Coronal Magnetic Field Topology from Total Solar Eclipse Observations

The shape of the solar corona was first characterized during total solar eclipses (TSEs) with hand-drawn sketches and early photographic techniques (e.g., Maunder 1899 and references therein). It was found to vary in a periodic manner correlated with the sunspot cycle (Darwin et al. 1889) as observed on the solar surface (Schwabe 1844). The later discovery that the sunspot cycle is in fact a magnetic cycle (Hale et al. 1919) implied that the variable shape of the corona was magnetic in nature. During solar minimum the corona was found to have large polar plumes (Saito 1958) that dominate the north and south poles, which originate from polar coronal holes (Munro & Withbroe 1972). The equatorial regions are dominated by streamers (Maunder 1899)—which tend to have a higher electron density, and thus a higher visible light coronal intensity (van de Hulst 1950). During solar maximum the coronal structure was less organized, with mostly radial features (e.g., Newkirk 1967 and references therein).

In recent years, the most common technique employed to measure the direction of the coronal magnetic field has been linear polarization observations of coronal emission lines with a coronagraph (e.g., Habbal et al. 2001; Gibson et al. 2017; Dima et al. 2019). These observations have remained somewhat limited in extent, however, as ground-based coronagraphic observations of polarized emission lines can only probe below ≈1.5 solar radii (R_⊙) due to scattering of sunlight by the Earth's atmosphere. Polarization observations of coronal emission lines also have to contend with the "Van Vleck" angle ambiguity (Van Vleck 1925), where structures curved more than 547 with respect to the radial direction from Sun center will have their polarization angles rotate by 90° from the true magnetic field direction. This ambiguity can lead to misleading identification of the field direction in highly curved structures, such as in active regions and the base of streamers.

Space-based coronagraphs often acquire broadband white-light polarization observations of the continuum corona (e.g., the and Heliospheric Observatory (SOHO)/LASCO Brueckner et al. 1995 and STEREO/COR Kaiser et al. 2008), which can be used to infer the line-of-sight electron density and F corona brightness (e.g., Woo & Habbal 1997; Habbal & Woo 2001; Morgan & Habbal 2007), but do not provide any information about the coronal magnetic field. The planned ASPIICS coronagraph on PROBA-3 (Galano et al. 2018) intends to observe the continuum corona between ≈1.1 and 3 R_⊙, while METIS/COR on the Solar Orbiter (Antonucci et al. 2017) will cover a variable distance range, where it can observe between ≈1.2 and 3.0 R_⊙ (1.6 to 4.1 R_⊙) at minimum (maximum) orbital distance. ASPIICS and METIS should provide images of the continuum coronal emission with substantially higher resolution, lower noise, and less scattered light compared to LASCO and STEREO, but will not surpass the radial extent and spatial resolution that is possible to achieve during a TSE.

Inferences of the coronal magnetic field topology are possible with extreme ultra-violet (EUV) observations (e.g., SDO/AIA; Lemen et al. 2012) as demonstrated by Schad (2017). However, they remain limited in spatial extent due to their collisionally excited nature (see Section 1 of Boe et al. 2020 for an in-depth discussion). SWAP on PROBA-2 can perform slightly better, sometimes detecting EUV emission (Fe x 17.4 nm) as far as 2 R_⊙ (Seaton et al. 2013), but only in regions with a relatively high electron density (n_e). METIS promises to do even better, possibly reaching 3 R_⊙ or more for the He ii 30.4 nm and H i 121.6 nm lines (Antonucci et al. 2017). Even with the improvements in EUV observations, their collisionally excited nature ($\propto {n}_{e}^{2}$) will cause them to be strongly weighted toward seeing regions with higher electron density. Observing the corona in the EUV is consequently useful for studying the magnetic loops above active regions and other structures in the low corona (≲1.5 R_⊙), but is not capable of fully revealing the magnetic field line structure for the entire corona. At present, only during a TSE is the entire corona—and its magnetic field structure—observable simultaneously from just above the photosphere out to at least 6 R_⊙.

The first attempts to model the coronal magnetic field were based on extrapolations of the photospheric magnetic field outward to an upper boundary, known as the "source surface," beyond which the coronal magnetic field was assumed to be radial. Originally proposed by Altschuler & Newkirk (1969) and Schatten et al. (1969), these so-called potential field source surface (PFSS) models continue to evolve. The source surface has traditionally been set at 2.5 R_⊙ from the Sun, but some more recent models have set the source surface as low as 1.5 R_⊙ (Asvestari et al. 2019) to attempt to best fit the open flux from coronal holes in the solar wind. More sophisticated models, such as magnetohydrodynamic (MHD) models (Mikić 1990; Mikić et al. 1999), have been developed in an attempt to more completely encompass the physics of the magnetized coronal plasma, but are more computationally expensive than PFSS models.

The photospheric magnetic observations used as the initial conditions for coronal models, such as those by the Michelson Doppler Imager (MDI) on SOHO (Scherrer et al. 1995) and the Helioseismic and Magnetic Imager (HMI) on SDO (Scherrer et al. 2012), rely on line-of -sight Zeeman polarization effects. These observations are highly uncertain for position angles greater than ≈45 to 75° from disk center (e.g., Altschuler & Newkirk 1969; Wiegelmann et al. 2017). Synoptic maps can compensate for this angle effect with solar longitude, provided the observations from an entire solar rotation are combined. However, this process assumes that the coronal magnetic field does not change substantially over a ≈28 day rotation period. Furthermore, observations of photospheric magnetic fields at the poles have never been possible given that every solar magnetic field observatory is located near the ecliptic plane. Even Ulysses (McComas et al. 1998), which reached a heliocentric latitude as high as ≈80°, did not have any remote sensing instruments (magnetic measurements in situ only yield the in situ field strength and direction). Solar Orbiter's PHI (Solanki et al. 2015) promises to observe the polar photospheric field for the first time enabled by its inclined orbit; though, it will only be able to see one pole at a time for a subset of its orbit.

PFSS, MHD, and other models are powerful ways to explore the magnetic field of the corona and the origin of the solar wind (see Wiegelmann et al. 2017 for a coronal modeling review), but to date these approaches have been largely unconstrained due to the absence of magnetic field measurements in the corona. Even the most complex MHD models, including a combination of coronal heating processes, are not capable of perfectly recreating the shape of the solar corona when compared with TSE observations (Mikić & Downs et al. 2018). Furthermore, no model has been quantitatively compared to the eclipse observations beyond a discussion of the general location and size of streamers and coronal holes.

In this work, we quantify the topology of the coronal magnetic field using TSE observations spread over almost two complete solar activity cycles. We use an existing database of broadband white-light eclipse images (see Section 2), and apply the rolling Hough transform (developed by Clark et al. 2014, see Section 3) to automatically trace the coronal magnetic field line structure. A detailed accounting of the observed features of the coronal magnetic field is given in Section 4. Implications for coronal magnetic field, existing models, and solar wind formation are discussed in Section 5. Concluding remarks are given in Section 6.

Quantifying the structure of the magnetic field in the midcorona (1.5 to 3 R_⊙) has remained somewhat elusive despite the advancement of ground- and space-based observatories (see Section 1), yet observing this region is crucial for understanding the origins of the solar wind. The easiest way to simultaneously observe the entire corona (from 1 to at least 6 R_⊙) with high spatial resolution and strong signal continues to be during a TSE. The many years of existing TSE observations thus present a unique opportunity to quantify the magnetic field topology of the corona throughout the solar cycle, and provide improved constraints for coronal modeling. We processed archival broadband white-light total solar eclipse images taken between 2001 and 2019 to infer the magnetic field angle relative to the radial direction for all solar latitudes (see Sections 3 and 4). These 14 eclipses span almost two complete solar cycles and offer a unique probe of the direction of the coronal magnetic field.

The entirety of the data used here was taken from an open source database of processed eclipse images,³ which comes from a variety of eclipse observers—both professionals and amateurs (see Table 1). All eclipse data were stacked using the technique described in Druckmüller (2009), and processed with the adaptive circular high-pass filter (ACHF) method (Druckmüller et al. 2006) to enhance the magnetic field structure in the corona. Examples of the white-light coronal data from each of the 14 TSEs are shown in Figure 1. Note that the data are not photometrically calibrated, so the intensity in the image is not representative of the relative intensity of the corona between eclipses. The processing also changes the brightness contrast between different heliocentric distances. Several of the white-light images have been previously used to study individual eclipses (e.g., Habbal et al. 2010, 2014; Alzate et al. 2017; Boe et al. 2018, 2020). For most eclipses, multiple cameras observed the corona at different angular scales. The panels in Figure 1 show each of the coronae below 2.5 R_⊙. Examples from the much wider field images (inside 5 R_⊙), albeit with lower resolution, are shown in Figure 2.

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Imbedded in this TSE data set is a record of the coronal magnetic field topology with high spatial resolution between 1 and over 6 R_⊙, which is quantified here for the first time. It is important to underscore that the corona is a three-dimensional source; all the TSE images are line-of-sight (LOS) integrations of the corona, resulting in a 2D image that has been collapsed into the plane of sky (POS). Since the coronal density drops sharply with heliocentric distance, the observed intensity of the corona is substantially weighted toward the field lines that are the closest to the solar limbs in the POS (perhaps ≈ ± 20° of the POS). Thus, the final observables presented in this paper should be considered as a density and angle weighted average of the coronal magnetic field in the vicinity of the POS.

To quantify the coronal magnetic field structure, we apply a line tracing algorithm developed by Clark et al. (2014) called the RHT. The RHT algorithm was originally used to trace the magnetic field of the Milky Way galaxy, and has been demonstrated with solar EUV data (Schad 2017). Specifically, Schad (2017) performed tests with the RHT algorithm indicating that it was a robust technique for detecting magnetic field line edges—even with low signal-to-noise ratio data. The implementation of the RHT algorithm to the eclipse data set is based on that described by Schad (2017).

First, the white-light eclipse images are processed with a Gaussian high-pass filter, and then blurred at the same resolution to enhance edges in the image and eliminate any low frequency structure that is not a direct result of the magnetic field topology. Next the image is converted to a binary scale before finally being processed with the RHT algorithm. An example of the procedure for an image from the 2019 TSE is shown in Figure 3. The RHT algorithm determines the angle of the magnetic field line independently for each pixel with respect to the image coordinates (as shown in Figures 3, 4, 5, and 6, see Figure 4 for the color scaling), by attempting to fit a line to every possible direction inside a box centered on each pixel in the binary data. It then reports the relative probability for every possible field angle direction, which is used to determine the uncertainty of each measurement, and the relative certainty between various pixels. This process is repeated independently throughout the image, so every pixel in the final image is a unique observation. The RHT algorithm does not connect features between pixels, so the lines seen in Figures 3–6 are not connected by the algorithm. Instead, the prevailing direction of the line is the same for the collection of pixels along the line, and thus the continuity of the features (i.e., consistent angle/color) indicates that the measurements are robust.

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Since the TSE data were acquired with a wide range of telescopes and detectors of varying quality (2001 and 2002 were done with film), each eclipse had to be manually processed to maximize the field line structure in the binary image. The Gaussian sigma of the high-pass filter ranged between σ_R = 0.002 and 0.05 R_⊙ (in pixel units), the RHT window size ranged from W_RHT = 0.08 to 1.0 R_⊙, and the RHT smearing radius ranged between R_RHT = 0.02 and 0.4 R_⊙ (see Clark et al. 2014 and Schad 2017 for a discussion of these parameters and how they affect the final results). In general, the final measured angle by the RHT algorithm is insensitive to the exact choice of parameters (i.e., σ_R, W_RHT and R_RHT). The main need for different parameters between eclipses is to maximize the resolution, while still capturing the coronal field topology to as large a distance as possible. The only problems occur when the structures are oversampled or undersampled. Oversampling occurs where the size of the RHT fitting box is smaller than the width of the field line in the binary image—causing the RHT algorithm to report angles perpendicular to the true field line angle. Undersampling occurs when the RHT fitting box is too large; in that case it sees several different field lines at the same time, and thus is not capable of resolving the field line structure. These effects necessitated changing the parameters manually to maximize the potential for the RHT algorithm to measure the field line angles in the largest parameter space possible.

The initial angles returned by the RHT algorithm are given with respect to the image frame (as shown in Figures 3–6). These angles are subsequently converted to be with respect to the radial direction. The angles relative to radial prevent any angle discontinuity effects (i.e., 0° versus 180°) and enables a direct comparison of structures throughout the corona for a single eclipse and between eclipses. In many cases, both narrow-field (typically 1–3 R_⊙) and wide-field (often up to ≥6 R_⊙) coronal data are available from the same eclipse. In a few cases, there were very high resolution data of the lowest regions in the corona, which show detailed structures ubiquitously (see Figure 6). When multiple images are available from a single eclipse, the data are stacked to generate a continuous measurement from the lunar limb to several solar radii. All RHT data from the wide-field images below 2 R_⊙ were removed from the stacked results, due to the poor spatial resolution compared to the narrow-field images.

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The combined data from each eclipse are shown in Figure 7 as histograms of the field line angle with respect to the radial direction. Figure 8 shows the same results as Figure 7, but in units of the relative uncertainty for the given bin (i.e., the statistical significance of the difference from radial). The uncertainties are propagated through the weighted mean inside each bin based on the reported RHT probabilities for each data point inside the bin.

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The direction of the coronal magnetic field with respect to the radial direction, for the individual eclipses shown in Figures 7 and 8, indicates a high variance in coronal structures over the solar cycle. Near solar minimum (roughly 2009 and 2019), the corona has two lobes of highly nonradial field lines between about 15 and 75° latitude in the north and south. The location of the nonradial lobe tends to move toward the equator with higher heliocentric distance, ending around 2.5 to 3 R_⊙. Near solar maximum (roughly 2001 and 2013), small pockets of nonradial field topology appear throughout the corona without the large-scale lobe structure as seen at solar minimum. These observations are consistent with the concept that the coronal magnetic field is dominated by dipolar and quadrupolar magnetic moments during solar minimum, while higher order moments begin to have a larger impact on the global coronal magnetic field at solar maximum. Indeed, the shape of the coronal magnetic field topology is strikingly similar for the eclipses near solar minimum, which is especially noticeable in the highest resolution TSE data (from 2008 and 2017) shown in Figure 6.

Averaging the individual eclipse data every two years across all position angles (Figure 9) shows the global divergence of the coronal magnetic field angle from the radial direction with the expansion of the solar wind. There does not appear to be an obvious solar cycle dependence in the average radial angle (for the entire corona), except for a slightly more nonradial angle during solar minimum in 2009 and 2019. It is clear from Figures 7–9, that the corona is consistently nonradial (≥10°) out to 3 R_⊙ or more, to a high confidence level. The distance at which the coronal field becomes statistically consistent with radial does move slightly outward for the more recent eclipses, but this effect is due to the increased spatial resolution of the recent eclipse observations that allow for more independent measurements of the coronal magnetic field inside the same bin (thus decreasing the uncertainty). At a distance of about 4 R_⊙, however, the field does become almost entirely radial. One notable exception is the highly nonradial structure at 6 R_⊙ in 2013, which is due to a coronal mass ejection (CME) that was previously reported by Alzate et al. (2017).

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The average deviation of the magnetic field from radial at different coronal latitudes over the past 20 years of observations is shown in Figure 10, where the field angle measurements between 1.5 and 3.0 R_⊙ are averaged for all eclipse data in 2 year periods. During solar minimum there are large, highly nonradial regions between 15 and 75° latitude, both in the north and south. While the midlatitudes have a highly nonradial field during solar minimum, the field lines above the poles are almost precisely radial. During solar maximum, there are more nonradial fields throughout the entire corona, but the difference from radial is substantially smaller than in the solar minimum lobes at midlatitudes. Throughout the solar cycle, there exist mostly radial fields above the solar equator.

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The coronal magnetic field topology is compared with the photospheric magnetic field in Figure 11. The photospheric magnetic field data in the top left panel of Figure 11 are a compilation of SOHO/MDI (Scherrer et al. 1995) and SDO/HMI (Scherrer et al. 2012) photospheric magnetic field observations. Level 2 synoptic maps were averaged in magnetic field strength at all longitudes for each Carrington rotation. For epochs where both MDI and HMI were operational, we averaged the two data sets. We then compared the photospheric magnetic field strength with the average magnetic field angle observed in the corona above, between 1.5 and 3.0 R_⊙. Every latitude bin for each eclipse (as in Figures 7 and 8) is compared to the same latitude in the photospheric magnetic field for the Carrington rotation of the given eclipse. The scatter of the correlation between the photospheric field strength and the coronal field angle is shown with different scalings in the remaining panels of Figure 11. The top right and bottom right panels are identical except for the scaling of the magnetic field strength to show different structures in the data (see Section 5). The bottom left panel shows the average magnetic field strength (absolute value) for different coronal field angles. The regions of the highest magnetic field curvature (≳20 degrees from radial) all have a weak average magnetic field below in the photosphere, while the regions of the highest magnetic field strength in the photosphere tend to result in a much more radial coronal magnetic field.

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As described in Section 4, the RHT algorithm is used here to infer the coronal magnetic field topology for 14 unique eclipse coronae—covering two solar activity cycles. These results establish that the coronal magnetic field is almost always nonradial below 4 R_⊙, and that the distribution of the nonradial fields is a function of latitude and solar cycle. Next, we give an overview of the inferred topology for the coronal magnetic field and discuss the implications of these results for coronal models and solar wind formation.

5.1. Fine-scale Structure and Coronal Radiality

5.2. Implications for PFSS and MHD Models

5.3. Implications for Solar Wind Formation

Application of the rolling Hough transform (RHT) to white-light TSE images, as done here for 14 eclipses between 2001 and 2019, has demonstrated that TSE observations continue to yield unique opportunities to quantify the topology of the coronal magnetic field throughout the solar cycle—thus providing valuable constraints for coronal modeling and solar wind formation. In particular, this work showed the following:

• 1.  
  The coronal magnetic field is highly structured down to the spatial resolution of the images, with topologically "open" field line structures visible at most latitudes throughout the corona for all solar eclipse images.
• 2.  
  The coronal field can evolve significantly in the midcorona (1.5 to 3 R_⊙) between different latitude regions over the solar cycle, often remaining nonradial out to about 4 R_⊙.
• 3.  
  The most nonradial regions in the midcorona occur above the weakest photospheric magnetic fields, whereas regions with the strongest fields are correlated with nearly radial field lines.
• 4.  
  These TSE observations yield essential constraints for coronal models. We find that the canonical source surface of 2.5 R_⊙ in PFSS models is not far enough from the Sun, and that the source surface is nonspherical.

The RHT algorithm and procedure presented here could be used in the future with coronagraphic observations, especially with the upcoming METIS and ASPIICS coronagraphs. Inferring the coronal magnetic field topology may consequently be possible for a much larger and almost continuous data set. The RHT tracing method could be useful for mapping the structures of CMEs in the corona and as a tool to empirically link in situ solar wind observations to sources on the Sun, without any assumptions whatsoever on the origin of the structure from just above the photosphere to at least the source surface region.

This work would not have been possible without the many years of hard work and dedication of all the eclipse observers, given in Table 1. Special thanks to Xudong Sun for suggesting the RHT algorithm for use with eclipse data. We thank the reviewer for exceptionally fair and insightful comments. Financial support was provided to B. Boe by AURA/NSO with grant N97991C to the Institute for Astronomy at the University of Hawaii. NASA grant NNX17AH69G and NSF grants AGS-1358239, AGS-1255894, and AST-1733542 to the Institute for Astronomy of the University of Hawaii. M.D. was supported by the Grant Agency of Brno University of Technology, project No. FSI-S-14-2290.