saito et al. (1977)). Several studies
have also been conducted attempting to separate the K- and F-components of the
solar corona (e.g. Boe et al. (2021a); Fainshtein (2009); Morgan & Habbal
(2007); Dürst (1982); Calbert & Beard (1972)). Most of the previous studies
involving TSE observations require some level of image processing as a result
of several factors - primarily the sharp decrease in brightness with radial
height from the Sun. For example, since the Moon and the Sun move relative to
each other with respect to the background stars, all TSE images need to be
coaligned and there are a number of different methods which can be used to do
this. One of the most common of these methods over the past decade or so has
been to use a modified phase correlation technique by Druckmüller (2009),
whereas others have used more manual methods such as using the drift of
background stars as they move relative to the TSE (Bemporad, 2020). There are
also many different image processing techniques that have been developed to
better reveal various structures and phenomena (Patel et al. (2022); Qiang et
al. (2020); Morgan & Druckmüller (2014); Druckmüller (2013); Byrne et al.
(2012); Druckmüllerová et al. (2011)). The outline of this paper is as
follows: section 1.1 describes the 2020 TSE in more detail, section 2
summarises the design of the instrument (2.1), and the calibrating,
processing, coaligning, and inversion to derive the coronal electron densities
(2.2.3 \- 2.2.7). Section 3 presents the results of the study, with sections 4
and 5 disseminating and concluding the results of this work, respectively.

###  1.1 14 December 2020 Total Solar Eclipse

The TSE on 14 December 2020 was observed by a team from Aberystwyth University
at a site in Neuquén province, Argentina. The observation site was located at
39∘42’ 40.4” S, 70∘23’ 57.6” W, and an altitude of $\approx$ 1,082 m (3,550
ft). Totality lasted $\approx$ 129 seconds with the time of maximum eclipse at
13:07:58 local time (16:07:58 UTC) and the apparent altitude of the Sun above
the horizon was $\approx$ 75∘. The weather conditions at the time of
observation were not optimal with intermittent cloud cover and very strong
gusts of up to 70 km/h which resulted in a lot of airborne dust. Furthermore,
a small, wispy cloud passed across the Sun’s disk for $\approx$ 12 seconds.
This had an adverse effect on the quality of the data but, despite the
conditions, the data captured during totality was still usable. Figure 1 shows
the location of the observation site (white marker) along with the cloud cover
at approximately the time of the eclipse. During this eclipse, a CME had
erupted from the eastern limb of the Sun $\approx$ 110 minutes before
totality, providing a truly unique opportunity to study CME dynamics right
down to the solar limb. The LASCO CME catalog (Gopalswamy et al., 2009) states
that the CME first appeared in the C2 field-of-view (FOV) at 15:12:10 UT with
an estimated linear speed of 437 km/s, mass of 3 $\times$ 1012 kg, and a
central position angle of 121∘. The CME is discussed in detail by Boe et al.
(2021c).

Figure 1: Satellite image of cloud cover above the observation site in Neuquén
province, denoted by the white marker, taken at approximately 13:30 local time
(Credit: Zoom Earth)

##  2 Instrument Design and Data Processing

###  2.1 The Coronal Imaging Polarizer (CIP)

The instrument used to observe the corona during the total solar eclipse was a
VL linear polarization imager designed and built at Aberystwyth University.
The Coronal Imaging Polarizer (CIP) was designed to be relatively cheap and
simple to build, easy to assemble, and lightweight in order to be able to
relocate quickly on the day of an eclipse if necessary. It consisted of an
objective lens, a VL band-pass
filter111https://www.thorlabs.com/thorproduct.cfm?partnumber=FBH520-10, a
linear
polarizer222https://www.thorlabs.com/thorproduct.cfm?partnumber=LPVISC100
housed in a rotating
mount333https://www.thorlabs.com/newgrouppage9.cfm?objectgroup_id=12829, and a
CMOS sensor444https://www.atik-cameras.com/product/atik-horizon-ii/. The
objective lens, an f-5.6, 200 mm focal length DSLR lens, can be adjusted to
give different fields of view if required. The VL band-pass filter has a
center wavelength of 520 nm (denoted by the vertical dashed black line in
Figure 2) where transmission is $>$ 90%, and a bandwidth (FWHM) of 10 nm.
Since the peak of the Sun’s emission is around 500 nm, the filter is well-
placed to collect the maximum amount of light possible.

Figure 2: Extinction ratio and transmissions for both the band-pass filter and
linear polarizer measured by their respective manufacturers. The dotted
horizontal lines correspond to the polarizer’s extinction ratio (blue) and
transmission (red) at the band-pass filter’s center wavelength (Data from
Thorlabs)

Traditionally, when taking VL polarization observations of the corona, a
polarizer is either manually rotated through a set number of polarization
angles (typically 3 - 5) or several different instruments are set up, each
designed to capture the light of a single polarizer orientation angle. In
contrast, CIP used a piezoelectrically-driven rotation mount from Thorlabs to
automatically rotate the polarizer through six different polarization angles
(0∘, 30∘, 60∘, 90∘, 120∘, and 150∘). The motorized rotation mount allows for
360∘ rotation with a maximum rotation speed of 430∘/second and has an accuracy
of $\pm$ 0.4∘, resulting in high precision and fast rotation. The energy
requirement of the motor is very low, needing only a maximum of 5.5 V DC input
with a typical current consumption of 800 and 50 mA during movement and
standby, respectively.

The camera used for CIP was the Horizon II - a 16 mega-pixel CMOS camera
developed by Atik. It uses the Panasonic MN34230 4/3” CMOS sensor with a 4644
x 3506 resolution and has a very low readout noise ($\sim$ 1 e-). It is
powered by a 12 V 2 A DC input and has a minimum exposure time of 18 $\mu$s
and unlimited maximum exposure. It is also cooled via an internal fan and can
maintain a $\Delta$T of -40∘C, meaning the camera can still maintain low
thermal noise even at high ambient background temperatures. It is a black-and-
white camera since using an RGB camera requires extra steps when processing
the data (e.g. demosaicking), and the correction and alignment must be done
individually for each RGB channel - see Bemporad (2020), for example. In order
for the instrument to run in the most efficient way possible, the data
collection process was fully automated. The rotation mount rotates the
polarizer to a specific angle, then the camera collects a sequence of images
of varying exposure times from 0.001 - 3 seconds, then the polarizer is
rotated to the next angle, and the camera would run through the same sequence
as before, and so on for all six polarization angles. The time taken for one
complete cycle of data collection is around 30 seconds which resulted in two
full data sets and a third partial data set.

Figure 3: The Coronal Imaging Polarizer (CIP) - A: f-5.6, 200 mm objective
lens. B: housing containing the rotation mount, polarizer, and VL filter. C:
Atik Horizon II 16 MP CMOS camera D: housing for the interface board of the
rotation mount

###  2.2 Data Processing

####  2.2.1 Linearity

The response of the imaging sensor to incoming light intensity is measured by
using an integrating sphere, where the level of illumination in the sphere is
measured by a photodiode (in Amps). The integrating sphere is not calibrated
to give the light levels in SI units but it is linear so if the photodiode
current doubles the light level in the sphere is doubled. The illumination of
the sphere was gradually increased by 1 $\mu$A until it reached the saturation
point. Five measurements were taken and an average intensity was calculated
for each photodiode reading. Figure 4 presents the results of this linearity
test and it clearly shows that the sensor used in CIP has a linear response.
The linearity begins to break down close to the saturation limit ($\approx$
4096 counts) which is to be expected and is accounted for when the images
taken at different exposure times are combined for each polarization angle
(Section 2.2.4). The linearity of the sensor was quantified using equation
(2):

$Linearity\,(\%)=\frac{(MPD+MND)}{MI}\times 100$ (2)

where MPD, MND, and MI are the maximum positive deviation, maximum negative
deviation, and maximum intensity, respectively. The maximum positive and
negative deviations are found from the line of best fit. This calculation was
performed twice - once including the last data point where the linearity
begins to break down, and a second time without including the aforementioned
data point. These calculations result in linearities of 2.99% and 0.96%,
respectively. Both of these values are excellent linearities since most CMOS
sensors have linearities on the order of several percent (Wang & Theuwissen,
2017).

Figure 4: Average pixel intensity as a function of the integrating sphere’s
photodiode reading (black dots) and the calculated line of best fit (dashed
line) giving an R2 value of 0.99956

####  2.2.2 Flat field and dark frame correction

Five flat field images were taken at an exposure time of 0.01s for each
polarization angle using an integrating sphere. A master flat field image was
then produced for each polarization angle using the open-source plugin
AstroImageJ - an example of one of these is seen in Figure 5 along with an
intensity profile taken at the midpoint of the image.

Figure 5: Master flat field image for a polarization angle of 0∘ (top) and a
normalized intensity profile taken at the midpoint of the image shown by the
red horizontal line (bottom)

Two different types of dark field images were attempted: firstly, setting the
exposure time of the camera to zero and taking several pictures to get an
average; and secondly, blocking the front of the instrument with the lens cap
and running the full eclipse sequence five times. Unfortunately, the Atik
Horizon II has a minimum exposure time of 18 $\mu$s so the first method was
not able to be done. For the second method, the instrument was taken to a dark
room with the lens cap taped over the lens. Several dark frames were taken in
this way for each polarization angle and exposure time used in the eclipse
sequence. It is expected that the mean values of the intensity (along with
their standard deviations) should be similar for each polarization angle and
exposure time and this is indeed what was seen. Figure 6 shows the dark noise,
$\sigma_{dark}$, calculated by taking the average standard deviation of all
dark frames taken at each exposure time and polarization angle. It is clear
that pixels in the detector follow the same pattern and behave uniformly
across all polarization angles.

Figure 6: Average dark noise for all polarization angles as a function of the
exposure time

Once the master flat field and dark frame images were produced, the raw
eclipse images were calibrated using equation (3):

$C_{\theta,t}=\frac{(R_{\theta,t}-D_{t})*m}{(F_{\theta}-D_{t})}$ (3)

where $C_{\theta,t}$ is the reduced image, $R_{\theta,t}$ is the raw image
taken at a specific polarization angle ($\theta$) and exposure time ($t$),
$D_{t}$ is the dark frame for that particular exposure, $F_{\theta}$ is the
flat-field for that particular polarization angle, and $m$ is the image-
averaged value of $(F_{\theta}-D_{t})$. At this stage, the scale of the image
was calculated. The radius of the Moon was found to be 255.5 $\pm$ 0.5 pixels
by fitting a circle to points plotted along the lunar limb using ImageJ. One
of the lowest exposure images was used to obtain this radius in order to
reduce the brightness from the lower corona to obtain the most accurate value
possible. The Moon’s apparent radius at the time of the eclipse was found to
be 1,000.145” using the Stellarium software (Zotti et al., 2021), therefore,
the full-resolution images provided a spatial resolution of 3.92 $\pm$ 0.01
arcsec/pixel.

####  2.2.3 Image coalignment

After the initial calibration of the raw images with the dark and flat frames,
they were then coaligned before the images with different exposure times could
be combined. This step is crucial because, although a motorized tracking mount
was used to track the solar center, there are still some factors that need to
be accounted for and corrected. The two main factors are the high winds at
ground level throughout the eclipse and the fact that the Moon is moving with
respect to the Sun. The effect of the wind on the tracking mount can be seen
in Figures 7 and 8. As discussed in section 1, there are several different
methods of coaligning eclipse images such as the use of phase correlation
(e.g. Druckmüller (2009)) or by using the positions of stars visible in the
exposures (e.g. Bemporad (2020)). In this study, three stars were visible in
the images taken at longer exposure times (see Figure 7) and were identified
using Stellarium to be HD 157056 (star 1), HD 157792 (star 2), and HD 158643
(star 3), respectively. The images were initially coaligned using the
brightest of these stars (HD 157056), which is star 1 in Figure 7, since its
drift in both the x- and y-direction was fairly consistent (Figures 7 and 8).
For the shorter exposure images, where the stars were not visible, it was
assumed that the star did not move much relative to its position in the 1 and
3 s exposures. Shifts in both the x- and y-pixels were calculated for each
image, based on their respective exposure times, using the equations found in
Figure 8 and all of the images were co-aligned accordingly. The images are
then re-binned through 8 $\times$ 8 pixel averaging to improve the signal
quality, which reduced the resolution of the images to 31.25 $\pm$ 0.01
arcsec/pixel.

Figure 7: Left: Locations of the three visible stars in a 3 s exposure time
image. Right: Locations of the brightest pixel in star 1 for all 1 and 3 s
exposures Figure 8: Pixel drifts in both the x- and y-coordinates for star 1.
The data points represent the pixel coordinate of the brightest pixel during
the 1 and 3 s exposures. The darker and lighter shaded regions show one and
two standard deviations, respectively

####  2.2.4 Image combination

The next step was to combine all the images of different exposure times for
each individual polarization angle. Firstly, all negative pixel values were
set to zero and the images were normalized by their respective exposure times
(DN/s) using equation 4:

$I_{p}^{\prime}=\frac{I_{p}}{\Delta t_{e}}$ (4)

where $I_{p}^{\prime}$ is the pixel intensity ($I_{p}$) for each polarization
angle $p$, normalised by exposure time ($\Delta t_{e}$). Since the eclipse
images have a very high dynamic range between the brightest and darkest
pixels, they need to be combined together in a way that takes the pixel value
itself into account. In the shorter exposure images, the inner coronal signal
is strong but the outer coronal signal is too weak whereas, in the longer
exposure images, the outer coronal signal is strong but the inner corona is
over-exposed. Thus, the images need to be combined in such a way that the
inner and outer corona are adequately exposed in the final image. This is done
by using a signal-to-noise ratio (SNR) cut based on the square root of the
intensity, thus providing a lower boundary wherein any pixels with a value
less than this threshold are excluded. The SNR cut used in this work was
$I>\sqrt{I}$ where $I$ is the intensity of the pixel. An upper threshold of
3,900 counts is also used since the linearity of the CMOS sensor breaks down
around this value (see Figure 4). This process was done for each image taken
at a given polarization angle which means that each composite image consists
of two sets of different exposure times.

####  2.2.5 Sky Brightness Removal

During a TSE, the local sky brightness is significantly lowered to $\approx
10^{-9}-10^{-10}$ B⊙ which allows unmatched VL observations of the corona out
to greater heliocentric heights than without a TSE. However, the sky
brightness during a TSE is not negligible and must be removed from the data.
As Figure 9 shows, a box was defined in each corner of each combined TSE
image. There is a considerable amount of noise present in the image as a
result of the poor weather conditions, despite the rebinning to improve the
signal-to-noise ratio. There is also an optical artifact that takes the form
of two parallel lines spanning the width of the image, the cause of which is
unknown but most probably instrumental. The mean intensity in each box was
then calculated and the process was repeated for each polarizer angle. The
result of this can be seen in Figure 10 which seems to show that the intensity
at the right-hand side of the image (corresponding to solar north) is greater
than that of the left-hand side (corresponding to solar south). A map is then
created to estimate the sky brightness at each point in the image by
interpolating the mean counts between each of the four boxes for each
polarizer angle separately. Two examples of these interpolated sky brightness
maps are shown in Figure 11 for 0∘ and 90∘. The polarization of the background
sky lies at around 10% of the overall polarization of the image and this
component is subsequently removed as a result of this sky brightness removal
step. Finally, the F-coronal component has a noticeable impact on $pB$ data
from heliocentric heights of $\sim$ 2.5 - 3.0 R⊙ (Boe et al., 2021b) but since
this study limited observations to a heliocentric height of 1.5 R⊙, the
F-coronal component can be neglected at such low heights.

Figure 9: Location of each box used to determine the sky brightness overlaid
on the combined 0∘ image. The image has not been rotated so that solar north
aligns with the top of the image. L, T, B, and R correspond to the left, top,
bottom, and right sides of the image, respectively Figure 10: Mean intensity
for each box in Figure 9 at each polarizer angle. The fit to equation 5 is
shown as a solid line for each of the four boxes where the results from the
top-left, top-right, bottom-left, and bottom-right boxes are shown in black,
red, green, and blue, respectively. The mean polarization of each box is also
expressed as a percentage of the total polarization

Figure 11: Examples of interpolated sky brightness maps subtracted from the
respective TSE images for polarizer orientation angles of 0∘(left) and
90∘(right)

####  2.2.6 Determining the Polarized Brightness

A standard inversion method used to acquire coronal electron densities was
initially developed by van de Hulst (1950) and further developed by Newkirk
(1967), Saito et al. (1977), Hayes et al. (2001), and Quémerais & Lamy (2002),
among others. The original model assumes both spherical symmetry and that the
$pB$ is produced purely by the Thomson scattering of photospheric light from
free coronal electrons and it is proportional to the integrated LOS density of
the electrons, as shown in Equation (1). Most similar studies use three or
four different polarization angles to determine the $pB$ \- typically -60∘,
0∘, 60∘ (Hanaoka et al., 2021), and 0∘, 45∘, 90∘, 135∘ (Vorobiev et al.,
2020). However, CIP captured images taken at different exposure times for six
different polarization angles (0∘, 30∘, 60∘, 90∘, 120∘, and 150∘). As a
result, in order to invert the calibrated intensities to find the $pB$ at each
pixel, an approach involving least squares fitting was used. For a polarizer
angle, $\theta_{i}$, where $i=0,1,2,...,n-1$ (with $n=6$ for this study), the
measured intensity, $I_{i}$, is described by Equation 5.

$I_{i}=I_{0}+a\cos{2\theta_{i}}+b\color[rgb]{0,0,0}\sin\color[rgb]{0,0,0}{2\theta_{i}}.$
(5)

where the coefficients to be fitted are the unpolarized background intensity,
$I_{0}$, and the polarized components, $a$ and $b$. The polarized brightness
is then given by

$pB=\sqrt{a^{2}+b^{2}}$ (6)

In order to find a solution for $I_{0}$ and $pB$ at each pixel, the squared
sum is then minimized as follows

$\sum_{i=0}^{n-1}[I_{i}-(I_{0}+a\cos{2\theta_{i}}+b\color[rgb]{0,0,0}\sin\color[rgb]{0,0,0}{2\theta_{i}})]^{2}$
(7)

This least squares fitting method was tested against the standard Mueller
matrix inversion method for three polarization angles (0∘, 60∘, and 120∘) and
it agreed within a few percent. As a result, CIP can theoretically take images
of the corona for any number of polarizer angles in the future.

####  2.2.7 Relative Radiometric Calibration

A relative radiometric calibration step converts the polarized brightness
measured by CIP (recorded as DN/s) to units relative to the mean solar
brightness (MSB). The instrument used in this step was the Mauna Loa Solar
Observatory’s (MLSO) COSMO K-Coronagraph (DOI: 10.5065/D69G5JV8) which
provides $pB$ data with a field-of-view from 1.05 to $\sim$ 3 R⊙ and a spatial
resolution of 11.3”. The 10-minute averaged K-Cor data was used for this
calibration step with the first observation occurring at 17:56:41 UTC and the
last at 18:10:20 UTC. Figure 12 shows the 10-minute averaged $pB$ observations
taken by MLSO’s K-Cor instrument on the day of the TSE, processed using the
Multi-scale Gaussian Normalization technique (Morgan & Druckmüller, 2014) in
order to enhance the fine-scale coronal structure, and the concentric rings
represent heliocentric heights of 1.0, 1.5, 2.0, and 2.5 R⊙, respectively. It
is clear to see that consistent $pB$ data is restricted to $\sim$ 1.7 R⊙with
the data extending further out in the equatorial regions. As a result, the
relative radiometric calibration in this study was limited to a heliocentric
height of 1.5 R⊙.

Figure 12: 10-minute averaged $pB$ observations taken by MLSO’s K-Cor
instrument on the day of the eclipse, processed using the Multi-scale Gaussian
Normalization technique, with the concentric circles corresponding to
heliocentric heights of 1.0, 1.5, 2.0, and 2.5 R⊙, respectively

The images from both CIP and K-Cor are then coaligned and an intensity profile
is taken for a thin slice of the corona at a specified height for both images.
Some examples of these intensity profiles can be seen in Figure 13 for a range
of heliocentric heights. A calibration factor is temporarily applied to the
CIP data in order to visualize both latitudinal profiles on the same axis
scale, which is denoted in each plot as CF. The mean ratio of the data from
both instruments is then computed at intervals of 0.025 R⊙between 1.1 and 1.5
R⊙and these values are shown in Figure 14. This linear increase in the
intensity ratio with height is then applied to the TSE data thus converting
the $pB$ observations taken with CIP from units of DN/s to units of solar
brightness ($B_{\odot}$). A cross-correlation was also performed on the
intensity profiles to find the angle needed to rotate the CIP data in order to
align it with that taken by K-Cor (i.e., solar north upwards).

Figure 13: Latitudinal distribution of intensity profiles from both CIP (red)
and K-Cor (black) for a range of heliocentric heights. A calibration factor
(CF), noted in each plot, is temporarily applied to the CIP data in order to
see both profiles on the same axis scale

Figure 14: Mean intensity ratio between the CIP and K-Cor intensity profiles
for each height interval (black) along with the linear fit used to perform the
relative radiometric calibration of the TSE data (red)

##  3 Results

Figure 15: Final MGN-processed $pB$ image of the December 14 2020 TSE after
relative radiometric calibration with respect to MLSO/K-Cor

Figure 15 shows the final calibrated $pB$ image of the December 14 2020 TSE.
It has been further processed using the MGN technique to enhance the fine-
scale coronal detail. Due to the poor viewing conditions at the observation
site adversely affecting the data, the data analysis has been limited to a
heliocentric height of 1.5 R⊙but the image in Figure 15 has been extended out
to 2 R⊙to show more of the corona.

Figure 16: Calibrated and corrected $pB$ profiles from CIP (black) and K-Cor
(red) for both the polar coronal hole and equatorial streamer

Figure 16 shows a comparison of $pB$ data as a function of heliocentric height
for the north polar coronal hole and west equatorial streamer chosen for this
study (at position angles 12∘ and 245∘ counter-clockwise from solar north,
respectively). As would be expected, the $pB$ is highest in the solar
equatorial region and lower in the polar region. It is clear to see that the
relative radiometric calibration with respect to MLSO/K-Cor is optimal at the
equator but the polar $pB$ observed by CIP is still greater than that of
K-Cor. This is believed to be due to the poor weather conditions at the time
of observation having a greater effect on the fainter coronal signal in the
polar regions.

Figure 17: Latitudinal distribution of $pB$ for heliocentric heights of 1.2
(black), 1.3 (red), 1.4 (blue), and 1.5 R⊙(green). Vertical dotted lines are
included to represent the position angles for the polar coronal hole (left),
CME (middle), and equatorial streamer (right).

Figure 17 shows the latitudinal distribution of the $pB$ for heliocentric
heights of 1.2, 1.3, 1.4, and 1.5 R⊙ in black, red, blue, and green,
respectively. The $pB$ decreases with increasing distance from the limb which
is to be expected, and all four latitudinal distributions have a similar shape
but there is a noticeable offset between corresponding maxima and minima as
heliocentric height increases. The dotted vertical lines represent the
position angles at which the densities are calculated for the polar coronal
hole and the equatorial streamer, as well as the position angle for the CME
where its presence becomes increasingly apparent in the 1.4 and 1.5 R⊙ $pB$
distributions.

Figure 18: Coronal electron densities as a function of heliocentric height for
the coronal hole (left) compared with Baumbach (1937), Doyle et al. (1999),
and Guhathakurta et al. (1999), and the equatorial streamer (right) compared
with Gibson et al. (1999), Liang et al. (2022), and Gallagher et al. (1999).
The black data points show the densities acquired by inverting the $pB$ and
the solid black line shows the fit to equation 9.

Figure 18 shows the coronal electron densities, derived by inverting the $pB$,
as a function of heliocentric height for both the polar coronal hole (left)
and equatorial streamer (right). Both plots also show comparisons to data from
previous works which agree very well with the data from this study. As
expected, the streamer’s density is greater than the coronal hole’s by an
average factor of $\sim$ 4 between 1.1 - 1.5 R⊙. For the coronal hole
comparison, density profiles from Baumbach (1937), Doyle et al. (1999), and
Guhathakurta et al. (1999) were used. The latter two in particular were
selected because they represent observations of polar coronal holes taken at
the start of solar cycle 24 (solar minimum) and, since the 2020 TSE occurred
exactly a year after the start of solar cycle 25, they are reasonable
comparisons to make. For the equatorial streamer, the densities obtained in
this study are compared with Gibson et al. (1999), Liang et al. (2022), and
Gallagher et al. (1999). Again, these are reasonable comparisons to make since
the compared works represent observations of a streamer (Gibson et al. (1999))
and equatorial regions (Liang et al. (2022); Gallagher et al. (1999)) taken at
or near solar minimum.

##  4 Discussion

###  4.1 Number of Polarizer Angles

Typically, similar studies and space-based coronagraphs use polarizer angles
of 0∘, 60∘, and 120∘ to infer the coronal electron density. As mentioned
previously, CIP is designed to efficiently take polarized observations for any
number of pre-defined polarizer angles. For this study, six polarizer angles
were chosen in an attempt to see if increasing the number of polarizer angles
leads to better constraints on the $pB$ observations and the inferred coronal
electron densities. Figure 19 shows the mean percentage difference between the
coronal electron densities derived using all six polarizer angles (henceforth
referred to as Angle Set A) and only the 0∘, 60∘, and 120∘polarizer angles
(Angle Set B) with the error bars representing the standard error in the mean.
The difference is shown as a function of the position angle between the
heliocentric heights 1.1 - 1.5 R⊙ and ranges from $-12.8\%$ to 9.44% with the
overall mean at 0.56%. For the position angles chosen to represent the coronal
hole and equatorial streamer, the mean difference is -8.26% and 8.04%,
respectively.

Figure 19: Mean difference between the coronal electron densities derived
using all six polarizer angles (Angle Set A) and only the 0∘, 60∘, and
120∘polarizer angles (Angle Set B). The mean difference ranges from $\sim$
-13% to 10% and the mean in the difference is also shown (red line). The error
bars show the standard error in the mean.

Figure 20 shows the difference in density between both angle sets A (red) and
B (black) as a function of heliocentric height for both the coronal hole
(left) and equatorial streamer (right). It is clear to see that the densities
found using set A (all polarizer angles) are lower in the coronal hole and
higher in the equatorial streamer in comparison to those found using only set
B (0∘, 60∘, 120∘). Furthermore, the difference between the two calculated
densities clearly becomes greater with increasing heliocentric height, which
is shown more clearly in Figure 21. Since the difference increases with
increasing heliocentric height, it is unfortunate that the weather conditions
on the day of the TSE were sub-optimal because it would be interesting to see
how this difference evolves beyond the inner coronal region. However, the
existence of this difference implies that changing the number of polarizer
angles used does provide a better constraint on the inferred coronal electron
densities, particularly for heliocentric heights above $\sim$ 1.5 R⊙. For the
remainder of the analysis of these results, set A (all polarizer angles) will
be used.

Figure 20: Difference in coronal electron densities between using $pB$ data
from Angle Sets A (red) and B (black) for both the coronal hole and equatorial
streamer Figure 21: Mean difference (expressed as a percentage) between using
$pB$ data from Angle Sets A and B for both the coronal hole (black) and
equatorial streamer (red) with the error bars representing one standard
deviation in both cases

###  4.2 Radial Density Fitting

Several studies (Saito et al., 1977; Guhathakurta et al., 1999; Hayes et al.,
2001; Thernisien & Howard, 2006) state that the radial dependence of the
coronal electron density can be expressed in the form of a polynomial:

$N_{e}(r)=\Sigma_{i}\alpha_{i}r^{-i}$ (8)

where $r$ is given in solar radii and $\alpha$ and $\beta$ are coefficients
fit to the data. Since the analysis of the data obtained in this study
extended from below $\sim$ 1.1 - 1.5 R⊙, three terms is sufficient to provide
a good fit to the data, thus, the equation used to fit the data and determine
the coefficients was:

$N_{e}(r)=ar^{-b}+cr^{-d}+er^{-f}$ (9)

The coefficients for both coronal features of interest are shown in table 1
and they agree well with previous studies.

Table 1: Coefficients for equation (9) | a | b | c | d | e | f
---|---|---|---|---|---|---
Coronal Hole |  1.12$\times$106 | 0.107 |  2.27$\times$108 | 10.4 |  1.50$\times$108 | 560
Equatorial Streamer |  -2.13$\times$109 | 6.56 |  1.43$\times$109 | 6.77 |  1.20$\times$109 | 6.77

##  5 Conclusion

The primary aim of this study was to present a new design for a lightweight
polarization instrument capable of observing using more polarization angles
than is typically used, called the Coronal Imaging Polarizer (CIP). The
instrument was designed and built at Aberystwyth University to observe the
polarized brightness of the solar corona during the December 14 2020 TSE for
six orientation angles of the linear polarizer. Due to the design of the
instrument, it is very easy to increase or decrease the number of polarization
angles depending on the time available during the totality phase of an
eclipse. One of the main design elements of CIP was that it would be
lightweight in order to be easily transported to an observing site. This
element was not only met but was also needed as the team had to relocate to a
new observing site on the morning of the eclipse. The new site was a few
hours’ drive from the original site and with the instrument itself weighing
only $\sim$ 1.59 kg, the bulk of the transporting was due to the tripod and
tracking mount. Consequently, the entire instrument was easily transported and
the simplicity of the design meant that the team could set it up very quickly
at the new observing site. However, the weather conditions at the new site
still impacted the data - primarily the strong gusts of wind which can be
clearly seen in Figure 7, along with airborne dust and high humidity. The
instrument was also designed to capture data fully autonomously and
efficiently and, again, this aim was met.

The raw VL images were successfully corrected using flat-field and dark frame
subtraction (see Section 2.2.2) and the corrected images were then manually
coaligned by tracking the drift of a star in the background of the data
throughout the duration of totality (see Section 2.2.3). The images were then
rebinned to be 8x8 times smaller in order to improve the signal-to-noise ratio
which might not be needed in future eclipses given better viewing conditions.
The images of different exposure times were combined by using a simple signal-
to-noise ratio cut to create composite images of the eclipse for each
individual angle of polarization (see Section 2.2.4). These composite images
were then combined to give the $pB$ image using a simple least squares fitting
method (see Section 2.2.6) and relative radiometric calibration was
successfully done by cross-calibration with MLSO’s K-COR (see Section 2.2.7).
The final $pB$ image was then inverted, assuming a locally spherically
symmetric corona, to produce radial density profiles for a polar coronal hole
and an equatorial streamer. These densities were then compared with previous
studies and were found to be in good agreement (see Section 3). Changing the
number of polarizer angles does have an effect on the inferred coronal
electron densities, however, due to the inclement weather conditions at the
time of observation, no meaningful comparison of the accuracy of more
polarizer angles could be performed. It is hoped that CIP can be sent to
observe future TSEs to provide better quality constraints on $pB$ and coronal
electron densities and build upon the results of this study, particularly with
regard to studying the impact of more polarization angles on the quality of
TSE data.

A special note of appreciation must go to the members of the team who braved
the COVID-19 pandemic in order to observe this eclipse, without whom this work
would not be possible. We acknowledge the valuable advice of the Solar Wind
Sherpa team of collaborators led by Prof Shadia Habbal at the University of
Hawaii. We also acknowledge studentship funding from the Coleg Cymraeg
Cenedlaethol, STFC grant ST/N002962/1 and STFC studentships ST/T505924/1 and
ST/V506527/1 to Aberystwyth University which made this instrument and work
possible. Some of the $pB$ data and coronal images used in this work are
courtesy of the Mauna Loa Solar Observatory, operated by the High Altitude
Observatory, as part of the National Center for Atmospheric Research (NCAR).
NCAR is supported by the National Science Foundation. This research has made
use of the Stellarium planetarium.

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