# The Solar Minimum Eclipse of 2019 July 2. III. Inferring the Coronal $T_{e}$
with a Radiative Differential Emission Measure Inversion

Benjamin Boe  Institute for Astronomy, University of Hawaii, Honolulu, HI
96822, USA  Cooper Downs  Predictive Science Inc., San Diego, CA, 92121, USA
Shadia Habbal  Institute for Astronomy, University of Hawaii, Honolulu, HI
96822, USA Benjamin Boe<EMAIL_ADDRESS>
###### Abstract

Differential Emission Measure (DEM) inversion methods use the brightness of a
set of emission lines to infer the line-of-sight (LOS) distribution of the
electron temperature ($T_{e}$) in the corona. DEM inversions have been
traditionally performed with collisionally excited lines at wavelengths in the
Extreme Ultraviolet (EUV) and X-ray. However, such emission is difficult to
observe beyond the inner corona (1.5 $R_{\odot}$), particularly in coronal
holes. Given the importance of the $T_{e}$ distribution in the corona for
exploring the viability of different heating processes, we introduce an analog
of the DEM specifically for radiatively excited coronal emission lines, such
as those observed during total solar eclipses (TSEs) and with coronagraphs.
This Radiative DEM (R-DEM) inversion utilizes visible and infrared emission
lines which are excited by photospheric radiation out to at least 3
$R_{\odot}$. Specifically, we use the Fe x (637 nm), Fe xi (789 nm), and Fe
xiv (530 nm) coronal emission lines observed during the 2019 July 2 TSE near
solar minimum. We find that despite a large $T_{e}$ spread in the inner
corona, the distribution converges to an almost isothermal yet bimodal
distribution beyond 1.4 $R_{\odot}$, with $T_{e}$ ranging from 1.1 to 1.4 in
coronal holes, and from 1.4 to 1.65 MK in quiescent streamers. Application of
the R-DEM inversion to the Predictive Science Inc. magnetohydrodynamic (MHD)
simulation for the 2019 eclipse validates the R-DEM method and yields a
similar LOS Te distribution to the eclipse data.

Solar corona (1483), Solar eclipses (1489), Solar coronal streamers (1486),
Solar cycle (1487), Solar coronal holes (1484), Solar optical telescopes
(1514)

††journal: ApJ

##  1 Introduction

Differential Emission Measure (DEM) inversion techniques combine observations
of multiple emission lines to infer the line-of-sight (LOS) distribution of
the electron temperature ($T_{e}$) in an optically thin plasma, such as the
solar corona. DEM inversions are commonly used to explore coronal
thermodynamics with emission lines at X-Ray (e.g. Pottasch 1964; Kepa et al.
2006, 2022) and Extreme Ultra Violet (EUV; e.g. Withbroe 1978; Hannah & Kontar
2012; Cheng et al. 2012; Cheung et al. 2015) wavelengths. EUV DEMs are a
useful tool for inferring $T_{e}$ in active regions (Aschwanden et al., 2015),
and the solar cycle dependence of $T_{e}$ in the corona (Morgan & Taroyan,
2017). X-ray DEMs have even been used to study the temperature of stellar
coronae (Güdel et al., 2001).

A significant limitation of DEM inversion analysis based on X-Ray and EUV
lines is that the lines are predominantly collisionally excited, which means
the emission is proportional to the density squared, and so they decline in
brightness rapidly at larger helioprojective distances. There can be radiative
excitation of EUV lines but that requires flux from the same collisionally
excited line lower down in the corona (the solar disk does not emit any
meaningful amount of EUV photons). Recent developments have proven that EUV
lines can be detected out to 2 $R_{\odot}$ with Proba-2/SWAP (Goryaev et al.,
2014) and 2.5 $R_{\odot}$ with GOES/SUVI (Seaton et al., 2021), but the EUV
lines are difficult to observe beyond 1.5 $R_{\odot}$, and the relative
contribution of resonant scattering is not as well characterized since its
source brightness (i.e., emission from the low corona) has spatial and
temporal variation. The visible and infrared coronal lines, on the other hand,
can be radiatively excited by photospheric emission and in turn, can often be
observed at helioprojective distances of up to 3.4 $R_{\odot}$ or more (see
Boe et al. 2022).

Another drawback of DEM inversion performed specifically with bandpasses in
the EUV is that multiple lines are present in each bandpass and so the
observed brightness leads to complex temperature response functions (e.g.,
O’Dwyer et al. 2010; Boerner et al. 2012). Solving a DEM with EUV observations
is therefore rather challenging; it requires non-trivial inversion methods
(e.g. Hannah & Kontar 2012; Guennou et al. 2012a, b; Aschwanden et al. 2015;
Morgan & Pickering 2019) and clever techniques to reduce the computationally
expensive operation (e.g., Kashyap & Drake 1998; Plowman et al. 2013; Cheung
et al. 2015; Pickering & Morgan 2019). (See Del Zanna & Mason (2018) for a
detailed overview of the history of DEM inversion methods and applications)

Outside of EUV and X-ray DEM inversions, the coronal electron temperature has
been inferred with emission line ratios at wavelengths in the ultraviolet from
space (e.g., Raymond et al. 1997; Ko et al. 2002) and in the visible and
infrared during total solar eclipses (e.g., Boe et al. 2020a, 2022; Del Zanna
et al. 2023). Without emission lines, the effective temperature has been
inferred using the density profile inferred from polarized brightness
measurements (e.g., Munro & Jackson 1977). However, the effective temperature
is a combination of the electron and proton temperatures, which will follow
significantly different profiles (see Esser et al. 1997). Isolating the
electron and proton temperatures using continuum-based analysis is thus not
possible without other data or assumptions (e.g., Doyle et al. 1999). The
electron temperature of the corona can also be deduced from in situ
measurements of ionic composition (e.g., Ko et al. 1996, 1997; Smith et al.
2003; Habbal et al. 2010; Landi & Testa 2014) since the ions will cease charge
exchange processes at some distance from the Sun (i.e., they freeze-in their
ionic state; see Boe et al. 2018; Gilly & Cranmer 2020). Unfortunately, any in
situ driven analysis lacks precise information about the coronal structure
(and freeze-in distance) that each solar wind parcel originated from. None of
these methods have provided a complete survey of the distribution of the
electron temperature throughout the corona, and in all but a few cases are
only able to probe very small portions of the corona. Only visible and
infrared emission line observations have shown the potential to quantify the
LOS $T_{e}$ distribution throughout the entire middle corona from below 1.5
$R_{\odot}$to perhaps as much as 6 $R_{\odot}$.

In this paper, we implement a new Radiative-DEM method, which is introduced
for lines that are radiatively excited (see Section 3). The R-DEM is applied
to observations of the Fe x (637.5 nm), Fe xi (789.2 nm), and Fe xiv (530.3
nm) emission lines from the 2019 total solar eclipse (TSE; see Section 2). We
also implement the R-DEM method on forward-modeled line emission from the
Predictive Science Inc. (PSI) Magnetohydrodynamic (MHD) simulation (see
Section 4.2) and compare the inversion results to the actual $T_{e}$
distribution along the LOS in the MHD model. We discuss the details of the
findings, including comparisons between the eclipse and model inferences and
actual LOS $T_{e}$ distribution in Section 5. Concluding remarks are given in
Section 6.

##  2 2019 July 2 Total Solar Eclipse

To explore the $T_{e}$ distribution in the corona with a new Radiative-DEM
inversion procedure (see Section 3), we use visible observations of the
brightness of Fe x (637.5 nm), Fe xi (789.2 nm) and Fe xiv (530.3 nm), which
were acquired during the 2019 July 2 TSE in Rodeo, Argentina. This eclipse
occurred very close to solar minimum when the corona was dominated by a
dipolar field component, as showcased by the white light eclipse image and the
magnetic field line extrapolation of the PSI MHD model (see Section 4.2) in
the top panels of Figure 1. The brightness of the emission lines was acquired
with narrowband telescope systems that have been deployed at a number of
recent TSEs (see Habbal et al. 2021; Boe et al. 2020a, 2018).

The temperature responses of the Fe x, Fe xi, and Fe xiv lines span coronal
electron temperatures ($T_{e}$) from 0.8 to 2.5 MK. The ionic equilibrium
curves that we use in this work are shown in the top panel of Figure 2. The
curves are generated from version 10 of the CHIANTI database (Dere et al.,
1997; Del Zanna et al., 2021), which have been interpolated from their
recorded spacing of $\Delta$ log(K) = 0.05. Higher $T_{e}$ plasma can exist in
active regions below 1.2 $R_{\odot}$, but the vast majority of coronal plasmas
that escape into the solar wind are found to have a coronal temperature
between 1 and 2 MK (Habbal et al., 2021). Further, this eclipse occurred near
solar minimum when there were no active regions on the Sun.

Figure 1: Top left: White light image of the 2019 TSE (first presented in Boe
et al. 2020b). Top right: PSI MHD model field line prediction of the eclipse
(see Section 4.2). The bottom two rows contain a collection of inferred (and
modeled) coronal properties in a cartesian representation of polar coordinates
from Boe et al. (2021, 2022). Bottom left: K corona brightness from the
eclipse (middle) and PSI MHD model (bottom). Bottom middle: Inferred ionic
density ratio of Fe x/Fe xi (middle) and Fe xiv/Fe xi (bottom). Bottom right:
Inferred $T_{e}$ from the Fe xiv/Fe x (middle) and Fe xiv/Fe xi (bottom) ionic
density ratios.  Figure 2: Top: Ionic equilibrium curves of Fe x, Fe xi, and
Fe xiv versus $T_{e}$ with values interpolated from CHIANTI. Bottom: Ionic
equilibrium density ratios vs $T_{e}$ from the curve above.

To isolate the line emission, we use one telescope with a bandpass centered on
the wavelength of the emission line, and another about 1 to 3 nm away from the
line to observe the background continuum brightness. The set of continuum
observations from this eclipse were used by Boe et al. (2021) (Paper I) to
study the K (electron) and F corona (dust) brightness using a novel color-
based inference method. The calibration for the continuum data was leveraged
off the Mauna Loa Solar Observatory’s (MLSO) K-coronagraph (K-Cor)111K-Cor
DOI: 10.5065/D69G5JV8; https://mlso.hao.ucar.edu/mlso_data_calendar.php
polarized brightness data from the same day as the eclipse. The inferred K
corona brightness from Paper I can be found in the middle left panel of Figure
1. Boe et al. (2022) (Paper II) then extended the calibration to the
simultaneous line emission observations to isolate the brightness of Fe x, Fe
xi, and Fe xiv.

In Paper II, we inferred the LOS integrated ionic column density ratio which
we will use in this work to perform the R-DEM inversion. Specifically, we
isolated the radiative component of the line emission observations by removing
the fraction of brightness that originated from collisional excitation based
on the PSI MHD model (see Section 4.2). The radiative line ratios were then
used to infer the integrated column density ratios of Fe x/Fe xi and Fe xiv/Fe
xi after accounting for the atomic radiation processes of each ion as well as
the effect of variable line-widths integrated over each bandpass at different
helioprojective distances (based on previous observations). Note that since
these ratios are integrated along the same LOS, they are equivalent both to
the column density ratio as well as the average volumetric density ratio along
each LOS.

The final inferred ionic density ratios are shown in the bottom middle panels
of Figure 1. These ionic density ratio maps clearly show that Fe xi is the
most abundant ion of the three. Fe x has a similar density in the low corona
(below 1.3 $R_{\odot}$) but declines relative to Fe xi at higher
helioprojective distances. As for the Fe xiv ion, it has a similar density to
Fe xi in the cores of the coronal streamers but is otherwise significantly
less abundant than Fe xi. Paper II then took these ionic equilibrium density
ratios and inferred $T_{e}$ in the corona, using the correspondence between
the ionic equilibrium and $T_{e}$ (i.e., the curves shown in the bottom panel
of Figure 2). The Fe xi/Fe x ratio in particular did not return a realistic
average $T_{e}$, likely due to the very shallow slope of the density ratio vs
$T_{e}$ curve. The other two $T_{e}$ inferences, from the Fe xiv/Fe x and Fe
xiv/Fe xi ionic density ratios, are shown in the bottom right panels of Figure
2. In this work, we implement a new Radiative-DEM procedure that uses the
corrected Fe x/Fe xi and Fe xiv/Fe xi inferred density ratios to infer the LOS
distribution of $T_{e}$ in the corona.

##  3 Radiative Differential Emission Measure

###  3.1 R-DEM Formalism

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

$\displaystyle DEM(T_{e})=N_{e}^{2}\frac{dh}{dT_{e}},$ (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, $\beta_{i}$, 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, $C_{i}(T_{e},N_{e},\lambda_{i})$,
multiplied with the DEM function. The brightness is then:

$\displaystyle\beta_{i}=Ab(Z)\int_{T_{e}}C_{i}(T_{e},N_{e},\lambda_{i})\
DEM(T_{e})\ dT_{e}.$ (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 Radiative-DEM (R-DEM), which
will replace the $DEM(T_{e})$ term in Equation 2, where

$\displaystyle RDEM(T)=N_{e}\frac{dh}{dT},$ (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., $C_{i}(T_{e},\lambda_{i}$)). 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 Fig. 2). The brightness of a radiatively excited line is then:

$\displaystyle\beta_{i}=Ab(Z)\int_{T_{e}}C_{i}(T_{e},\lambda_{i})\
RDEM(T_{e})\ dT_{e}.$ (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 $\Delta T$ centered
on temperature $T_{0}$,

$\displaystyle REM(T)=\int_{T_{0}-\frac{\Delta T}{2}}^{T_{0}+\frac{\Delta
T}{2}}RDEM(T)\ dT,$ (5)

which represents the amount of emitting plasma at the specified temperature
range in units of cm-2.

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,

$\displaystyle\frac{n_{j}}{n_{k}}=\frac{\beta_{j}\ A_{k}\ g_{l,j}\ g_{u,k}\
{\nu_{k}}^{3}}{\beta_{k}\ A_{j}\ g_{u,j}\ g_{l,k}\ {\nu_{j}}^{3}},$ (6)

where, for ions $j$ and $k$, $n$ is the number density, $\nu$ 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 $\beta$ 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 $\approx 1.5$ $R_{\odot}$). 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,

$\displaystyle\frac{\beta_{j}}{\beta_{k}}$ $\displaystyle=\frac{n_{j}\ A_{j}\
g_{u,j}\ g_{l,k}\ {\nu_{j}}^{3}}{n_{k}\ A_{k}\ g_{l,j}\ g_{u,k}\
{\nu_{k}}^{3}}$ $\displaystyle=\frac{Ab(Z_{j})\int C_{j}(T_{e},\lambda_{j})\
RDEM(T_{e})\ dT_{e}}{Ab(Z_{i})\int C_{k}T_{e},\lambda_{k})\ RDEM(T_{e})\
dT_{e}},$

which reduces to,

$\displaystyle\frac{n_{j}}{n_{k}}=\frac{Ab(Z_{j})\int C^{\prime}_{j}(T_{e})\
RDEM(T_{e})\ dT_{e}}{Ab(Z_{k})\int C^{\prime}_{k}(T_{e})\ RDEM(T_{e})\
dT_{e}}.$ (7)

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 ratio of the ionic density ratios have 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.

###  3.2 R-DEM Inversion Procedure

In this work, we will infer the R-DEM with the Fe x, Fe xi, and Fe xiv line
observations. We thus apply Equation 7 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 7). 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.minimize222https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.minimize.html#scipy.optimize.minimize
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 7 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 $\approx$ 1.3
$R_{\odot}$) 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.

##  4 Results

###  4.1 Eclipse R-DEM

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 $R_{\odot}$, in
steps of 0.2 $R_{\odot}$, inside of an 8-degree by 0.18 $R_{\odot}$ 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.

Figure 3: Collection of R-DEMs for the coronal holes at the poles of the Sun.
The top panel contains the white light image of the corona with a set of
windows identified (8 degrees by 0.18 $R_{\odot}$, steps of 0.2 $R_{\odot}$).
The bottom panels show the R-DEM results for the North Pole (left) and South
Pole (right) where the top panels are from the eclipse data (see Section 4.1),
the middle panels are from the MHD forward modeled lines (see Section 4.2),
while the bottom panels are the actual R-DEMs in the MHD model (see Section
4.2.1). Every R-DEM inside each window is shown as a bold line, while the
variance of the R-DEM distributions inside the window is shown as the shaded
bands around each R-DEM curve.  Figure 4: Same as Figure 3, except for windows
centered on the streamers at the East (left) and West (right) equator. The
white light image has the same orientation as Figure 1.

Each R-DEM is normalized by the integral of the R-DEM overall $T_{e}$ 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.

Figure 5: Top left: Mean $T_{e}$ from the R-DEM inversion from the eclipse
lines. Bottom left: Standard deviation of the R-DEM around the mean $T_{e}$.
Middle: Same as left, for the MHD forward modeled R-DEM inversion. Right: Same
as left, for the actual R-DEM distribution in the MHD model.  Figure 6: Radial
traces of the R-DEMs from the eclipse data (solid red line), inferred from the
MHD model (dashed blue line), and the actual MHD model (dotted black line).
The traces are taken from the median average inside a 15 degree wedge centered
on the cardinal direction indicated in the title of each panel. The filled
bands show the 1-$\sigma$ standard deviation of the R-DEM distributions within
the wedge.  Figure 7: Latitudinal traces of the R-DEM median (solid lines) and
standard deviation (shaded bands) at a selection of distances from the eclipse
data (top panel), inferred from the MHD model (middle panel), and the actual
R-DEM in the MHD model (bottom panel). The traces are taken as the median
average R-DEM at a set of fixed helioprojective distances within a band of 0.1
$R_{\odot}$ of each distance (i.e., $\pm 0.05$ $R_{\odot}$).

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 to 1.65 MK. The
overall $T_{e}$ values in the coronal holes rise considerably from around 1.1
MK at 1.1 $R_{\odot}$, up to almost 1.4 MK by about 1.4 $R_{\odot}$. The
streamers have a similar behavior, where they also rise in $T_{e}$ up to about
1.4 $R_{\odot}$. 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 $R_{\odot}$, 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 $R_{\odot}$, but becoming only a
very small portion by 2.6 $R_{\odot}$.

The width of the R-DEMs along each LOS indicates a large spread in the $T_{e}$
throughout the corona (roughly $\pm$ 0.1 to 0.3 MK) below about 1.4
$R_{\odot}$. However, beyond that distance, both the coronal holes and
streamers appear almost entirely isothermal along the LOS with R-DEM widths
less than $\pm$ 0.05 MK (see Section 5.2).

###  4.2 PSI MHD R-DEM

Next, we test the effectiveness of the R-DEM procedure with the state-of-the-
art Magnetohydrodynamic (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 Appendix A, 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 $R_{\odot}$.

####  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
substantially wider $T_{e}$ distribution with more fine-scale structures (see
Appendix A). 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
vs inferred model results is shown in the top left panel of Figure 8, where
the inversion result is $4.6\%$ $\pm$ $7.2$ 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 $\pm 0.2$ MK. Along the southeast streamer
edge, for example, the R-DEM inversion correctly finds widths between $\pm$
0.3 and 0.4 MK.

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 which 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 depending on the temperature
response curves of the instrument (Guennou et al., 2012a, b). 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.

Figure 8: Scatter plot comparisons between the various R-DEM and two line
$T_{e}$ results from the eclipse, the PSI MHD forward-modeled lines and the
actual LOS average of $T_{e}$ in the model.

##  5 Discussion

In Section 3.1, we introduced the radiative analog of a DEM, called an R-DEM.
We then applied this method in Section 4.1 to the Fe x, Fe xi, and Fe xiv data
from the 2019 TSE (see Section 2), as well as to the forward-modeled emission
lines from the PSI MHD prediction for this eclipse in Section 4.2. We compared
the inferred model R-DEM distribution to the actual distribution of the MHD
model in Section 4.2.1 to verify the accuracy of the inversion procedure. Now,
we will discuss the R-DEM inversion results from both the eclipse and model
and compare them to the actual LOS $T_{e}$ distribution in the model.

###  5.1 The $T_{e}$ of Coronal Holes and Streamers

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 $R_{\odot}$), 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 to 1.4
MK, where the values begin low, rise with helioprojective distance, then
plateau at 1.4 MK by around 1.4 $R_{\odot}$. These values are somewhat higher
than the estimate of 1.0 to 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 to 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 $R_{\odot}$) 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 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
to 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 $R_{\odot}$. 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 more shallow 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
which 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.

###  5.2 On the Isothermality of the Middle Corona

One notable attribute of the eclipse R-DEM inversion results is that there is
a large temperature spread ($\pm$ 0.2 to 0.35 MK) in the lower corona (i.e.
the yellow lines at 1.2 $R_{\odot}$ 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 $\approx$ 1.4 $R_{\odot}$, which is highlighted by the low
$T_{e}$ standard deviation of the R-DEM (bottom left panel of Fig. 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 $\pm 0.2$) beyond 1.4 $R_{\odot}$, 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 $R_{\odot}$.

The eclipse R-DEM inversion is indicating that (at the time of this eclipse)
the corona below about 1.4 $R_{\odot}$ 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 which 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 $R_{\odot}$, 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 $R_{\odot}$, 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, b), 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.

###  5.3 Testing the PSI MHD Simulation

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 $7.6\%$ $\pm$ $12.5$
higher than the model inversion, and $9.9\%$ $\pm$ $14.0$ 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 under-
heating of the polar magnetic field lines in the Wave Turbulence Driven
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 large 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 $R_{\odot}$. 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 towards the south from 1.2 out
to 2.2 $R_{\odot}$. 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 further 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).

###  5.4 Comparison to Two-Line Inferences

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 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
$6.7\%$ $\pm$ $9.0$ ($6.5\%$ $\pm$ $9.4$) 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 $2.1\%$ $\pm$ $5.1$ ($1.9\%$ $\pm$
$6.0$) 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 dataset. As for the eclipse observations, the Fe
xiv/Fe x $T_{e}$ average is $1.3\%$ $\pm$ $2.7$ higher than the mean R-DEM
inversion values (middle right panel), whereas the Fe xiv/Fe xi $T_{e}$
average is $1.2\%$ $\pm$ $4.7$ 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 Appendix A). 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 $R_{\odot}$). 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.

##  6 Conclusions

We have introduced the analog of a Differential Emission Measure for
radiatively excited visible emission lines in the corona, which we call an
R-DEM. The R-DEM methodology required a slight modification from traditional
DEMs made with collisionally excited lines in the EUV and X-rays (see Section
1), which we detailed in Section 3.

We then applied the new R-DEM method to data from the 2019 total solar eclipse
observations of Fe x, Fe xi, and Fe xiv (see Section 2) as well as to the same
forward-modeled lines from the PSI MHD model for this eclipse (see Section
4.2). We then compared the actual LOS distribution of $T_{e}$ in the model to
the inferred R-DEM, finding that the R-DEM method with only three emission
lines was able to accurately reproduce the average $T_{e}$ values in the model
(see Section 4.2.1). Moreover, the R-DEM method was able to recover the width
of the distribution in regions of the PSI MHD model that had a LOS $T_{e}$
standard deviation greater than about 0.2 MK but was unable to distinguish the
$T_{e}$ distribution over finer scales (see Section 5.2). Likely, this effect
is caused by the broad $T_{e}$ responses of these lines, and the limit of only
three used in the inversion. Future studies should incorporate additional
lines to improve the temperature resolution of the R-DEM.

Given the demonstrated ability of this three-line R-DEM inversion to recover
wide $T_{e}$ distributions, it is interesting that the eclipse R-DEM results
indicate that most of the corona has a very small standard deviation of the
LOS $T_{e}$. In the low corona, below about 1.3 $R_{\odot}$, there are wide
distributions in the $T_{e}$ values, but beyond that distance, the corona
seems effectively isothermal. The coronal hole plasma is remarkably isothermal
at about 1.1 to 1.4 MK, while the quiescent streamers at the equator are
closer to 1.4 to 1.65 MK. There is clearly variation in the temperature
between the quiescent streamers and the coronal holes, but the LOS variation
of each structure is negligible.

This finding of near-isothermality supports the work of Habbal et al. (2010,
2021), who found that both the coronal and solar wind plasmas are strongly
weighted to the charge state of Fe xi, which corresponds to about 1.3 MK.
These results are also broadly consistent with Del Zanna et al. (2023), who
performed DEM and two-line $T_{e}$ inferences during the same eclipse with EUV
and airborne Infrared observations (Samra et al., 2022a). These consistent
findings provide strong constraints for models of the solar wind.
Specifically, that relatively isothermal temperatures can be used for solar
wind model initialization outside of the lower corona (above $\approx$ 1.4
$R_{\odot}$). Combining solar wind models and in situ ionic composition data
with this R-DEM inversion method, applied to observations of visible and
infrared coronal emission lines, could have immense potential for advancing
our understanding of coronal heating and the exact nature of the link between
the corona and solar wind.

The configuration of the corona during the 2019 eclipse, in particular, is an
example of a near-perfect solar minimum corona. Future work could use this
R-DEM method to test if there is a larger LOS $T_{e}$ standard deviation
during periods closer to solar maximum, or if the corona remains essentially
isothermal. R-DEM inversions would also be highly valuable when the corona has
been disturbed by a coronal mass ejection. For example, Boe et al. (2020a)
found changes in the LOS average $T_{e}$ due to a passing CME during the 2017
eclipse with only a 2-line $T_{e}$ inversion, but was unable to probe the LOS
behavior with only two emission lines. Similar observations with three or more
lines could be used to probe the complete temperature structure of an erupting
CME in the middle corona (between 1.5 to 6 $R_{\odot}$).

The comparison between the R-DEM inversions of the eclipse and PSI MHD model
also enabled a test of the model predictions. Similar tests had already been
performed by Boe et al. (2021) and Boe et al. (2022) for this specific
simulation using the inferred K corona and two-line $T_{e}$ inferences
respectively. In those papers as well as in this work (i.e., Section 5.3), the
PSI MHD simulation is found to be an excellent match to the eclipse data.
There is perhaps a very slight underestimation of the average $T_{e}$ values
by the model, but the differences are typically less than $10\%$. Still, these
slight differences might point towards additional areas of improvement for
coronal heating formulation in the MHD model. Further, the small discrepancies
between the MHD model and eclipse data highlight the complexity of fine-scale
structures in the corona that the model is not yet able to perfectly match.
Regardless, the accuracy of the MHD simulation predictions is excellent.

The two-line ratio $T_{e}$ inferences (from Boe et al. 2022) were then
compared to the R-DEM inversions of both the eclipse and MHD model (see
Section 5.4). In all cases, the inferred $T_{e}$ from both the Fe xiv/Fe x and
Fe xiv/Fe xi ratios are able to rather accurately infer the correct average
R-DEM $T_{e}$, except for lower regions of the corona (below 1.3 $R_{\odot}$)
where there is wider temperature distribution including a mixture of both
colder ($<$1 MK) and hotter ($>$1.5 MK) plasmas along the LOS. Thus,
observations with only Fe xiv and either Fe x or Fe xi can still be reliably
used to infer the average LOS temperature.

As demonstrated in this work, visible and infrared emission lines are powerful
tools for inferring the physical state of the corona out to at least 2.8
$R_{\odot}$. In Boe et al. (2022), the Fe xi line was observed as far as 3.4
$R_{\odot}$, a distance limited by the size of the detector rather than by a
loss of signal. At future eclipses, instruments should be optimized to observe
these emission lines to significantly larger helioprojective distances, which
would be incredibly valuable for studying the formation and evolution of the
solar wind in the corona, especially since these ions should freeze-in well
below 5 $R_{\odot}$ (Boe et al., 2018; Gilly & Cranmer, 2020). It may be
possible to measure visible line emission out to 10 $R_{\odot}$ during
eclipses, which could be directly compared to spacecraft plasma measurements
from future perihelia of the Parker Solar Probe (Fox et al., 2016). Indeed,
Antonucci et al. (2023) recently demonstrated with Metis on Solar Orbiter that
the UV H i Ly $\alpha$ emission can be detected beyond 6 $R_{\odot}$. The
eclipse-based observations thus continue to provide a key tool for studying
the plasma properties of the corona and nascent solar wind.

The R-DEM method introduced here could be applied beyond eclipse observations
as well. The new data from the ground-based UCoMP coronagraph (Tomczyk et al.,
2021) in particular, would be well suited for an application of this R-DEM
method. UCoMP can observe the same, and several additional, visible and near-
infrared lines. With a wide selection of emission lines, the R-DEM method
should be able to infer the $T_{e}$ distribution below $\approx 1.5$
$R_{\odot}$. Additionally, the R-DEM method may be applied to the high
spatial-resolution coronal observations by DKIST (Rimmele et al., 2020).
Studies with DKIST and UCoMP in the lower corona could be used to explore the
effect of collisional excitation on the R-DEM inversion method (see Section
4.1), especially since both will observe the collisionally sensitive Fe xiii
doublet (see Section 5.4). While DKIST is not as well suited for studying the
formation of the solar wind compared to the eclipse observations, it could
infer the R-DEM at exceptionally high spatial resolution in prominence
cavities, polar plumes, jets, CMEs, etc., which could be of high value to
multiple areas of solar physics.

Finally, recent demonstrations of the radiative self-excitation of the 17.1 nm
and 19.3 nm EUV lines beyond 2 $R_{\odot}$ (Seaton et al., 2021) suggest that
future work could even apply this R-DEM technique to EUV lines farther out in
the corona – albeit after accounting for the complicated resonant scattering
process driven by the collisional excitation of the same line lower down in
the corona. Comparisons between similar inversions using lines all across the
electromagnetic spectrum would be valuable for constraining the temperature
distribution of coronal plasmas at a wide range of heliocentric distances and
even could be applied to collisionless astrophysical plasmas generally.

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).

##  Appendix A Actual R-DEM of the PSI MHD Model

The PSI/MAS MHD model is resolved at exceptionally high spatial resolution
compared to previous global models (see Mikić et al. 2018), and generally at a
higher spatial resolution than the eclipse data presented in this work.
Specifically, the latitude and longitudinal mesh spacing is 0.009 radians
($\approx$ 0.5 degrees) except within 20 degrees of the poles, where the mesh
spacing is 0.0105 radians ($\approx$ 0.6 degrees). The radial mesh spacing is
highly variable depending on the distance from the Sun, where the mesh spacing
is less than 0.01 $R_{\odot}$ below 1.6 $R_{\odot}$, $\approx$ 0.025
$R_{\odot}$ at 2 $R_{\odot}$, and $\approx$ 0.05 $R_{\odot}$ at 3 $R_{\odot}$.
Consequently, the spatial resolution of the model is considerably finer than
the eclipse data in most of the corona, which has mesh points of 0.02
$R_{\odot}$ by 3 degrees (in position angle). Further, each LOS integration
through the model combines a multitude of bins so even when the model voxels
are larger than the data resolution, their intersection probes the corona at a
finer resolution.

For the purposes of comparing with the R-DEM inversions in this work, we
integrated the actual R-DEM in the model with the same polar coordinate
binning on the POS projection from the Earth’s perspective and the same
$T_{e}$ bins used for the R-DEM inversion, which have a width of 0.01 MK (see
Section 3.2). Since the actual R-DEM was integrated over these bins, the
resulting integrals are Radiative Emission Measures (R-EM; i.e., Equation 5).
A collection of these R-EM maps are shown in Figure 9 for a selection of
temperatures from 0.8 to 1.9 MK. An animation showing every R-EM from 0.6 to
2.6 MK is available in the online version of the article. These R-EM maps show
the amount of plasma at a given $T_{e}$ along each LOS, in units of $cm^{-2}$
(see Equation 5). In this representation, the densities are not normalized for
each LOS, as they were in Figures 3 and 4. In Section 4.2.1), we concluded
that even this binned $T_{e}$ resolution is much higher than is possible to
resolve using the three-line R-DEM used in this work, still the R-EM maps are
interesting for revealing the finer details in the MHD simulation.

Figure 9: Selection of radiative emission measures from the PSI MHD model
ranging from 0.8 to 1.9 MK. Each panel shows the integrated R-EM over a range
of 0.01 MK. An animation with every 0.01 MK slice of the R-EM from 0.6 to 2.6
MK is available in the online version of the article.

The model R-EMs show that there is a small density of plasma in the coronal
holes at $T_{e}$ values below about 1 MK, especially the southern coronal
hole, and at the base of the corona below 1.2 $R_{\odot}$. Around 1 to 1.3 MK,
the plasma density is somewhat uniformly distributed throughout the corona,
roughly proportional to the bulk density. Above 1.5 MK, the plasma becomes
increasingly confined to closed field lines in the streamers, and the extent
of plasma at higher helioprojective distances decreases. The highest $T_{e}$
plasma (around 2 MK) in the south-eastern streamer overlaps with regions that
had very low $T_{e}$ values as well.

In comparisons with the eclipse data, we found that the average $T_{e}$ in the
model inversion is close to the eclipse data. However, the width of the model
R-DEM inversion is considerably larger in some regions, particularly in the
Eastern streamer. It is not clear why the model has a wider distribution of
temperature in certain regions than the eclipse data indicates, but is likely
related to the precise physical mechanism of the Wave-Turbulence-Driven (WTD)
approach to heat the corona in the model (Lionello et al., 2014; Downs et al.,
2016; Mikić et al., 2018) and the inherent resolution of the global model
grid, which naturally cannot resolve structures and contrasts below a certain
size. To deduce the exact cause of these differences would require additional
simulations where the precise assumptions and boundary conditions of the model
are varied, which is beyond the scope of this work.

## References

  * Antonucci et al. (2023) Antonucci, E., Downs, C., Capuano, G. E., et al. 2023, Physics of Plasmas, 30, 022905
  * Arnaud & Raymond (1992) Arnaud, M., & Raymond, J. 1992, ApJ, 398, 394
  * Aschwanden et al. (2015) Aschwanden, M. J., Boerner, P., Caspi, A., et al. 2015, Sol. Phys., 290, 2733
  * Aschwanden & Nitta (2000) Aschwanden, M. J., & Nitta, N. 2000, ApJ, 535, L59
  * Boe et al. (2021) Boe, B., Habbal, S., Downs, C., & Druckmüller, M. 2021, ApJ, 912, 44
  * Boe et al. (2022) —. 2022, ApJ, 935, 173
  * Boe et al. (2020b) Boe, B., Habbal, S., & Druckmüller, M. 2020b, ApJ, 895, 123
  * Boe et al. (2020a) Boe, B., Habbal, S., Druckmüller, M., et al. 2020a, ApJ, 888, 100
  * Boe et al. (2018) —. 2018, ApJ, 859, 155
  * Boerner et al. (2012) Boerner, P., Edwards, C., Lemen, J., et al. 2012, Sol. Phys., 275, 41
  * Cheng et al. (2012) Cheng, X., Zhang, J., Saar, S. H., & Ding, M. D. 2012, ApJ, 761, 62
  * Cheung et al. (2015) Cheung, M. C. M., Boerner, P., Schrijver, C. J., et al. 2015, ApJ, 807, 143
  * Chitta et al. (2023) Chitta, L. P., Seaton, D. B., Downs, C., DeForest, C. E., & Higginson, A. K. 2023, Nature Astronomy, 7, 133
  * Del Zanna et al. (2021) Del Zanna, G., Dere, K. P., Young, P. R., & Landi, E. 2021, ApJ, 909, 38
  * Del Zanna & Mason (2018) Del Zanna, G., & Mason, H. E. 2018, Living Reviews in Solar Physics, 15, 5
  * Del Zanna et al. (2023) Del Zanna, G., Samra, J., Monaghan, A., et al. 2023, ApJS, 265, 11
  * Dere et al. (1997) Dere, K. P., Landi, E., Mason, H. E., Monsignori Fossi, B. C., & Young, P. R. 1997, A&AS, 125, 149
  * Downs et al. (2016) Downs, C., Lionello, R., Mikić, Z., Linker, J. A., & Velli, M. 2016, ApJ, 832, 180
  * Doyle et al. (1999) Doyle, J. G., Teriaca, L., & Banerjee, D. 1999, A&A, 349, 956
  * Esser et al. (1997) Esser, R., Habbal, S. R., Coles, W. A., & Hollweg, J. V. 1997, J. Geophys. Res., 102, 7063
  * Fox et al. (2016) Fox, N. J., Velli, M. C., Bale, S. D., et al. 2016, Space Sci. Rev., 204, 7
  * Gilly & Cranmer (2020) Gilly, C. R., & Cranmer, S. R. 2020, ApJ, 901, 150
  * Goryaev et al. (2014) Goryaev, F., Slemzin, V., Vainshtein, L., & Williams, D. R. 2014, ApJ, 781, 100
  * Güdel et al. (2001) Güdel, M., Audard, M., Magee, H., et al. 2001, A&A, 365, L344
  * Guennou et al. (2012a) Guennou, C., Auchère, F., Soubrié, E., et al. 2012a, ApJS, 203, 25
  * Guennou et al. (2012b) —. 2012b, ApJS, 203, 26
  * Habbal et al. (1993) Habbal, S. R., Esser, R., & Arndt, M. B. 1993, ApJ, 413, 435
  * Habbal et al. (1995) Habbal, S. R., Esser, R., Guhathakurta, M., & Fisher, R. R. 1995, Geophys. Res. Lett., 22, 1465
  * Habbal et al. (2010) Habbal, S. R., Morgan, H., Druckmüller, M., & Ding, A. 2010, ApJ, 711, L75
  * Habbal et al. (2021) Habbal, S. R., Druckmüller, M., Alzate, N., et al. 2021, ApJ, 911, L4
  * Hannah & Kontar (2012) Hannah, I. G., & Kontar, E. P. 2012, A&A, 539, A146
  * Kashyap & Drake (1998) Kashyap, V., & Drake, J. J. 1998, ApJ, 503, 450
  * Kepa et al. (2022) Kepa, A., Sylwester, B., Siarkowski, M., & Sylwester, J. 2022, ApJ, 927, 19
  * Kepa et al. (2006) Kepa, A., Sylwester, J., Sylwester, B., Siarkowski, M., & Stepanov, A. I. 2006, Solar System Research, 40, 294
  * Ko et al. (1997) Ko, Y.-K., Fisk, L. A., Geiss, J., Gloeckler, G., & Guhathakurta, M. 1997, Sol. Phys., 171, 345
  * Ko et al. (1996) Ko, Y. K., Fisk, L. A., Gloeckler, G., & Geiss, J. 1996, Geophys. Res. Lett., 23, 2785
  * Ko et al. (2002) Ko, Y.-K., Raymond, J. C., Li, J., et al. 2002, ApJ, 578, 979
  * Laming (2015) Laming, J. M. 2015, Living Reviews in Solar Physics, 12, 2
  * Landi & Testa (2014) Landi, E., & Testa, P. 2014, ApJ, 787, 33
  * Lionello et al. (2014) Lionello, R., Velli, M., Downs, C., et al. 2014, ApJ, 784, 120
  * Matsumoto & Suzuki (2014) Matsumoto, T., & Suzuki, T. K. 2014, MNRAS, 440, 971
  * Mikić et al. (2018) Mikić, , Z., Downs, C., et al. 2018, Nature Astronomy, 2, 913
  * Morgan & Pickering (2019) Morgan, H., & Pickering, J. 2019, Sol. Phys., 294, 135
  * Morgan & Taroyan (2017) Morgan, H., & Taroyan, Y. 2017, Science Advances, 3, e1602056
  * Munro & Jackson (1977) Munro, R. H., & Jackson, B. V. 1977, ApJ, 213, 874
  * O’Dwyer et al. (2010) O’Dwyer, B., Del Zanna, G., Mason, H. E., Weber, M. A., & Tripathi, D. 2010, A&A, 521, A21
  * Pickering & Morgan (2019) Pickering, J., & Morgan, H. 2019, Sol. Phys., 294, 136
  * Plowman et al. (2013) Plowman, J., Kankelborg, C., & Martens, P. 2013, ApJ, 771, 2
  * Pottasch (1964) Pottasch, S. R. 1964, Space Sci. Rev., 3, 816
  * Raymond et al. (1997) Raymond, J. C., Kohl, J. L., Noci, G., et al. 1997, Sol. Phys., 175, 645
  * Riley et al. (2019) Riley, P., Linker, J. A., Mikic, Z., et al. 2019, ApJ, 884, 18
  * Rimmele et al. (2020) Rimmele, T. R., Warner, M., Keil, S. L., et al. 2020, Sol. Phys., 295, 172
  * Samra et al. (2022a) Samra, J. E., Madsen, C. A., Cheimets, P., et al. 2022a, ApJ, 933, 82
  * Samra et al. (2022b) Samra, J. E., Marquez, V., Cheimets, P., et al. 2022b, AJ, 164, 39
  * Seaton et al. (2021) Seaton, D. B., Hughes, J. M., Tadikonda, S. K., et al. 2021, Nature Astronomy, 5, 1029
  * Smith et al. (2003) Smith, E. J., Marsden, R. G., Balogh, A., et al. 2003, Science, 302, 1165
  * Tomczyk et al. (2021) Tomczyk, S., Landi, E., Berkey, B., et al. 2021, in AGU Fall Meeting Abstracts, Vol. 2021, 2089
  * Verdini et al. (2010) Verdini, A., Velli, M., Matthaeus, W. H., Oughton, S., & Dmitruk, P. 2010, ApJ, 708, L116
  * Weber et al. (2005) Weber, M. A., Schmelz, J. T., DeLuca, E. E., & Roames, J. K. 2005, ApJ, 635, L101
  * Withbroe (1978) Withbroe, G. L. 1978, ApJ, 225, 641