The Solar Minimum Eclipse of 2019 July 2: II. The First Absolute Brightness Measurements and MHD Model Predictions of Fe X, XI and XIV out to 3.4 $R_{\odot}$

Since the continuum data were already calibrated into solar disk brightness units ($B_{\odot}$, see Boe et al. 2021a), via the Mauna Loa Solar Observatory’s (MLSO) K-coronagraph (K-Cor)²²2K-Cor DOI: 10.5065/D69G5JV8; https://mlso.hao.ucar.edu/mlso_data_calendar.php, we used the Off-band to calibrate the On-band data. First, we measured the sky brightness at the center of the Moon during totality in each composite eclipse image (both On- and Off-band) and subtracted it to set the correct zero point. Next, we took the pixel flux in a region in the corner of each On- and Off-band image $>3.5$ $R_{\odot}$, and set them equal to each other in each image pair. In doing so, we assumed that the line emission is negligible at that distance given the dominance of the F-corona emission over both the K corona and line emission (see Boe et al. 2021a). Any remaining K-corona at that distance would further wash out the presence of negligible line emission. Once the On- and Off-band pairs have been self-calibrated, we added back the expected Earthshine brightness on the moon of $2.5\times 10^{-10}\pm 1.5\times 10^{-10}B_{\odot}$ (Agrawal, 2016). Once the data were transformed into solar disk brightness units, the continuum (Off-band) images were then subtracted from the On-band data to isolate the brightness of each emission line.

Finally, we performed a correction to the line emission data based on the expected line widths and known bandpasses. The bandpasses were designed to be broader than the line widths, but the bandpasses are not a perfect boxcar function (see Fig 3), so we had to account for the effect that the shape of the bandpass had on the final integrated line brightness.

To determine the expected line widths, we used the expected effective ion temperatures ($T_{\text{eff}}$). $T_{\text{eff}}$ is a measure directly determined by the line width according to the following equation:

where $m_{i}$ is the mass of the ion, $k$ is Boltzmann’s constant, $c$ is the speed of light, $\lambda$ is the line wavelength, and $\Delta\lambda$ is the Doppler width, defined by the standard deviation of the Gaussian line (i.e. $\sigma$). Note that the full width at half maximum (FWHM) is $2.355\ \sigma$ for a Gaussian function. Equation 1 and an estimate of $T_{\text{eff}}$ then allows one to determine the expected width for an arbitrary line. $T_{\text{eff}}$ is related to the ion temperature, with additional broadening due to turbulent and non-thermal motions. Since we only need an estimate of the line width, and not the actual ion temperature, $T_{\text{eff}}$ is sufficient for our purposes.

For our estimate of $T_{\text{eff}}$ throughout the corona, we used a number of previously published estimates. We used UVCS/SOHO measurements of $T_{\text{eff}}$ of the Mg x 62.6 nm and O vi 103.7 nm lines, as reported by Esser et al. (1999), which probed helioprojective distances beyond 2.2 $R_{\odot}$. For lower helioprojective distances, we used line width observations from the LASCO-C1 coronagraph of Fe xiv 530.3 nm (Mierla et al., 2008), which we converted to $T_{\text{eff}}$ using equation 1. We also used EUV observations from EIS/Hinode observations, reported as the effective ion velocity (i.e., $v_{\text{eff}}=c\ \Delta\lambda/\lambda$) of the Fe xi 18.8 nm line (Hahn & Savin, 2013). We then fit an exponential function to this set of data using Scipy curve$\_$fit (as shown in the top panel of Figure 5), which gave $T_{\text{eff}}=(0.13\pm 0.01)\ e^{(2.03\pm 0.08)R}$, with $R$ given in units of $R_{\odot}$, and $T_{\text{eff}}$ in MK. With the fit to $T_{\text{eff}}$, we computed the expected FWHM for each observed emission line, as shown in the middle panel of Figure 5.

To determine the correction factor needed for each line, we had to determine the relative transmission of the line and continuum signals integrated over the bandpasses. The line transmission efficiency was determined as the integrated product of a Gaussian function of the expected line width with the known bandpass for each line. The efficiency of transmitting the K+F corona signal, the integral of which was calibrated to the solar disk brightness (see Section 2.1), was found by integrating a flat continuum source over the bandpass since the K+F corona continuum is effectively flat over the narrow bandpasses. The ratio of these integrals gives the correction factor for each line, $C_{i}$, written as:

where $b_{i}$ is the bandpass transmission function (see Fig. 3) for line $i$ with a Gaussian line profile $G_{i}$ defined by the line width of $\Delta\lambda$ that will vary with helioprojective distance. This correction factor accounts for the relative transmission of the bandpasses, and converts the $B_{\odot}$ unit used with the K+F corona, into an analogous unit relative to the Gaussian line profiles. That is, the conversion translates the line signal into solar disk brightness units integrated over the Gaussian line profile.

As expected, in regions where the line widths are substantially smaller than the bandpass, this correction approached the limit where the line is a delta function at the center of the bandpass. As the line grows in width, it started to cover regions of the bandpass where the transmission is lower, and so the relative transmission of the line changes. Only in the outer corona beyond about 2 $R_{\odot}$ does the increasing line width start to become noticeable in the calibration correction. The rather small changes in the calibration corrections for different FWHM also strengthens the validity of our assumptions, since even rather large changes in the assumed $T_{\text{eff}}$ leads to exceptionally small changes in the calibration correction factor. At future TSEs, we intend to deploy slightly larger bandpasses (perhaps 1 nm) to avoid this behavior altogether at larger helioprojective distances.

The photometrically calibrated line emission data in solar disk brightness units integrated over the Gaussian line profiles (as described in Sections 2.1 and 2.2) are shown in the left panels of Figure 6. They have been sliced into a cartesian representation of polar coordinates for analysis purposes.This slicing also helps to increase the signal-to-noise (SNR) ratio in the outer corona, since higher coronal projected distances have the signal averaged from more pixels in each bin. The data is only utilized in pixels where the SNR $>2$ of the photometric data, as determined by the propagation of photometric errors for both the On- and Off-band frames. We also overplot contours of powers of 10 of brightness in units of the solar disk brightness.

Although one could convert these brightness values into absolute units by using the solar spectrum irradiance at the wavelength of the given line, here we chose to leave the values in units of solar disk brightness. The main motivation for this choice is that these lines are radiatively excited in the corona by photospheric photons, so the absolute energy irradiated by the ions will be tied to the incoming radiation from the Sun. Since we are interested in studying the physical properties of the corona, it makes more sense to consider the lines relative to the photospheric spectrum to remove any wavelength dependent effects. Furthermore, to infer the relative ionic abundances (and thus $T_{e}$) one must also account for the incident photon flux in the corona, which is achieved through the solar brightness units (see Section 3.2).

The latitudinal differences of the line emission are best illustrated in Figure 7, where we show traces of the line brightnesses at the fixed helioprojective distances of 1.2, 1.5, 2, and 2.5 $R_{\odot}$. The radial drop off of the line brightness is most evident in Figure 8, where radial traces of the line emission are shown for a set of latitudinal regions.

It is clear from both Figures 7 and 8 that Fe xi is by far the brightest emission line of the three. In fact, it is the only line which retains high signal-to-noise ratio data until 3.4 $R_{\odot}$, with a brightness of about $5\times 10^{-10}B_{\odot}$. The extent of the Fe xi emission data was limited more by the size of the detector used for the observations than by the strength of the line emission. In the future, we plan to use a wider field of view with a larger detector to enable observations of line emission out to even higher helioprojective distances.

In streamers, the brightness of Fe xi ranges from just over $10^{-6}B_{\odot}$ below 1.2 $R_{\odot}$, and falls to about $5\times 10^{-9}B_{\odot}$ by 3 $R_{\odot}$. The spatial distribution of Fe xi emission roughly correlates with the K-corona (electron scattering) emission inferred for this eclipse (see Boe et al. 2021a). The brightness of the K corona is 2 to 3 times brighter than Fe xi throughout the corona, but the line brightness drops to only a small fraction (about $10\%$) of the total continuum K + F corona brightness beyond about 2 $R_{\odot}$.

The Fe x emission behaves somewhat similarly to Fe xi, albeit with a considerably lower brightness. It reaches only about $5\times 10^{-7}B_{\odot}$ below 1.2 $R_{\odot}$, and fades to about $10^{-10}B_{\odot}$ by 3 $R_{\odot}$. The Fe x emission also has a much less pronounced variation with solar latitude, especially at distances below $\approx 1.5$ $R_{\odot}$, where the brightness of the streamers and coronal holes is comparable. Above 1.5 $R_{\odot}$, on the other hand, the brightness variation of Fe x looks much more similar to Fe xi shifted down by an order of magnitude.

By contrast, the Fe xiv emission shows the largest difference between coronal holes and streamers, with fine-scale variations throughout the streamers. It is almost as bright as Fe xi in the core of streamers below 1.5 $R_{\odot}$, but fades very quickly at higher elongations. Indeed, Fe xiv is the only Fe line that is virtually undetectable above the noise in some regions as low as 2 $R_{\odot}$, while maintaining a higher brightness in streamers out to as much as 2.8 $R_{\odot}$. Since Fe xiv emission originates from higher temperature plasmas ($T_{e}>$ 1.5 MK, see Fig. 3), the brightness variation implies a very low proportion of high temperature plasmas at the poles of the Sun and outside the cores of streamers.

The absolutely calibrated Fe line emission from all three lines enables the inference of the density weighted average $T_{e}$ for each LOS in the corona by comparing their relative brightness under the assumption of photoexcitation alone. These optical emission lines do have a component of collisional excitation in the low corona below $\approx$ 1.2 $R_{\odot}$, as shown analytically by Habbal et al. (2007), and for this eclipse specifically via the PSI MHD model (see Section 4 and Fig 6). To correct for the collisional emission, we multiply each emission line by the fractional amount of light that is expected to be from radiative excitation alone (from the MHD model). Since we will use the line ratio to infer $T_{e}$, this correction is somewhat independent of the exact density in the model and rather corrects for the relative intrinsic sensitivity that the specific emission line has to collisions.

Here we apply the methodology of Boe et al. (2020a), who used the calibrated ratio of Fe xiv/Fe xi from data acquired at multiple sites during the 2017 TSE, and expand it to line ratios including Fe x.

A given ionic abundance ratio ($n_{j}/n_{k}$) can be related to the intensity ratio of line emission $I$, for two lines $j$ and $k$ as in the following equation:

where the $A$ values are the Einstein coefficients for spontaneous emission, $g_{u}$ and $g_{l}$ are the statistical weights for the higher and lower energy level of the given transition, $\nu$ is the frequency of the light emitted by the transition, the $\epsilon$ values indicate the photometric efficiency of the telescope systems (and Earth’s atmosphere), and the $\rho(\nu)$ terms are the volumetric photon energy density in the corona where the ions are being excited. The $A$, $g$ and $\nu$ constants used for this work are shown in Table 1.

Equation 3 can be simplified since the product $\rho(\nu)\ \epsilon$ for each line is accounted for in the absolute photometric calibration of the line brightness into solar disk brightness units (see Section 3.1). The $\rho(\nu)$ terms are technically composed of the solar spectral energy distribution (accounted for via the solar disk brightness unit) and the volumetric scattering geometry of the corona and extended solar disk. However, these volumetric considerations will be identical for any two lines integrated over the same LOS, hence they disappear in a line ratio. Equation 3 then reduces to:

where the $\beta$ terms refer to the brightness of the emission lines in units of the solar disk brightness, rather than the absolute intensity of the lines (so $\beta_{i}=I_{i}\ \rho(\nu_{i})^{-1}\ \epsilon_{i}^{-1}$).

Next, we convert the ionic density ratio from Equation 4 to an inference of $T_{e}$ based on the ionic abundances as a function of $T_{e}$ from CHIANTI (see Section 2 and Fig. 3). With the Fe x, Fe xi and Fe xiv observations, we calculate three different line ratio temperatures for each possible combination. The resulting ionic density ratios and $T_{e}$ maps are shown in Figure 9. We only display pixels where the SNR $>2$ for both lines used in each line ratio $T_{e}$ inference. These panels are shown out to a heliocentric distance of 2.8 $R_{\odot}$, rather than 3.4 $R_{\odot}$ as in Figure 6. The lower distance was used for this figure because the signal of the Fe xiv line specifically was exceptionally weak beyond this distance, and so the inferred line ratios would not be robust for distances beyond 2.8 $R_{\odot}$.

We find that Fe $10^{+}$ (Fe xi) is the most abundant ion throughout the corona, even in higher $T_{e}$ regions. Fe $9^{+}$ (Fe x) is comparable to Fe $10^{+}$ in the low corona below 1.3 $R_{\odot}$, but is 3 to 5 times less abundant beyond that distance. Fe $10^{+}$ is always more abundant than Fe $13^{+}$ (Fe xiv) everywhere in the corona. In fact, Fe $13^{+}$ is never more than $\approx 50\%$ as abundant as Fe $10^{+}$, even in the hottest regions at the core of the Eastern streamer. Nevertheless, Fe $13^{+}$ is up to 2 times as abundant as Fe $9^{+}$ in the high $T_{e}$ regions. The Fe $13^{+}$ abundance drops dramatically in open field regions however, often to less than $10\%$ the abundance of Fe $9^{+}$. The result that Fe $10^{+}$ is the most abundant ion in the corona supports Habbal et al. (2010b, 2021), who found it to be the most abundant Fe ion in the solar wind and is the brightest Fe emission line in the corona, regardless of solar cycle.

The $T_{e}$ values inferred from the Fe xiv/Fe xi and Fe xiv/Fe x line ratios are consistent with each other, as demonstrated by the direct comparison between these $T_{e}$ inferences for each LOS in the top left panel of Figure 10. The Fe xiv/Fe x inferred $T_{e}$ is only 2.5$\%$ higher than the Fe xiv/Fe xi $T_{e}$ value on average. The root-mean-square (RMS) variance of the average is 3.7 $\%$, so the two methods are statistically equivalent. Both inferences range from about 1.25 – 1.4 MK in the coronal holes and about 1.5 – 1.65 MK in the equatorial streamers. The streamers show a large $T_{e}$ spatial variability where the streamer cores are hotter (about 1.6 MK) than the boundaries (1.5 MK). There are also pockets of lower temperature plasmas (1.3 MK) at the base of the streamers below 1.2 $R_{\odot}$. The thickness of the streamers (as visualized by the $T_{e}$ spatial distribution) also decreases at larger helioprojective distances, while the coronal holes dominate more of the corona with a roughly uniform temperature.

The Fe xi/Fe x inferred $T_{e}$ is quite different from the two line ratios involving Fe xiv, where it infers a much larger range of temperatures. This line ratio indicates that the coronal hole $T_{e}$ is closer to 1.1 MK with distinct fine-scale plumes of cool plasma emerging into the corona. The core of the streamers then is over 1.8 MK in this inference, while the outer corona is quite a bit higher in temperature than the other line ratio inferences. While the Fe xi/Fe x ratio is an interesting map for showcasing small temperature variations in the coronal hole plumes, it is not a very reliable measure of the temperature. One reason for this large spread of $T_{e}$ is that the slope of the Fe xi/Fe x ionic abundance ratio as a function of $T_{e}$ is considerably shallower than the other line ratios (see Fig. 4). Therefore, a small change in the emission ratio leads to a large change in the inferred temperature. The temperature response range of the Fe x and Fe xi curves alone also does not probe the range above about 1.5 MK effectively; consequently this line ratio is unreliable anywhere outside the coolest regions in the corona. The large difference between the Fe xi/Fe x and the other line ratio $T_{e}$ inferences perhaps implies that the corona is not isothermal along a single LOS, and so the LOS average is not identical for different line ratio inferences. Additionally, average $T_{e}$ inferences from two line ratios may be biased toward finding temperatures in-between the peak ionization of the temperature response functions (i.e. Fig. 4) as demonstrated in the EUV by Weber et al. (2005). We intend to explore a more complex $T_{e}$ inference using all three (or more) of these visible emission lines simultaneously in a future study, as it is beyond the scope of this work.

Assuming local thermodynamic equilibrium, the total line emissivity of a parcel of plasma depends on the local electron density, temperature, and radiation field near line center, originating from both radiative and collisional excitation. To tackle this problem, we use the CHIANTI 10 database and software package (as described in Section 2) and its framework for adding photo-excitation to the level population calculations (Young et al., 2003). The contribution function for each Fe line individually is computed on a 3D grid of density, temperature, and solar radius. The radial distance is then used to approximate the local radiation field at these wavelengths by using the observed solar spectral irradiance from the International Space Station (Meftah et al., 2018). This lookup table for the contribution function is then used to calculate the local emissivity at each point along the LOS in the forward modeling computation.

For the abundance factor we adopt the “hybrid” coronal abundances (Schmelz et al., 2012), which match the radiative loss function used in the thermodynamic MHD calculation. We can also turn off the photoexcitation flags in the CHIANTI calculation to compute contribution functions based on collisions alone and use these in a separate forward modeling experiment. This enables us to compare the relative importance of collisional excitation for each LOS in the forward modeled line emission. Finally, to compare the model calculations with the eclipse observations, we convert the LOS integrated emission into solar disk brightness units by using the solar spectral irradiance and accounting for the angular size of the solar disk, which had a radius of about 943 arc-seconds at the time of the eclipse (Earth was at aphelion).

Radially flattened versions of the model predicted Fe x, Fe xi, and Fe xiv line emission are shown in the bottom panels of Figure 2. The photometric brightness predictions of all three lines are shown in the middle panels beside the observed emission in Figure 6, with the same cartesian representation of polar coordinates as the eclipse data in the left panels. In the right panels of the figure, we show the percentage of the emission that originates from collisional excitation compared to the total brightness of each line. Collisional excitation is important in the low corona inside streamers, but the photoexcitation becomes the dominant excitation process everywhere beyond 1.2 $R_{\odot}$. We use the modeled fraction of emission that is radiative to correct the emission observations specifically for the line ratio inferred $T_{e}$ values (as done in Section 3.2). However, it is possible that our $T_{e}$ inferences are not as robust at the base of streamers. This region is commonly explored through EUV observations, since EUV emission is predominantly collisionally excited, so it is not crucial for these eclipse observations to robustly probe the collisionally dominated regions of the corona. Regardless, our $T_{e}$ results are generally consistent with the values inferred via EUV observations during periods close to solar minimum (e.g., Morgan & Taroyan 2017).

We thank Judd Johnson and Pavel Štarha, who acquired the narrowband data at the Rodeo, Argentina observing site during the 2019 July 2 total solar eclipse.

The K-Cor data were 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.

Observables presented in this paper and other eclipse data from our group can be found at: https://www.ifa.hawaii.edu/SolarEclipseData/. The PSI MHD model eclipse predictions can be found here: https://www.predsci.com/corona/

Financial support was provided to B. B. by the National Science Foundation under Award No. 2028173. S. H. and the 2019 eclipse expedition were supported under NASA grant NNX17AH69G and NSF grant AST-1733542 to the Institute for Astronomy of the University of Hawaii. C. D. was supported by the NASA Heliophysics Supporting Research and Living With a Star programs (grants 80NSSC18K1129 and 80NSSC20K0192). M. D. was supported by the Grant Agency of Brno University of Technology, project No. FSI-S-20-6187.