The Solar Minimum Eclipse of 2019 July 2. III. Inferring the Coronal T_e with a Radiative Differential Emission Measure Inversion

The traditional DEM is defined as (see Withbroe 1978; Del Zanna & Mason 2018)

$$\begin{eqnarray}&&\mathrm{DEM}({T}_{e})={N}_{e}^{2}\displaystyle \frac{{dh}}{{{dT}}_{e}},\end{eqnarray} \tag{ 1 }$$

where N_e is the electron density (squared since it is causing collisional excitation of the ions). The DEM represents the density-squared-weighted T_e distribution along a given LOS, in units of cm⁻⁵ K⁻¹.

The brightness, β_i , of an emission line i from an ion of element Z with elemental abundance Ab(Z) can be related to the integral of the temperature over the contribution function, C_i (T_e , N_e , λ_i ), multiplied by the DEM function. The brightness is then

$$\begin{eqnarray}&&{\beta }_{i}={Ab}(Z){\int }_{{T}_{e}}{C}_{i}({T}_{e},{N}_{e},{\lambda }_{i})\ \mathrm{DEM}({T}_{e})\ {{dT}}_{e}.\end{eqnarray} \tag{ 2 }$$

The traditional DEM formulation only works for collisionally excited lines, which are the main type of line emission one can usually expect at X-ray and EUV wavelengths (see Section 1). The contribution function is thus a complicated interdependence of the nature of the given ionic line transition, the electron density (since the lines are collisionally exited), and the temperature dependence of the ionic species.

For the visible lines observed during the eclipse, on the other hand, the lines are almost entirely radiatively excited and so a different formalism is required. We thus introduce a slightly modified R-DEM, which will replace the DEM(T_e ) term in Equation (2), where

$$\begin{eqnarray}&&\mathrm{RDEM}(T)={N}_{e}\displaystyle \frac{{dh}}{{dT}},\end{eqnarray} \tag{ 3 }$$

which will have units of cm⁻² K⁻¹. Unlike the collisional DEM inversion, the R-DEM inversion has a significantly simpler contribution function as it does not depend on the electron density (i.e., C_i (T_e , λ_i )). Rather, it is only a combination of the straightforward radiative ionic transition parameters based on radiative excitation from the photospheric emission combined with the temperature response of the ion (i.e., the curves in the top panel of Figure 2). The brightness of a radiatively excited line is then

$$\begin{eqnarray}&&{\beta }_{i}={Ab}(Z){\int }_{{T}_{e}}{C}_{i}({T}_{e},{\lambda }_{i})\ \mathrm{RDEM}({T}_{e})\ {{dT}}_{e}.\end{eqnarray} \tag{ 4 }$$

One can also calculate the radiative emission measure, or R-EM, as the integral of the R-DEM over a defined temperature range of ΔT centered on temperature T₀,

$$\begin{eqnarray}&&\mathrm{REM}(T)={\int }_{{T}_{0}-\tfrac{{\rm{\Delta }}T}{2}}^{{T}_{0}+\tfrac{{\rm{\Delta }}T}{2}}\mathrm{RDEM}(T)\ {dT},\end{eqnarray} \tag{ 5 }$$

which represents the amount of emitting plasma at the specified temperature range in units of square centimeters.

Next, we simplify the inversion procedure given the particular properties of radiative excitation by using calibrated ionic column density ratios rather than the absolute emissivity. In Paper II, we used the ratio of the brightness between two radiative lines to infer the relative ionic density with the following equation:

$$\begin{eqnarray}&&\displaystyle \frac{{n}_{j}}{{n}_{k}}=\displaystyle \frac{{\beta }_{j}\ {A}_{k}\ {g}_{l,j}\ {g}_{u,k}\ {{\nu }_{k}}^{3}}{{\beta }_{k}\ {A}_{j}\ {g}_{u,j}\ {g}_{l,k}\ {{\nu }_{j}}^{3}},\end{eqnarray} \tag{ 6 }$$

where for ions j and k, n is the number density, ν is the frequency of the line emission, A is the Einstein coefficient for spontaneous emission, and g is the statistical weight for the given energy level. Equation (6) assumes that the observed brightness β has been calibrated in solar disk units (i.e., B_⊙), thus removing the spectral dependency on the photospheric spectrum (see Boe et al. 2020a, 2022). It also assumes the brightness to be caused entirely by radiative excitation. The ionic density ratios as computed in Paper II already removed the collisional excitation component from the brightness using the ratio of the collisional and radiative excitation predicted by the PSI MHD simulation (which is only important in the streamers below ≈1.5 R_⊙). The calibrated ionic density ratios inferred in Paper II using Equation (6) for Fe x/Fe xi and Fe xiv/Fe xi are shown in the bottom-middle panels of Figure 1.

To express the R-DEM inversion in terms of the ionic density ratio, we take the ratio of the R-DEM brightness definition from Equation (4) and combine it with Equation (6), giving

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{{\beta }_{j}}{{\beta }_{k}} & = & \displaystyle \frac{{n}_{j}\ {A}_{j}\ {g}_{u,j}\ {g}_{l,k}\ {{\nu }_{j}}^{3}}{{n}_{k}\ {A}_{k}\ {g}_{l,j}\ {g}_{u,k}\ {{\nu }_{k}}^{3}}\\ & = & \displaystyle \frac{{Ab}({Z}_{j})\displaystyle \int {C}_{j}({T}_{e},{\lambda }_{j})\ \mathrm{RDEM}({T}_{e})\ {{dT}}_{e}}{{Ab}({Z}_{i})\displaystyle \int {C}_{k}{T}_{e},{\lambda }_{k})\ \mathrm{RDEM}({T}_{e})\ {{dT}}_{e}},\end{array}\end{eqnarray} \tag{ 7 }$$

which reduces to

$$\begin{eqnarray}&&\displaystyle \frac{{n}_{j}}{{n}_{k}}=\displaystyle \frac{{Ab}({Z}_{j})\int C{{\prime} }_{j}({T}_{e})\ \mathrm{RDEM}({T}_{e})\ {{dT}}_{e}}{{Ab}({Z}_{k})\int C{{\prime} }_{k}({T}_{e})\ \mathrm{RDEM}({T}_{e})\ {{dT}}_{e}}.\end{eqnarray} \tag{ 8 }$$

The new contribution function, $C^{\prime} ({T}_{e})$, has now been simplified to remove all dependence on the line formation itself and instead is solely the temperature dependence of the ionic equilibrium (i.e., the curves in the top panel of Figure 2). The contribution function can be simplified in this way since the ionic density ratio has already accounted for the incident radiation and resonant excitation physics—a process that cannot be applied with traditional DEMs given the dependence of collisional excitation on density.

In this work, we will infer the R-DEM with the Fe x, Fe xi, and Fe xiv line observations. We thus apply Equation (8) to both the Fe xiv/Fe xi and Fe x/Fe xi ionic density ratios and fit a density distribution of the RDEM(T_e ) that satisfies both ratios simultaneously (note that the elemental abundance terms disappear since we are only using Fe lines). That is, we use Fe xi as the reference density, scale the other densities by the inferred ionic density ratio, then solve for what R-DEM would best fit the two ratios of the integrals (i.e., Equation (8)). Specifically, we use 201 bins of temperature ranging from 0.6 to 2.6 MK, with a spacing of 0.01 MK, and use the standard Python scipy optimize.minimize ⁴ function to fit the distribution of density coefficients. This optimization tool implements a Broyden–Fletcher–Goldfarb–Shanno algorithm, which is an iterative quasi-Newton method (default for minimize). The optimization procedure stops when it reaches the relative minimum of the difference between the inferred R-DEM and the observed ionic density ratios.

We initialized the temperature distribution as a uniform temperature distribution across the entire range from 0.6 to 2.6 MK. The function then varies the coefficients of each R-DEM bin to attempt to fit Equation (8) for both the Fe xiv/Fe xi and Fe x/Fe xi ionic density ratios simultaneously. Typically, the procedure terminated after about 100 iterations, with a maximum number of iterations at 310 for the eclipse data and 445 for the MHD modeled lines (we set the maximum number of iterations to 2000 to prevent premature termination). The resulting distribution provides the R-DEM for each LOS. A simple fit is sufficient for this R-DEM inversion since the temperature response functionality of the bandpasses shown in the top panel of Figure 2 is simple. Consequently, the combination of bandpass integrals quickly converges on a solution—even if only using these two ionic density ratios.

As with all optically thin plasmas, this inversion determines the integral over all structures along the LOS, which may well have different temperature distributions. The final inverted R-DEM should be thought of as the density-weighted temperature distribution along the entire LOS rather than for a well-defined volume of plasma.

In this inversion, we are not taking into account any geometric LOS or limb-darkening effects on the radiative excitation of the lines. Along each LOS, the emission lines will experience similar dynamics from changes in the size and apparent horizon of the extended solar disk. Thus, most of these effects are removed by considering the ratio of the lines independently for each LOS, unless there is a dramatic asymmetry in the distribution of plasma temperatures along the LOS. There will be slight differences in the limb-darkening profile at different wavelengths, but these changes should only lead to small differences in the final line emission. As described in Papers I and II, we did account for limb darkening when calibrating the absolute brightness of the lines, so we have already removed the limb-darkening effects somewhat. Finally, any effects of this kind will be most pronounced in the lowest part of the corona, which is also where we have to make corrections for collisional excitation using the MHD model. Clearly, the lower corona (below ≈1.3 R_⊙) is not as robustly probed by this particular R-DEM inversion compared to the outer corona. Since EUV DEMs already probe the lower corona regularly, the primary focus of this R-DEM inversion method is to infer the temperature distribution farther out in the corona, which is otherwise exceptionally difficult to measure.

A collection of RDEM(T_e ) inferences for the observed emission lines (see Section 2) is shown in the top panels of Figures 3 and 4 for samples from the poles and equator, respectively. Each panel contains multiple sets of R-DEMs at different helioprojective distances ranging from 1.2 to 2.2 R_⊙, in steps of 0.2 R_⊙, inside of an 8° by 0.18 R_⊙ window centered on the various latitudinal regions. The R-DEM curves are colored according to the distance range they were taken at, as shown in the legend. Every individual R-DEM inversion result inside each distance range and latitudinal direction is shown as faint lines, while the median average of all R-DEMs in the window is shown as bold lines in the figure.

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Each R-DEM is normalized by the integral of the R-DEM over all T_e bins to show the relative R-DEM distribution along the LOS rather than the density of the corona. The coronal density drops rapidly with distance, so for visualization, it makes more sense to present the results in this manner. These R-DEMs could be scaled by an inferred electron density of each region given a coronal density inversion using the polarized or total brightness of the K corona (a K corona map is shown in Figure 1). However, we are not concerned with the absolute density of the corona here; rather, we are interested in the relative T_e distribution throughout the corona.

To create a representative map of the R-DEM distributions, we took the weighted mean of the R-DEM from each LOS to infer an average T_e and the standard deviation (i.e., width) of the RDEM(T_e ) distribution around the mean. The resulting maps are shown in the left panels of Figure 5. We then show a series of radial traces of the R-DEM at different latitudes in Figure 6 and latitudinal traces at a series of radial distances in Figure 7.

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We find that the electron temperature generally varies from about 1.1 to 1.4 MK in the coronal holes, while the streamers are closer to 1.4–1.65 MK. The overall T_e values in the coronal holes rise considerably from around 1.1 MK at 1.1 R_⊙, up to almost 1.4 MK by about 1.4 R_⊙. The streamers have a similar behavior, where they also rise in T_e up to about 1.4 R_⊙. On the POS, the streamers have considerable spatial variation, with multiple distinguishable stalks, whereas the coronal holes have much less spatial variation. Below 1.4 R_⊙, there are a number of small cool plumes that extend into what appear to be closed field lines at the base of both the streamers and coronal holes.

The cores of the streamer have the highest T_e , especially in the eastern streamer, with values up to 1.65 MK. The western streamer is cooler in general, other than a closed loop and smaller stalk on the northwest side. The western streamer has a gap in the middle where the temperature is almost indistinguishable from the coronal holes, despite this region having higher electron densities (i.e., see the white-light image and K corona map in Figure 1). The width of the streamers also decreases considerably with height, occupying more than half of the corona at 1.2 R_⊙ but becoming only a very small portion by 2.6 R_⊙.

The width of the R-DEMs along each LOS indicates a large spread in the T_e throughout the corona (roughly ±0.1–0.3 MK) below about 1.4 R_⊙. However, beyond that distance, both the coronal holes and streamers appear almost entirely isothermal along the LOS with R-DEM widths less than ±0.05 MK (see Section 5.2).

Next, we test the effectiveness of the R-DEM procedure with the state-of-the-art MHD simulation of the PSI/MAS model prediction (Mikić et al. 2018) of this eclipse. Papers I and II used this MHD simulation to forward model the K corona and line emission, respectively, and in both cases, the model was found to rather accurately predict the eclipse observations. The forward-modeled line emission can now be used to test the validity of the R-DEM approach by comparing the R-DEM inferred from the forward-modeled lines to the true LOS T_e distribution directly from the model. Using the same procedure described in Section 3.1, examples of the R-DEM inversion results are shown in Figures 3–7. In the same figures, we include the actual R-DEM from the MHD model in order to test the inversion process. In the Appendix, we further discuss the details of the R-DEM distribution inside the PSI MHD model.

The inferred T_e distribution from the MHD model is very similar to the eclipse R-DEM in general, although the model-inferred R-DEM has somewhat more variation in T_e throughout the POS and along the LOS. The eastern streamer in particular has a large T_e variance on the southern edge, which persists beyond at least 2.8 R_⊙.

4.2.1. Testing the R-DEM Inversion

To test the new R-DEM methodology, we compare the results from the actual LOS T_e distribution in the PSI MHD model with the inversion results from the forward-modeled lines. The actual R-DEMs, showcased in Figures 3–7, are quite similar to the inversion results, though the actual R-DEM shows a substantially wider T_e distribution with more fine-scale structures (see the Appendix). Still, the average T_e value is rather well recovered by the R-DEM inversion. A direct comparison between the average T_e in the actual versus inferred model results is shown in the top-left panel of Figure 8, where the inversion result is 4.6% ± 7.2% higher than the actual R-DEM distribution. Further, while the inversion tends to underestimate the width of the R-DEM distribution, it is able to recover realistic R-DEM widths when the actual R-DEM is wider than about ±0.2 MK. Along the southeast streamer edge, for example, the R-DEM inversion correctly finds widths between ±0.3 and 0.4 MK.

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It is not surprising that the inversion is unable to resolve the fine-scale T_e structures and smaller R-DEM widths since it only used three emission lines that have a somewhat broad temperature response compared to the shape of the actual R-DEM distribution, which reduces the precision that is possible to achieve. This effect is similar to the isothermal bias caused by two-line ratios (Weber et al. 2005) and, for some DEMs, depends on the temperature response curves of the instrument (Guennou et al. 2012a, 2012b). Future observations of additional lines such as Fe xiii (1074.7 nm) would increase the temperature resolution of a similar R-DEM inversion. Thus, any inferred R-DEM widths below about 0.2 MK are not necessarily accurate in the inversions presented here. Instead, smaller widths should be considered as below that sensitivity limit.

The novelty of this work is not the finding of any particularly unexpected T_e values but rather that we have inferred the T_e values at a higher spatial resolution than has ever been achieved beyond the low corona (below 1.5 R_⊙) and that we have inferred the LOS variance of T_e (discussed in Section 5.2). EUV-based observations cannot constrain the electron temperature as far out as we have here, particularly in coronal holes where the signal is especially low. This high spatial resolution of the inference allows unique findings about both coronal holes and streamers.

We find the coronal holes to have electron temperatures of about 1.1–1.4 MK, where the values begin low, rise with helioprojective distance, then plateau at 1.4 MK by around 1.4 R_⊙. These values are somewhat higher than the estimate of 1.0–1.3 MK (for the same helioprojective distances) by Habbal et al. (1993), who reviewed a large number of earlier coronal hole T_e inferences using X-ray, EUV, and white-light data. These lower coronal hole T_e values are also consistent with many years of in situ charge state data of coronal holes (Smith et al. 2003; Habbal et al. 2010). One reason for this slight difference may not be due to actual differences in the electron temperature but rather due to changes in the ionic equilibrium estimates used to infer T_e . For example, the Fe ions used in this work had their equilibrium curves shifted to a higher temperature by approximately 0.1–0.15 MK in the more recent estimates in CHIANTI compared to older, commonly used computations such as Arnaud & Raymond (1992). These small changes in the ionic equilibrium estimates are sufficient to explain the small discrepancy between the coronal holes in this work compared to older results. In all cases, coronal holes are found to be rather isothermal (other than a slight increase in T_e up to 1.4 R_⊙) and consistent in the inferred T_e values regardless of the inference method or period of the solar cycle, so it seems reasonable that improved ionic calculations are the only variable in this case.

As for the streamers, we find considerable complexity in the structure of the T_e . Unlike the coronal holes, it is difficult to give a singular value of the average streamer temperature, as it varies significantly at high spatial resolution. Overall, the T_e values in streamers range from about 1.4 to 1.65 MK, and we describe some of the fine-scale features in Section 4.1. These T_e values are relatively consistent with streamer observations from Raymond et al. (1997), who used UVCS observations (mostly of collisional lines) to find a rough temperature of about 1.6 MK. However, they did not perform a full T_e inversion. Indeed, Raymond et al. (1997) used lines from several different elements, so the observed brightness is convolved with the elemental abundance, which will vary due to the first ionization potential (FIP) effect (see Laming 2015). In this work, we used three lines of Fe, which removes any FIP effect dependence on the T_e inference. Similarly, Ko et al. (2002) found T_e values near 1.5 MK in "quiet-Sun" streamers and higher-T_e plasma above active regions (3 MK), but there were no active regions on the Sun during this particular eclipse. Additionally, Ulysses ionic composition data yielded 1.3–1.8 MK near the solar equator at solar minimum (Smith et al. 2003), which can be assumed to have originated from coronal streamers similar to the ones seen in this eclipse.

The contrast between the east and west streamers is a great example of the importance of these observations, specifically in showcasing the diversity of streamer dynamics even near solar minimum when no active regions are present on the Sun. There are turbulent structures inside the eastern streamer visible in the white-light image (top panel of Figure 1; see Habbal et al. 2021), which correspond to the highest T_e values anywhere in the corona during this eclipse. The western streamer, on the other hand, has a core with a much lower T_e that is almost indistinguishable from the coronal holes despite having a significantly higher density. These very different streamer temperatures indicate that there are never truly "quiet-Sun" streamers but rather different levels of activity and dynamics that can create quite unique structures. Chitta et al. (2023) recently found similar fine-scale structures, complexity, and diversity of coronal streamers using white-light and EUV observations along with MHD modeling—albeit without any T_e measurement.

Both the coronal holes and streamers have a slight rise in T_e from the base of the corona out to about 1.4 R_⊙. The cause of this ubiquitous T_e rise is not entirely clear, but there are two likely effects that could be contributing to the observed effect. One explanation might be a temperature-dependent scale-height effect. Aschwanden & Nitta (2000) described how, for an unresolved combination of hydrostatic loops, the higher-temperature loops will have larger scale heights. Thus, higher-temperature structures will have a shallower density gradient and will increasingly dominate the emission at larger distances. The same principle should apply to hydrodynamic flows with different base temperatures as well. The other possibility is that the temperature rise may actually be a change in the T_e of the plasma as it flows outward. Simulations of coronal heating often show a rise in temperature regardless of the exact heating mechanism used (e.g., Habbal et al. 1995; Verdini et al. 2010; Matsumoto & Suzuki 2014). Additionally, as with all optically thin plasmas, there are LOS effects from overlapping structures that may change the inferred temperature depending on the exact structures present in the corona. Disentangling these effects is not a trivial endeavor and is beyond the scope of this work. Nevertheless, these observations provide constraints for benchmarking future coronal heating simulations.

One notable attribute of the eclipse R-DEM inversion results is that there is a large temperature spread (±0.2–0.35 MK) in the lower corona (i.e., the yellow lines at 1.2 R_⊙ in Figures 3, 4, and 7), with both much higher and lower T_e values than found with the two-line ratios (see Figure 1). This spread then disappears at higher distances. In fact, all of the eclipse R-DEMs converge to an almost isothermal T_e value (spread of <0.05 MK) by ≈1.4 R_⊙, which is highlighted by the low T_e standard deviation of the R-DEM (bottom-left panel of Figure 5).

The R-DEM inversion from the model has a somewhat wider T_e standard deviation throughout compared to the eclipse inversion, with the most dramatic spread occurring on the southern edge of the eastern streamer (see Figures 5 and 7). As discussed in Section 4.2.1, this three-line R-DEM inversion is not sensitive to distributions with widths less than about 0.2 MK but it is capable of detecting wider distributions. It is likely that if the actual distribution was significantly wider, then the R-DEM inversion should have detected it given its performance on the forward-modeled data. Our findings thus imply that in most of the corona, the emission-weighted temperature is near isothermal (to at least ±0.2) beyond 1.4 R_⊙, albeit with T_e differences between coronal holes and streamers on the POS. However, it is possible that an unknown systematic error related to photometric noise or other optical issues may be suppressing the inferred R-DEM width, especially since we do not have a high T_e precision with only three emission lines. Nevertheless, the isothermal nature of the corona appears at quite a low helioprojective distance where the line-emission signals were strong, and the MHD inversion results were able to recover wide R-DEMs in the eastern streamer out to 2.8 R_⊙.

The eclipse R-DEM inversion is indicating that (at the time of this eclipse) the corona below about 1.4 R_⊙ was a mixture of various structures along the LOS, which have a wide range of temperatures (i.e., streamers and coronal hole plumes overlapping on the POS), which is expected near the Sun given the wide range of T_e found in DEMs that have been achieved with EUV and X-ray observations (e.g., Pottasch 1964; Morgan & Taroyan 2017). There could also be scale-height effects contributing to this inference, as discussed in Section 5.1.

Beyond 1.4 R_⊙, the outer corona becomes an almost isothermal plasma at the root of the solar wind. This finding supports the recent work of Habbal et al. (2010, 2021), who found similar behavior in the corona and solar wind regardless of the phase of the solar cycle. We do find some small T_e standard deviation between the coronal structures but that spread is entirely on the plane of sky (POS) rather than along each LOS, and the majority of the corona becomes increasingly dominated by the highly isothermal coronal holes as the corona transitions into the solar wind.

Similar findings were reported by Del Zanna et al. (2023), who performed a collection of DEM and two-line ratio T_e inferences from data taken during this same eclipse. In particular, they found a rather isothermal T_e in the western streamer at 1.08 R_⊙ and a wider DEM at the same distance in the eastern streamer. The Del Zanna et al. (2023) analysis incorporated infrared spectra from AIR-Spec (Samra et al. 2022a, 2022b), an airborne spectrometer flown during the eclipse, finding rather consistent two-line T_e values for a number of infrared and EUV lines, supporting the concept that the corona was largely isothermal. However, that study was limited in its spatial coverage and elongation compared to the results presented here. Future work should incorporate additional lines as well as data from different phases of the solar cycle to better explore the behavior of the coronal R-DEM.

Since the mean of the R-DEM inversion with the forward-modeled MHD lines is a close fit to the mean of actual temperature distribution in the model (see Section 4.2.1), we can reliably use the R-DEM results from the eclipse data to benchmark the PSI MHD simulation itself. The top middle and right panels of Figure 8 show direct comparisons between the eclipse-inferred R-DEM average T_e with the model R-DEM inversion (top middle) and the actual T_e mean in the model (top right). Both comparisons are remarkably similar, with the eclipse-inferred mean being somewhat higher than the model mean for the majority of LOSs. Specifically, the eclipse inversion is 7.6% ± 12.5% higher than the model inversion and 9.9% ± 14.0% higher than the actual model mean T_e . These comparisons indicate that the model is very close to reproducing the correct temperature distribution, with perhaps a slight underestimation of T_e on average. The model-inferred R-DEM also has slightly more temperature variability throughout the corona than the eclipse data, which is essentially isothermal at each distance (see Figure 7). The variability is especially notable in the coronal holes, where the model has a significant peak at the north pole (about 1.4 MK) compared to the minima in the northeast (1.1 MK) and southwest (1 MK). These small differences between the model and eclipse data are consistent with Paper II, which found that the Fe xi and Fe xiv lines were brighter in the coronal holes than the model predicted as well. Those differences were interpreted as an underheating of the polar magnetic field lines in the wave-turbulence-driven (WTD) heating model or possibly some other limitations in the MHD modeling approach.

The streamer core temperatures in the model inversion are generally close to the values in the eclipse data; however, the width and location of the streamer stalks are somewhat different. The eastern streamer in particular has a much larger latitudinal extent in the model and has two distinct peaks of T_e whereas the eclipse data only has one prominent peak with a few smaller stalks on the southern side. The model inversion also predicts that the eastern streamer T_e should decrease considerably with distance from the Sun, whereas the eclipse inversion maintains a relatively similar temperature out to at least 2.2 R_⊙. Both the model and the data show the width of the eastern streamer decreasing considerably at larger distances.

The MHD model makes even better predictions about the behavior of the western streamer, both in terms of the overall T_e values and a drift of the center of the streamer stalk from the northwest toward the south from 1.2 out to 2.2 R_⊙. Similar to the eastern streamer, the model does somewhat overestimate the width of the western streamer. Even though the trends in the western streamer are similar between the model and data, the model has the precise latitudinal location of the streamer somewhat farther south than in the eclipse data. This small difference in latitude may be explained by slight inaccuracies in the magnetic boundary conditions of the model, which incorporate synoptic magnetic measurements from the Sun–Earth line and estimates of the polar magnetic flux, both of which are imperfect proxies for the full-Sun–surface-flux distribution at a given instant of time (e.g., Riley et al. 2019).

The R-DEM mean T_e maps are rather similar to both the Fe xiv/Fe x and Fe xiv/Fe xi two-line ratio inferences of the T_e (bottom-right panels of Figure 1) for both the eclipse data and the MHD model. A collection of direct comparisons between both the model and eclipse two-line ratio inferences to their respective R-DEM means are shown in Figure 8. The two-line ratio inferences from the forward-modeled lines match the MHD average temperature quite well, with Fe xiv/Fe x (Fe xiv/Fe xi) returning a T_e value that is 6.7% ± 9.0% (6.5% ± 9.4%) higher than the actual LOS average (bottom-left panels). Similarly, the Fe xiv/Fe x (Fe xiv/Fe xi) ratio inference finds an average value that is 2.1% ± 5.1% (1.9% ± 6.0%) higher than the full R-DEM inversion of the MHD model (bottom-middle panels). In all cases, the average values are within the scatter of the collection of all LOS in the data set. As for the eclipse observations, the Fe xiv/Fe x T_e average is 1.3% ± 2.7% higher than the mean R-DEM inversion values (middle-right panel), whereas the Fe xiv/Fe xi T_e average is 1.2% ± 4.7% lower than the R-DEM mean (bottom-right panel).

Based on these comparisons, it is clear that the R-DEM inversion is able to better reproduce the actual LOS average temperature than the two-line ratios, but the simple two-line ratios are accurate to better than 10%. So, older eclipse data with only two lines can still be reliably used to infer the average LOS T_e (e.g., Boe et al. 2020a). Still, there are tails of points at lower T_e in the two-line comparisons that diverge from the R-DEMs (i.e., the left side of the distributions in the middle and bottom panels of Figure 8), which correspond to the low regions of the corona that had a high standard deviation of the R-DEM T_e distribution. These low helioprojective regions are very likely a mix of colder open-field regions and hotter closed-field regions overlapping on the LOS, as illustrated by the R-EM of the PSI MHD model (see Figure 9 in the Appendix). These more complicated regions are not well resolved by a simple two-line average. Fortunately, this effect is rather confined to the lower corona (below 1.3 R_⊙). These lower regions of the corona are also where there is a considerable amount of collisional excitation in these lines. We isolated the radiative component of the observed brightness based on the collisional excitation fraction predicted by the MHD model (see Section 4.1; Paper II), but there could be uncertainties or systematic errors induced by this procedure. Future work should include observations of collisionally excited lines, such as the Fe xiii 1079.8 nm line (in addition to the radiative Fe xiii 1074.7 nm line), which could be used to observationally isolate the collisional excitation component without the need for the MHD model to perform the R-DEM inversion.