# The Evolution of Ion Charge States in Coronal Mass Ejections

J. Martin Laming Space Science Division, Code 7684, Naval Research Laboratory,
Washington DC 20375, USA  Elena Provornikova The Johns Hopkins University
Applied Physics Laboratory, Laurel, MD 20723, USA  Yuan-Kuen Ko Space Science
Division, Code 7684, Naval Research Laboratory, Washington DC 20375, USA

###### Abstract

We model the observed charge states of the elements C, O, Mg, Si, and Fe in
the coronal mass ejections (CMEs) ejecta. We concentrate on “halo” CMEs
observed in situ by ACE/SWICS to measure ion charge states, and also remotely
by STEREO when in near quadrature with Earth, so that the CME expansion can be
accurately specified. Within this observed expansion, we integrate equations
for the CME ejecta ionization balance, including electron heating
parameterized as a fraction of the kinetic and gravitational energy gain of
the CME. We also include the effects of non-Maxwellian electron distributions,
characterized as a $\kappa$ function. Focusing first on the 2010 April 3 CME,
we find a somewhat better match to observed charge states with $\kappa$ close
to the theoretical minimum value of $\kappa=3/2$, implying a hard spectrum of
non-thermal electrons. Similar, but more significant results come from the
2011 February 15 event, although it is quite different in terms of its
evolution. We discuss the implications of these values, and of the heating
required, in terms of the magnetic reconnection Lundquist number and anomalous
resistivity associated with CME evolution close to the Sun.

††journal: ApJ

##  1 Introduction

Coronal Mass Ejections (CMEs) are arguably the most important component of
space weather. These extreme events of solar activity can efficiently
accelerate energetic particles and produce geomagnetic storms. Understanding
the eruption process of CMEs, in particular physical mechanisms that heat
plasma and accelerate particles in the early stages of CME evolution, is
crucial to understanding and forecasting the CME component of space weather.

Ion charge states detected in situ at 1 AU provide crucial insight into the
heating and evolution of coronal mass ejections (CMEs) near the Sun. Different
ionization stages of different elements “freeze-in” at different heights in
the solar atmosphere, typically within about 6 $R_{\sun}$, (though denser
material in prominences can freeze-in beyond this distance Rivera et al.,
2019), and no longer change as electron density rapidly decreases with height.
This allows one to reconstruct the various episodes of heating and expansion.
Some of these charge states are uniquely produced by magnetic reconnection
associated with CMEs, where high charge states indicate the presence of high
electron temperatures, and so offer a novel observational approach to
understanding this process, particularly its role in electron heating and
acceleration.

Our analyses of CME ejecta ion charge states began with an initial exploration
in Rakowski, et al. (2007), a spheromak model for a CME by Rakowski et al.
(2011), and breakout and flux-cancellation modeling of CMEs by Lynch et al.
(2011), while Lepri et al. (2012) revealed spatial distribution of heating in
the 2005 January 20 CME observed by both the Advanced Composition Explorer
(ACE) and Ulysses. These works quantified thermal energy input during early
CME evolution and showed the importance of magnetic reconnection for CME
heating, but many questions still remain to be answered. In particular, many
details of the reconnection process resulting in different charge states
distributions for CMEs of different speeds remain to be understood. Besides
energy partitioning between bulk kinetic energy, magnetohydrodynamic (MHD)
waves, and heat, details of the electron heating and acceleration remain to be
validated observationally.

In this work we revisit the modeling of CME ejecta charge states with a
considerably improved model, incorporating the effects of non-Maxwellian
distribution functions. Our CMEs are carefully chosen so that the expansion
and morphological evolution can be properly specified. Integration of
equations for time-dependent ionization balance within this evolution for
different degrees of electron heating, and for different electron distribution
functions (taken here as $\kappa$ distributions), reveals a preference for
non-Maxwellian electron distribution with plausible degrees of heating. We
further discuss these results in terms of modeling of the electron
distribution from the Fokker-Planck equation, and the parameters such as
Lundquist number, anomalous resistivity and plasmoid instability required.
Section 2 below summarizes the theoretical context for this work. Section 3
outlines the time dependent ionization balance calculations and implementation
of the non-Maxwellian electron distributions. Section 4 introduces the data
and CME events we study, while section 5 describes the results of our
modeling. Section 6 continues with a discussion of implications of the
results. Section 7 presents supporting MHD simulations of magnetic
reconnection with multiple plasmoids. Section 8 presents the conclusions of
the study.

##  2 Theoretical Context

###  2.1 Plasmoid Instability

While the model CME eruption geometries differ, most scenarios invoke magnetic
reconnection above and/or below the ejected plasma. Reconnection below the CME
may impart kinetic energy to the CME, and is arguably the most likely source
of heat for the CME plasma and hence the high Fe charge states detected in-
situ. Murphy et al. (2011) discuss a range of other possible heating
mechanisms and are able to reject all of them except for heating by small-
scale magnetic reconnection in the CME current sheet.

However, it is widely believed that the fundamental component of a CME is a
magnetic flux-rope (Chen & Krall, 2003; Linton & Moldwin, 2009; Liu et al.,
2010; Vourlidas et al., 2013), which may arise independently of the
reconnection, or be formed by the plasmoid instability in high Lundquist
number reconnection. Simulations and theory are now starting to clarify
exactly how magnetic reconnection can heat or accelerate electrons and ions in
this manner. In recent studies of magnetic reconnection, the role of the
secondary plasmoid instability of the highly elongated reconnection current
sheets has been under investigation (Shibata and Tanuma, 2001; Huang &
Bhattacharjee, 2010; Cassak and Drake, 2009; Landi et al., 2015; Arnold et
al., 2021).

Plasmoid (or magnetic island) formation is important to this work as a means
of electron heating (Drake et al., 2006) and as a constraint on the rate of
reconnection. Loureiro et al. (2007) have shown that the criteria for onset of
the plasmoid instability is that the plasma Lundquist number $S\equiv 4\pi
LV_{A}/(\eta c^{2})\gtrsim 10^{4}$, where $L$ is the length of the current
sheet, $V_{A}$ is the Alfvén speed in reconnection inflow and $\eta$ is the
plasma resistivity. This implies that slow Sweet-Parker like reconnection in
the regime of large $S$ never happens because the current sheet is always
disrupted into plasmoids. Pucci & Velli (2014) argue that the “ideal” tearing
mode takes over when the current sheet reaches a thickness $\delta/L\sim
S^{-1/3}$, which is by two orders of magnitude thicker than the Sweet-Parker
scaling $\sim S^{-1/2}$. Uzdensky & Loureiro (2016) give a slightly different
criterion, and Comisso et al. (2016) provide a more general theory of plasmoid
formation. A number of simulations by Huang & Bhattacharjee (2010), Lapenta
(2008), Daughton et al. (2009), and Loureiro et al. (2012) demonstrated
plasmoid generation and $S$-independent or weakly dependent reconnection rates
for $S>S_{crit}\sim 10^{4}$, where $S_{crit}$ is the critical Lundquist
number. Uzdensky et al. (2010) show that the reconnection rate in the plasmoid
regime ($S>10^{4}$) is fast and independent of $S$, $cE_{eff}\sim
S_{crit}^{-1/2}V_{A}B_{0}$. However plasmoid generation in three dimensions
may be more complex (Daughton et al., 2011; Wyper & Pontin, 2014). Plasmoid
formation, ejection and merging has been observed in CMEs by Song et al.
(2012), and in laboratory experiments by Hare et al. (2017); Suttle et al.
(2016). The reconfiguring of the solar corona by reconnection during a CME is
studied by van Driel-Gesztelyi et al. (2014).

###  2.2 Energy Partitioning

Currently no strong prediction exists for how much plasma heating should be
expected as a function of CME kinetic energy. In the case of laminar (i.e.
Sweet-Parker) reconnection, Priest (2014) gives an estimate that thermal
energy released should be equal to the kinetic energy in the incompressible
case, increasing as $2\rho_{o}/\rho_{i}-1$ in compressible reconnection where
$\rho_{i,o}$ are the densities of the inflow and outflow respectively.

In the case of reconnection diffusion region restricted to a length $L_{i}$ of
a magnetic field reversal extending over length $L_{e}$ (see Fig. 1), Priest
(2014) gives

$\frac{L_{i}}{L_{e}}=\frac{\rho_{i}^{1/2}}{\rho_{e}^{1/2}}\frac{1}{S_{e}}\frac{1}{M_{e}^{1/2}M_{i}^{3/2}}=\frac{\rho_{e}^{1/4}}{\rho_{i}^{1/4}}\frac{1}{S_{e}}\frac{1}{M_{e}^{2}}$
(1)

where a factor involving the densities has been reinstated,
$M_{e,i}=v_{e,i}/V_{Ae,i}$ are Mach numbers in terms of the Alfvén speeds at
exterior and interior inflow regions, and $S_{e}=4\pi L_{e}V_{Ae}/\eta c^{2}$
is the exterior Lundquist number. The second equality comes from taking
$B_{i}\sim B_{e}$, $v_{i}\sim v_{e}$ so that
$M_{e}/M_{i}\sim\sqrt{\rho_{e}/\rho_{i}}$, the outflow kinetic energy as a
fraction of the inflow magnetic energy is

$\frac{\rm outflow~{}KE}{\rm
inflow~{}magnetic~{}energy}=\frac{1}{2}\left(\frac{\rho_{e}^{1/4}\rho_{i}^{3/4}}{\rho_{o}}\right)\frac{1}{S_{e}M_{e}^{2}}.$
(2)

The remaining energy is taken to be heat, giving a ratio

${\frac{\rm thermal~{}energy}{\rm
outflow~{}KE}}=2S_{e}M_{e}^{2}\frac{\rho_{o}}{\rho_{e}^{1/4}\rho_{i}^{3/4}}-1.$
(3)

Adopting $S_{e}\sim 10^{4}$ as the threshold for plasmoid instability implies
$M_{e}\sim 10^{-2}$ and

${\frac{\rm thermal~{}energy}{\rm
outflow~{}KE}}=2r\left(\frac{\rho_{i}}{\rho_{e}}\right)^{1/4}\frac{S_{e}}{10^{4}}-1$
(4)

where $r=\rho_{o}/\rho_{i}$. If $M_{i}>\beta$, a slow mode shock can develop
in the inflow to the diffusion region, leading to $1<r<2.5$. The kinetic
energy comes only from the $L_{i}$ diffusion regions, while the magnetic
islands that separate them contribute only heat. These arguments do not
account for any energy in waves (c.f. Kigure et al., 2010). We anticipate that
Alfvén waves should be part of the kinetic energy budget, while magnetosonic
waves are part of the thermal energy. Stronger compression leads to more heat,
as in Provornikova et al. (2016, 2018).

Figure 1: Schematic diagram of a reconnection current sheet with
characteristic external scale length $L_{e}$, and interior length of diffusion
region (shown in gray) of $L_{i}$. The exterior and interior flow velocities
are $v_{e}$ and $v_{i}$ respectively. Magnetic islands form either side of the
diffusion regions, and are the sites of electron heating as they evolve and
merge.

An important feature of this heating will be the partitioning of energy
between electrons and ions. A large fraction of the released magnetic energy
in flares may be channeled into high energy electrons, protons (Lin & Hudson,
1971; Emslie et al., 2004, 2005) and other ion species (Mason, 2007), as well
as waves and other forms of kinetic energy. The X-ray emission spectra reveal
that energetic electrons in flares typically consist of a hot component in the
range of 20-30 keV with higher energy powerlaw tails (Holman et al., 2003).
Over-the-limb observations of flare energy release reveal that almost all
electrons in the flaring site can be accelerated to became part of the
energetic component, whose pressure approaches that of the magnetic field
(Krucker et al., 2010). Glesener et al. (2013) suggest that flare-accelerated
electrons collisionally deposit their energy into plasma heating in the CME to
at least 11 MK.

###  2.3 Anomalous Resistivity

Observationally, conditions in CME current sheets may be closer to the
threshold for plasmoid generation than are usually thought. Observational
signatures of drifting and pulsating structures in solar flare associated
current sheets have been interpreted by Kliem et al. (2000) and Karlický
(2004) as evidence of embedded secondary plasmoids. Takasao et al. (2012) also
reported EUV observations of plasmoid emergence, collisions and injections in
the rising phase of the limb flare. For typical coronal parameters ($L\sim
10^{8}$ cm, $V_{A}\sim 10^{8}-10^{10}$ cm s-1, $\eta\sim 10^{-16}$ s-1), $S$
is about $10^{12}-10^{14}$, and is plausibly even larger at a large scale
current sheet with larger $L$ behind a CME. However these estimates assume a
Spitzer electrical resistivity. A more accurate idea of the Lundquist number
may come from using an anomalous resistivity, increased from the classical
(i.e. Spitzer) value. Anomalous resistivity is thought to arise in conditions
where plasma turbulence can scatter current carrying electrons, and can do so
more effectively than the Coulomb collisions between electrons and ions
Lazarian & Vishniac (1999). It may be estimated by replacing the
isotropization frequency by Coulomb collisions with the relevant wave-particle
interaction rate.

Wu & Huang (2009) give quiescent values of the anomalous resistivity of
$\times 10^{7}$ the Spitzer value, with transient values as high as $\times
10^{9}$ in a background plasma temperature of $10^{7}$ K using solutions of
the one dimensional Vlasov equation designed to model the Buneman instability
in a reconnecting current sheet, and therefore model electrons in Langmuir
turbulence. Lin et al. (2007) estimate a much higher resistivity from the
observed width of current sheets trailing CMEs, as high as $10^{12}$ times the
classical resistivity.

Incorporating such an anomalous resistivity brings the Lundquist number down
to $S\sim 10^{5}$, much closer to the threshold for the production of magnetic
islands of about $10^{4}$. Thus the increasing Lundquist number with CME speed
in this range much more plausibly suggests increasing production of magnetic
islands within the current sheet, such that the associated electron
acceleration can become more important as the CME speed increases.

###  2.4 Electron Acceleration

Models of electron acceleration in reconnecting current sheets behind the
erupting CME make definite predictions about the non-Maxwellian electron
distribution (e.g. Drake et al., 2006; Liu et al., 2009; Guo et al., 2016).
Drake et al. (2006) give analytic and numerical treatments. In the weakly
driven case, where the back reaction of the accelerated electrons can be
neglected, the power law index of the electron distribution at high energies
can be related to the geometry of the magnetic islands formed in the current
sheet. In strongly driven cases, this dependence is lost, and the electron
plasma $\beta=8\pi n_{e}kT_{e}/B^{2}$ plays a more important role in limiting
the acceleration, in the sense that higher plasma $\beta$ restricts the energy
acquired by the electrons. When the back pressure of the accelerated particles
approaches that of the magnetic field, island contraction is quenched and
particle acceleration ceases. Liu et al. (2009) give numerical simulations in
the test particle limit, and find similar electron spectra to Drake et al.
(2006). Oka et al. (2010a, b); Karlický & Bárta (2011) also explore the
electron heating in plasmoids with different explanations for the
acceleration. Stronger electron acceleration is also found in magnetic islands
with strong compression, resulting from the reconnection of relatively
unsheared field lines and hence with a weak guide field (Guidoni et al., 2016;
Li et al., 2018; Provornikova et al., 2016).

More recently, Drake et al. (2013) and Montag et al. (2017) have given Fokker-
Planck treatments of electron acceleration by a Fermi mechanism in coalescing
magnetic islands. Montag et al. (2017) incorporate the effects of
compressibility and a guide field. Their incompressible limit recovers the
result of Drake et al. (2013), but compressible plasma allows for harder
electron spectra, and hence more energy going into the electrons. Lower
ambient plasma $\beta$ would lead to potentially more magnetic field
destruction, and hence higher plasma compression in the current sheet and
stronger electron acceleration, reinforcing the idea (Drake et al., 2006;
Provornikova et al., 2016, 2018) that lower ambient plasma $\beta$ leads to
stronger heating. The high energy power law component does not extend to
infinite energy, but is limited by the time available for particle
acceleration. The presence of a guide field also limits the degree of
compression in the current sheet, and further limits the energy in electrons.

Arnold et al. (2021) study electron acceleration using an MHD model of
reconnection incorporating fluid ions and electrons, as well as particle
electrons to model the acceleration process. As the Lundquist number increases
in the range $1.2\times 10^{7}$ to $6.1\times 10^{9}$ the electron spectra
become harder, but only with electron velocity power law index decreasing from
about 3.3 to 3. However, Arnold et al. (2021) vary the Lundquist number
through the system size, not the magnetic field or plasma $\beta$. Larger
systems produce more reconnected flux and hence more extended power laws.

##  3 Ionization and Recombination of CME Ejecta

The heating and acceleration of electrons during reconnection through magnetic
islands in current sheets behind erupting CMEs in the corona will have direct
observational consequences in the ion charge states measured at 1 AU. The
charge states evolve through ionization and recombination as a function of
temperature and density up to heights of 3 to 6 solar radii heliocentric
distance but are “frozen in” thereafter. Thus in situ measurements of ion
charge state distributions hold a unique potential for diagnosing the
conditions throughout the CME eruption.

We model the charge states within the CME ejecta for a variety of ions using
an adaptation of the BLASPHEMER (BLASt Propagation in Highly EMitting
EnviRonment) code (Laming & Grun, 2002, 2003; Laming & Hwang, 2003), which
given the temperature and density history of a Lagrangian plasma parcel
determines the ionization balance as it expands in the solar wind. The density
$n_{iq}$ of ions of element $i$ with charge $q$ is given by

$\displaystyle\frac{dn_{iq}}{dt}$
$\displaystyle=n_{e}\left(C_{ion,q-1}n_{i~{}q-1}-C_{ion,q}n_{iq}\right)+$
$\displaystyle
n_{e}\left(C_{rr,q+1}+C_{dr,q+1}\right)n_{i~{}q+1}-n_{e}\left(C_{rr,q}+C_{dr,q}\right)n_{iq}$

where $C_{ion,q},C_{rr,q},C_{dr,q}$ are the rates for electron impact
ionization (including autoionization following inner-shell excitation),
radiative recombination and dielectronic recombination respectively, out of
the charge state $q$. These rates are the same as those used in the ionization
balance calculations of Bryans et al. (2006), with more recent updates given
in Rakowski, et al. (2007) and Laming & Temim (2020). The electron density
$n_{e}$ is determined from the condition that the plasma be electrically
neutral. The ion and electron temperatures, $T_{iq}$ and $T_{e}$ are coupled
by Coulomb collisions by

$\displaystyle\frac{dT_{iq}}{dt}$ $\displaystyle=$
$\displaystyle-0.13n_{e}\frac{\left(T_{iq}-T_{e}\right)}{M_{iq}T_{e}^{3/2}}\frac{q^{3}n_{iq}/\left(q+1\right)}{\left(\sum_{iq}n_{iq}\right)}\left(\frac{\ln\Lambda}{37}\right)$
(6) $\displaystyle-\frac{4}{3}\frac{\gamma_{iq}U_{w}}{n_{iq}k_{\rm B}}$

and

$\displaystyle\frac{dT_{e}}{dt}$ $\displaystyle=$
$\displaystyle\frac{0.13n_{e}}{T_{e}^{3/2}}\sum_{iq}\frac{\left(T_{iq}-T_{e}\right)}{M_{iq}}\frac{q^{2}n_{iq}/\left(q+1\right)}{\left(\sum_{iq}n_{iq}\right)}\left(\frac{\ln\Lambda}{37}\right)$
(7)
$\displaystyle-\frac{T_{e}}{n_{e}}\left(\frac{dn_{e}}{dt}\right)_{ion}-\frac{2}{3n_{e}k_{\rm
B}}\frac{dQ}{dt}+\sum_{iq}\frac{4}{3}\frac{\gamma_{iq}U_{w}}{n_{e}k_{\rm B}}.$

Here $M_{iq}$ is the atomic mass of the ions of element $i$ and charge $q$ in
the plasma, and $\ln\Lambda\simeq 28$ is the Coulomb logarithm. The first term
on the right hand side of each equation represent temperature equilibration by
Coulomb collisions. The term in $dQ/dT$ in the electron temperature equation
represents plasma energy losses due to ionization and radiation. Radiation
losses can be taken from Summers & McWhirter (1979). They are generally
negligible here. The term
$-\left(T_{e}/n_{e}\right)\left(dn_{e}/dt\right)_{ion}$ gives the reduction in
electron temperature when the electron density increases due to ionization.
Recombinations, which reduce the electron density, do not result in an
increase in the electron temperature in low density plasmas, since the energy
of the recombined electron is radiated away (in either radiative or
dielectronic recombination), rather than being shared with the other plasma
electrons as would be the case for three-body recombination in dense plasmas.
The terms in $\gamma_{iq}U_{\omega}$ represent heating of individual plasma
components by turbulence, as opposed to bulk heating of the plasma itself as
is usually assumed. Electron or ion heating can be modeled with growth or
damping rates ($\gamma_{iq}$) and turbulent energy densities ($U_{\omega}$)
calculated from quasi-linear plasma theory. Here we simply put
$\gamma_{iq}U_{\omega}$ for electrons only equal to a fraction of the gain in
kinetic and gravitational energy of the expanding CME.

The background densities and temperatures evolve according to the observed
hydrodynamic expansion, in an operator splitting method. The expansion is
specified by observations described in Section 4 and described in more detail
in Section 5. The density varies as

$n\propto\left\\{\begin{array}[]{ll}1/\left(r-R_{\sun}\right)^{2}v_{exp},&{\rm
for}\quad r\leq r_{\rm Alf}\\\ 1/r^{3},&{\rm for}\quad r>r_{\rm
Alf}\end{array}\right\\}$ (8)

which captures the expansion at increasing velocity $v_{exp}$ from an
explosion at a point on the solar surface for heliocentric radii, $r$, within
the solar wind/CME Alfvénic surface, $r_{\rm Alf}$. For $r>r_{\rm Alf}$ an
approximate $1/r^{3}$ expansion takes over once the solar wind is essentially
hydrodynamic, no longer accelerating, and the plasma $\beta>1$. In this regime
$T\propto n^{2/3}$ as per adiabatic expansion. The starting density is
determined from in-situ observations at 1 AU extrapolated back to the solar
corona.

Extension of the non-equilibrium ionization modeling to incorporate a non-
Maxwellian electron distribution as predicted by for example Drake et al.
(2006) is a key part of this work. Other electron heating mechanisms are not
expected to energize the whole population of electrons, as the Fermi
acceleration in magnetic islands is expected (Drake et al., 2006) and observed
(Krucker et al., 2010) to do. We use expressions for power law index
analytically derived by Drake et al. (2006) and Montag et al. (2017) to
connect to magnetic island geometry. For $F\propto v^{-\gamma}$ Montag et al.
(2017) find

$\gamma=\frac{3}{2}-5\frac{\nu}{R^{2}}\frac{\dot{n}}{n}+\sqrt{\left(\frac{3}{2}-5\frac{\nu}{R^{2}}\frac{\dot{n}}{n}\right)^{2}+30\frac{\nu}{R^{2}}\frac{c_{Aup}}{L}}.$
(9)

Here $\nu=\left(\Omega_{e}\pi/4\right)\delta
B^{2}\left(k_{||}=\Omega_{e}/v_{||}\right)/B^{2}$ is the electron pitch angle
scattering rate in terms of the electron gyrofrequency, $\Omega_{e}$, and the
magnetic field fluctuations $\delta B$ at parallel wavevector
$k_{||}=\Omega_{e}/v_{||}$. In other terms $R=2\dot{n}/3n-\dot{B}/B$ ($n$ is
the plasma density and $B$ is the background magnetic field) and $c_{Aup}$ is
the upstream Alfvén speed. In incompressible conditions, $\dot{n}=\partial
n/\partial t=0$ and equation 9 reduces to the result of Drake et al. (2013). A
similar expression is found by le Roux et al. (2015, their equation 65).

The index depends on the degree of plasma compression and pitch angle
scattering frequency in plasmoid dominated reconnection. The latter depends on
the intensity of waves produced by reconnection and their reflection near the
current sheet. Plasmoid formation, merging, oscillation and ejection during
reconnection process inevitably produce waves propagating away from the
reconnection site. Frequencies and wavelengths of generated waves are related
to temporal and spatial scales of plasmoid dynamics. Provornikova et al.
(2018) showed that due to gradients in magnetic field and plasma density near
the current sheet, magnetosonic waves produced in reconnection undergo
reflection back towards the current sheet, increasing the pitch angle
scattering, and hardening the electron spectrum. Reflection is more efficient
in low-$\beta$ plasma leading to increased particle acceleration. Wave
frequencies would be determined by the dimensions of magnetic islands in the
outflow. In such conditions, we should expect strong particle acceleration.

Figure 2: Examples of the full $\kappa$ distribution (thick solid), the underlying Maxwellian distribution (thick dashed) and the approximate $\kappa$ function expressed as a sum of Maxwellians (thin solid) for $\kappa=$ 30 (top), 10 (middle) and 3 (bottom) as a function of $x=v/v_{t}$.  Table 1: 2010 April 3 CME Off-Limb Plasma Parameters Radius ($R_{\sun}$) |  $V_{A}$ (km s-1) | $\int V_{A}dl$ | $n_{e}$ | $S=\int V_{A}dl\sigma/4\pi c^{2}$ | $S^{\prime}$
---|---|---|---|---|---
1.1 | 1000 | 100 | $3\times 10^{9}$ | $6\times 10^{11}$ | $1\times 10^{4}-3.6\times 10^{7}$
1.5 | 300 | 220 | $2\times 10^{8}$ | $1\times 10^{12}$ | $8\times 10^{3}-2.4\times 10^{7}$
2.0 | 270 | 355 | $3\times 10^{7}$ | $2\times 10^{12}$ | $6\times 10^{3}-1.8\times 10^{7}$
2.5 | 250 | 480 | $1\times 10^{7}$ | $2.5\times 10^{12}$ | $5\times 10^{3}-1.5\times 10^{7}$
3.0 | 200 | 580 | $6\times 10^{6}$ | $3\times 10^{12}$ | $4\times 10^{3}-1.2\times 10^{7}$
5.0 | 200 | 980 | $1\times 10^{6}$ | $5\times 10^{12}$ | $3\times 10^{3}-1\times 10^{7}$

We model the electron distribution function as a $\kappa$-function, which
offers a means of interpolating between a quasi-Maxwellian core distribution
and a power law tail at high velocities. We express the $\kappa$-function in
terms of Maxwellians as follows,

$\displaystyle f_{\kappa}\left(v\right)$ $\displaystyle=$
$\displaystyle\left[1+v^{2}/2\kappa v_{t}^{2}\right]^{-\kappa}$ (10)
$\displaystyle\simeq$
$\displaystyle\big{[}\exp\left(-v^{2}/2v_{t}^{2}\right)+\exp\left(-v^{2}\left(1+5/\kappa^{3}\right)/6\right)/2\kappa$
$\displaystyle+\exp\left(-v^{2}\left(1+5/\kappa^{3}\right)/18\right)\times
4/\kappa^{4}\big{]}$ $\displaystyle/\left[1+1/2\kappa+4/\kappa^{4}\right]$

so that an atomic reaction rate for given $\kappa$ and temperature, $T$ is
given by

$\displaystyle\sigma\left(\kappa,T\right)$ $\displaystyle=$
$\displaystyle\frac{\Gamma\left(\kappa\right)}{\Gamma\left(\kappa-3/2\right)\kappa^{3/2}}\frac{1}{1+2/\kappa+4/\kappa^{4}}$
$\displaystyle\times\bigg{\\{}\sigma_{\rm M}\left(T\right)$ $\displaystyle+$
$\displaystyle\sqrt{\frac{3}{1+5/\kappa^{3}}}\frac{\sigma_{M}\left(3T/\left(1+5/\kappa^{3}\right)\right)}{2\kappa}$
(11) $\displaystyle+$
$\displaystyle\sqrt{\frac{9}{1+5/\kappa^{3}}}\frac{\sigma_{M}\left(9T/\left(1+5/\kappa^{3}\right)\right)}{4\kappa^{4}}\bigg{\\}}$

where $\Gamma\left(\kappa\right)$ is the Gamma function and
$\sigma_{M}\left(T\right)$ is the Maxwellian rate coefficient evaluated at
temperature $T$. Such an expansion is useful, because it allows ionization and
recombination rates appropriate for a $\kappa$ distribution to be constructed
from rates readily available in tabulations calculated by integrating the
cross sections over a Maxwellian. Hahn & Savin (2015) give a considerably more
detailed expansion of the $\kappa$ distribution in terms of Maxwellians, but
use a slightly different definition of $\kappa$. Our approximations for the
$\kappa$ functions (thin sold lines) are compared in Fig. 2 with the true
$\kappa$ function (thick solid lines) and the underlying Maxwellian (dashed
lines), where $x=v/v_{t}$, and consider that our simpler approach is clearly
adequate for the task at hand. With the identification of $\kappa=\gamma/2$ in
equation 9, the values of $\kappa$ (or $\gamma$) can be interpreted in terms
of reconnection parameters.

The energy of the $\kappa$-distribution is divided with proportion $5/2\kappa$
in the high energy tail and the remainder in the quasi-thermal bulk. As
$\kappa\rightarrow 5/2$ the energy of the distribution becomes infinite. This
is resolved by invoking a cut off at high energies. In fact such a cut off is
necessary because otherwise energy would accumulate in electrons with
velocities greater than the speed of light, $c$, a clearly unphysical
situation. We implement this by setting a minimum value of $\kappa=3$ when
partitioning the energy between bulk and tail portions of the distribution
function. For $\kappa=2.5$ and 2 this implies high velocity cutoffs at 27.4
and 14.2 times the thermal velocity of the bulk electrons, respectively, which
for a bulk electron temperature of $\sim 10^{6}$ K, mean $1\times 10^{10}$ and
$5\times 10^{9}$ cm s-1 respectively, i.e. suitably below $c$, and consistent
with simulations in Arnold et al. (2021). This is important in evaluating the
collisional relaxation of the electron distribution back to a Maxwellian, i.e.
the heating of the “bulk” plasma electrons by the suprathermal tail, but less
so in calculating ionization and recombination rates because the high energy
electrons have smaller cross sections for these processes. We assume that the
electron $\kappa$-distribution holds while the plasma is being heated by the
reconnection, followed by relaxation back to a Maxwellian when this heating
ceases. Within our model this happens when the CME acceleration ceases. In
terms of the other timescales connected with the plasma evolution, this
relaxation back to a Maxwellian is essentially “instantaneous”.

Figure 3: Plots of the Fe charge state distribution for Day of Year (DOY)
95.08 - 97.00 during the 2010 April 3 ICME, observed at ACE during 2010 April
5-7. During DOY 95.83 - 96.42, the dominant Fe charge state is 16+, indicating
extra heating and ionization beyond the typical solar wind value of 8+ - 9+.

##  4 Observations

###  4.1 Introduction

In selecting the useful events our first criteria were observations of high
charge states within the ICME, a CME eruption site observed in the solar disk,
and a likely post-CME current sheet with outflowing blobs. Accordingly, we
looked for events when the Solar TErrestrial RElations Observatory (STEREO)
was in near quadrature with Earth, so that the STEREO-Sun-Earth angle was
between 65 and 115 degrees. This corresponds to dates between 2010 February 1
and 2012 March 1, a period which is further restricted by Advanced Composition
Explorer/Solar Wind Ion Composition Spectrometer (ACE/SWICS) data availability
after August 2011. Therefore the promising events appear to be those on 2010
April 3, 2010 May 23, 2011 February 15, 2011 May 25 and 2011 June 2.

###  4.2 2010 April 3 Event

We concentrate first on the 2010 April 3 event. The eruption was observed by
the STEREO Extreme Ultraviolet Imager (EUVI) of the Sun Earth Connection
Coronal and Heliospheric Investigation (SECCHI; Howard et al., 2008) and the
Extreme ultraviolet Imaging Telescope (EIT) on the Solar and Heliospheric
Observatory (SOHO) by Liu et al. (2011), by Xie et al. (2012) who also use
WIND/WAVES radio data and by Seaton et al. (2011) who also include EUV
observations from the Sun Watcher with Active Pixels and Image Processing
(SWAP; Berghmans et al., 2006; De Groof et al., 2008) on the PROBA2
spacecraft. The CME is followed out further into the heliosphere using white
light images from SECCHI by Liu et al. (2011), Mishra et al. (2020), and Xie
et al. (2012). We use these works to define the expansion of the flux rope
containing the CME ejecta, which is seen to erupt with a speed of about 64 km
s-1 (Seaton et al., 2011), starting from a heliocentric distance of 1.2
$R_{\sun}$ (Seaton et al., 2011; Zuccarello et al., 2012), with electron
density $2\times 10^{9}$ cm-3, beginning at about 8:50 UT on 2010 April 3.
This is actually a few minutes before the onset of the flare observed by GOES.
Mishra et al. (2020) pick up the CME and flux rope expansion from about 2.5
$R_{\sun}$. The flux rope continues to accelerate at 0.05 km s-2 and expand
until a speed of about 700 km s-1 is reached at a heliocentric radius of about
10 $R_{\sun}$. Other authors give the maximum speed as 800 km s-1 (Liu et al.,
2011), $\sim 900$ km s-1 (Möstl et al., 2010; Wood et al., 2011), or $\sim
1000$ km s-1 (Xie et al., 2012). The ICME at Earth orbit is observed with
speeds 550 - 700 km s-1 (Rouillard et al., 2011; Liu et al., 2011; Möstl et
al., 2010; Rodari et al., 2019; Xie et al., 2012).

We characterize coronal plasma into which the CME expands using simulation
results provided by Predictive Sciences at
https://www.predsci.com/hmi/data_access.php. The electron density and Alfvén
speed at various heliocentric radii are given in Table 1, along with various
estimates of the Lundquist number. The first is the “classical” value, $S$,
assuming an electrical conductivity given by its Spitzer value,
$\sigma=\omega_{pe}^{2}/4\pi\nu_{c}$ where $\omega_{pe}$ is the electron
plasma frequency and $\nu_{c}$ is the Coulomb collision frequency. The second
range of $S^{\prime}$ is given by replacing $\nu_{c}$ in the Spitzer value by
the local electron plasma or lower-hybrid frequencies.

Table 2: 2011 February 15 CME Off-Limb Plasma Parameters Radius ($R_{\sun}$) |  $V_{A}$ (km s-1) | $\int V_{A}dl$ | $n_{e}$ | $S=\int V_{A}dl\sigma/4\pi c^{2}$ | $S^{\prime}$
---|---|---|---|---|---
1.1 | 1500 | 150 | $6\times 10^{8}$ | $9\times 10^{11}$ | $3\times 10^{4}-2\times 10^{8}$
1.5 | 750 | 450 | $8\times 10^{6}$ | $2\times 10^{12}$ | $6\times 10^{4}-4\times 10^{8}$
2.0 | 500 | 700 | $2\times 10^{6}$ | $4\times 10^{12}$ | $4\times 10^{4}-3\times 10^{8}$
2.5 | 500 | 950 | $7\times 10^{5}$ | $5\times 10^{12}$ | $3\times 10^{4}-2\times 10^{8}$
3.0 | 500 | 1200 | $4\times 10^{5}$ | $6\times 10^{12}$ | $2\times 10^{4}-2\times 10^{8}$
5.0 | 300 | 1800 | $1\times 10^{5}$ | $1\times 10^{13}$ | $1.3\times 10^{4}-1\times 10^{8}$

The ICME related to this eruption is studied in-situ using data from the
Advanced Composition Explorer (ACE), WIND, SOHO and STEREO (Rodari et al.,
2019; Möstl et al., 2010; Liu et al., 2011). We focus on the ACE/SWICS charge
state observations. In Fig. 3 we show 24 epochs (i.e. integrated 2 hour
observations) of Fe charge state distributions recorded by ACE/SWICS. The
first panel shows data from Day Of Year (DOY) 95.08, April 5 01:55 UT, and is
dominated by slow speed solar wind where Fe charge states 8+ - 9+ are the most
abundant. At DOY 95.83 - 96.42 (panels 9 - 18) the dominant Fe charge state is
now 16+, presumably as a result of heating and ionization in the reconnection
current sheet as the CME erupts. The later panels show a return to Fe charge
states around 8+ - 9+, typical of the slow speed solar wind. A concentration
of Fe in the 16+ charge state can be suggestive of strong heating followed by
recombination (Gruesbeck et al., 2011). This Ne-like charge state has weak
recombination rates into the next lower charge state due to the closed shell
structures, and so in recombining conditions, Fe charges bottleneck here.
However for this event (c.f. the 2011 February 15 event below), although Fe
can be explained in this way, the other elements, C, O, Mg, and Si do not
follow the same trend, being heavily overionized. In fact modeling these five
elements simultaneously is highly constraining, and is important to our
eventual conclusions.

Table 3: Model Fits 2010/04/03 CME |  | $\kappa$
---|---|---
epoch | year DOY time | $\infty$ | $10^{4}$ | $10^{2}$ | 10 | 4.5 | 4 | 3.5 | 3.1 | 2.6 | 2.0 | 1.6
10 | 2010 95 19:58 heat | 0.31 | 0.26 | 0.26 | 0.32 | 0.50 | 0.60 | 0.80 | 1.25 | 1.50 | 1.70 | 8.0
| $\Sigma\Delta f_{q}^{2}$ | 0.38 | 0.40 | 0.39 | 0.38 | 0.37 | 0.36 | 0.35 | 0.35 | 0.34 | 0.33 | 0.35
11 | 2010 95 21:58 heat | 0.21 | 0.23 | 0.24 | 0.29 | 0.45 | 0.55 | 0.70 | 1.15 | 1.30 | 1.45 | 6.2
| $\Sigma\Delta f_{q}^{2}$ | 0.26 | 0.27 | 0.27 | 0.26 | 0.23 | 0.23 | 0.22 | 0.21 | 0.20 | 0.21 | 0.23
12 | 2010 95 23:58 heat | 0.21 | 0.23 | 0.23 | 0.28 | 0.45 | 0.55 | 0.70 | 1.10 | 1.25 | 1.45 | 6.2
| $\Sigma\Delta f_{q}^{2}$ | 0.33 | 0.30 | 0.29 | 0.29 | 0.29 | 0.29 | 0.28 | 0.27 | 0.26 | 0.26 | 0.26
13 | 2010 96 01:58 heat | 0.24 | 0.26 | 0.26 | 0.32 | 0.50 | 0.60 | 0.80 | 1.25 | 1.50 | 1.70 | 8.6
| $\Sigma\Delta f_{q}^{2}$ | 0.32 | 0.31 | 0.31 | 0.30 | 0.29 | 0.29 | 0.28 | 0.27 | 0.26 | 0.27 | 0.27
14 | 2010 96 03:58 heat | 0.29 | 0.32 | 0.32 | 0.40 | 0.65 | 0.75 | 1.05 | 1.65 | 1.85 | 2.30 | 13.
| $\Sigma\Delta f_{q}^{2}$ | 0.34 | 0.34 | 0.34 | 0.33 | 0.32 | 0.32 | 0.31 | 0.30 | 0.29 | 0.29 | 0.29
15 | 2010 96 05:58 heat | 0.26 | 0.28 | 0.28 | 0.34 | 0.55 | 0.65 | 0.85 | 1.25 | 1.55 | 1.85 | 9.4
| $\Sigma\Delta f_{q}^{2}$ | 0.28 | 0.28 | 0.28 | 0.27 | 0.26 | 0.26 | 0.25 | 0.25 | 0.23 | 0.23 | 0.24
16 | 2010 96 07:58 heat | 0.27 | 0.29 | 0.30 | 0.36 | 0.55 | 0.65 | 0.90 | 1.45 | 1.65 | 1.95 | 10.4
| $\Sigma\Delta f_{q}^{2}$ | 0.30 | 0.30 | 0.30 | 0.29 | 0.27 | 0.26 | 0.25 | 0.24 | 0.22 | 0.23 | 0.24
17 | 2010 96 09:58 heat | 0.23 | 0.24 | 0.24 | 0.30 | 0.50 | 0.55 | 0.75 | 1.20 | 1.35 | 1.55 | 6.9
| $\Sigma\Delta f_{q}^{2}$ | 0.30 | 0.33 | 0.33 | 0.30 | 0.27 | 0.26 | 0.24 | 0.24 | 0.22 | 0.23 | 0.26
18 | 2010 96 11:58 heat | 0.22 | 0.24 | 0.24 | 0.29 | 0.45 | 0.55 | 0.75 | 1.15 | 1.30 | 1.50 | 6.4
| $\Sigma\Delta f_{q}^{2}$ | 0.42 | 0.45 | 0.45 | 0.42 | 0.39 | 0.37 | 0.36 | 0.35 | 0.34 | 0.35 | 0.39

###  4.3 2011 February 15 Event

The 2011 February 15 CME was observed in white light by both STEREO satellites
and SOHO/LASCO (Gopalswamy et al., 2012) and additionally by Wind/WAVES (Jing
et al., 2015). With the flare starting at 01:46:50 UT, and peaking at 01:54:08
UT (Gopalswamy et al., 2012), the CME is observed to appear in the LASCO C2
coronagraph (at 2.2 R☉ heliocentric distance) at 02:24 UT, already traveling
at its final speed of $\sim 1000$ km s-1. The average speed between the solar
surface and 2.2 R☉ is 500 km s-1 and the minimum acceleration is 0.5 km s-2.
We find the best matches to the observed charge states for an initial
acceleration of 1.4 km s-2, going to zero when the speed reaches 1000 km s-1.
This acceleration matches well with that given by Maričić et al. (2014), from
analysis of STEREO/SECCHI white light and EUV data. These authors analyze
three CMEs erupting form the same active region, and Temmer et al. (2014)
investigate the interaction between the 2011 February 14 and 15 events in more
detail.

We take an initial height for the CME flux rope of 0.07 R☉ above the solar
surface, (1.07 R☉ heliocentric distance), (Cho et al., 2013; Jing et al.,
2015), and an initial velocity of 75 km s-1 from Kay et al. (2017). The
initial density is $5\times 10^{9}$ cm-3, specified to match that measured at
10.5 R☉ (Temmer et al., 2014). This also matches the density measured at 1 AU,
after expanding according to equation 8 with $r_{\rm Alf}=10$ R☉. The off-limb
electron density and Alfvén speed with estimate of the Lundquist number are
given in Table 2 similarly for the 2010 April 3 event.

Figure 4: Variation of ionization balance with heliocentric distance for, from
top C, O, Mg, Si and Fe, with electron temperature and density profiles given
at the bottom, for the 2010 April 03 CME ejecta epoch 16 (from Table 3). The
left panels give the case for Maxwellian electron distributions. The right
panels are the same but for $\kappa=2.6$ electron distributions. The broader
charge state distributions are clearest for Si and Fe. The CME expansion
velocity is superimposed on the top panel, to be read on the right hand axis.
Selected charge states are labeled. Others can be inferred by comparsion with
Fig. 5.

Figure 5: Left: Final charge state distributions for C, O, Mg, Si and Fe, at 1
AU in epoch 16 in ICME ejecta from the 2010 April 03 CME event assuming
Maxwellian electron distributions. The model is shown as a histogram, and the
data are shown as filled black circles, with 2$\sigma$ uncertainties from the
level 2 ACE data given by the radius. Right: The same but for $\kappa=2.6$
electron distributions. The difference in modeled charge states distribution
between two cases is most clearly seen for Fe and Si.

For the X2.2 flare associated with the 2011 February 15 CME, we have estimated
the photoionization rate of Fe${}^{15+}\rightarrow$ Fe16+ to compare with
collisional ionization. Using various power laws or thermal bremsstrahlung for
the radiation spectrum, we find the ratio of rates for
photoionization/collisional ionization to be about 0.25% at 2 $R_{\sun}$
(assumed flare location at 1 $R_{\sun}$), and a maximum of about 10% at 6
$R_{\sun}$. Given that the charge states freeze in quickly for this event,
this is not high enough to concern us (errors on atomic data are typically or
order 10%, so we neglect this extra complication here. Clearly, for events
like the Halloween flare of 2003 at X28 - X45, photoionization will be more
important. The 2010 April 3 event had a much less luminous flare, at B7.4, so
we do not consider photoionization there at all.

###  4.4 Other Events

The 2010 April 3 and 2011 February 15 are the best observed events for our
purposes. The 2010 May 23 CME is studied by Lugaz et al. (2012), who
concentrate on its interaction with a following CME on 2010 May 24. Song et
al. (2012) identify merging magnetic islands and an ensuing Type III radio
burst which is an indicator of electron acceleration, precisely the mechanism
under study in our paper. However this particular merger occurred at
heliocentric radii greater than $4R_{\sun}$, just beyond the region where
charge states are typically frozen-in, and so is unlikely to affect the
evolution of the ionization balance of the CME ejecta. For the 2011 May 25 and
2011 June 2 CMEs, we are unaware of analysis of data early in the CME
evolution, while they are still accelerating. CME height-time plots beginning
at about 3 $R_{\sun}$ are given by Gopalswamy et al. (2019) for the 2011 June
2 event, but their main interest is in the shock wave, Type II radio bursts
and the sustained gamma ray emission, while Palmerio et al. (2017) are more
focused on the magnetic structure of the flux rope.

##  5 Models

###  5.1 2010 April 3 Event

We model the charge state distributions for C, O, Mg, Si, and Fe at each epoch
by integrating Equations 8, 6, and 7 within background hydrodynamic expansion
taken from the observations cited above. A Lagrangian plasma parcel starts
from 1.2 $R_{\sun}$ heliocentric distance with initial speed of 64 km s-1, and
an acceleration of 0.05 km s-1 leading to a final speed of 1000 km s-1. The
initial electron density is taken to be $2\times 10^{9}$ cm-3, which when
expanded to 1 AU from 6.5 $R_{\sun}$ by $1/r^{3}$ gives the observed in situ
density of $\sim 5-10$ cm-3 (Liu et al., 2011; Möstl et al., 2010; Xie et al.,
2012; Rodari et al., 2019). The initial temperature is typical of the corona,
$1.15\times 10^{6}$K. These parameters mean that the CME acceleration, and by
assumption the heating of its ejecta, extend to radii well beyond that where
the charge states freeze-in. Thus the entire charge state fraction evolution
takes place in a non-Maxwellian electron distribution function.

The heating of electrons is assumed to be a fraction of the gain in kinetic
and gravitational energy of the CME ejecta. We vary model parameters until
quantity $\Sigma\Delta f_{q}^{2}=\Sigma\left(f_{q,obs}-f_{q,mod}\right)^{2}$
is minimized. This is the sum of differences between observed and modeled
charge state fractions squared, taken over all ions of all five elements
considered. A number of other choices might be possible here, but we adopt
this prescription for its simplicity, and for its relative weighting towards
the more populated charge state fractions observed. One could weight the sum
of differences squared by the observational errors, or weight each element
rather than each observed charge state equally. We point out here that the
observed charge state fractions we are dealing with are not independent
variables, since the charge state fractions for any element must add up to
unity. Therefore the usual statistical approaches and theorems regarding least
squares minimization will not be valid. Rather than rely on such techniques,
we analyze a sample of charge state observations from this and other CMEs with
different evolutionary histories to understand how results are distributed.

Table 4: Model Fits 2011/02/15 CME |  | $\kappa$
---|---|---
epoch | year DOY time | $10^{6}$ | $10^{4}$ | $10^{2}$ | 10 | 4.5 | 4 | 3.5 | 3.1 | 2.6 | 2.0 | 1.6
9 | 2011 49 16:43 heat | 0.12 | 0.12 | 0.12 | 0.12 | 0.15 | 0.17 | 0.22 | 0.40 | 0.35 | 0.10 | 0.11
| $\Sigma\Delta f_{q}^{2}$ | 0.29 | 0.29 | 0.28 | 0.25 | 0.19 | 0.17 | 0.15 | 0.14 | 0.15 | 0.21 | 0.49
10 | 2011 49 18:43 heat | 0.11 | 0.11 | 0.11 | 0.12 | 0.15 | 0.17 | 0.22 | 0.35 | 0.29 | 0.11 | 0.11
| $\Sigma\Delta f_{q}^{2}$ | 0.45 | 0.45 | 0.45 | 0.40 | 0.30 | 0.27 | 0.21 | 0.16 | 0.18 | 0.32 | 0.64
11 | 2011 49 20:44 heat | 0.10 | 0.10 | 0.103 | 0.10 | 0.13 | 0.15 | 0.19 | 0.35 | 0.29 | 0.10 | 0.11
| $\Sigma\Delta f_{q}^{2}$ | 0.42 | 0.43 | 0.41 | 0.38 | 0.31 | 0.28 | 0.24 | 0.21 | 0.22 | 0.29 | 0.67
12 | 2011 49 22:44 heat | 0.11 | 0.11 | 0.11 | 0.12 | 0.14 | 0.15 | 0.18 | 0.60 | 0.23 | 0.12 | 0.11
| $\Sigma\Delta f_{q}^{2}$ | 0.42 | 0.42 | 0.41 | 0.37 | 0.31 | 0.30 | 0.29 | 0.30 | 0.31 | 0.30 | 0.54
15 | 2011 50 04:44 heat | 0.10 | 0.10 | 0.10 | 0.10 | 0.13 | 0.14 | 0.15 | 0.25 | 0.22 | 0.10 | 0.11
| $\Sigma\Delta f_{q}^{2}$ | 0.39 | 0.39 | 0.39 | 0.35 | 0.30 | 0.27 | 0.23 | 0.17 | 0.18 | 0.32 | 0.82
18 | 2011 50 10:44 heat | 0.11 | 0.11 | 0.11 | 0.12 | 0.14 | 0.15 | 0.15 | 0.35 | 0.30 | 0.12 | 0.11
| $\Sigma\Delta f_{q}^{2}$ | 0.28 | 0.28 | 0.28 | 0.25 | 0.21 | 0.20 | 0.18 | 0.23 | 0.22 | 0.22 | 0.49
19 | 2011 50 12:45 heat | 0.12 | 0.12 | 0.12 | 0.13 | 0.14 | 0.16 | 0.20 | 0.35 | 0.35 | 0.12 | 0.11
| $\Sigma\Delta f_{q}^{2}$ | 0.14 | 0.14 | 0.14 | 0.15 | 0.17 | 0.19 | 0.25 | 0.32 | 0.29 | 0.19 | 0.39
20 | 2011 50 14:45 heat | 0.12 | 0.12 | 0.12 | 0.13 | 0.14 | 0.18 | 0.20 | 0.40 | 0.35 | 0.12 | 0.11
| $\Sigma\Delta f_{q}^{2}$ | 0.35 | 0.35 | 0.34 | 0.35 | 0.41 | 0.44 | 0.49 | 0.58 | 0.38 | 0.44 | 0.62
21 | 2011 50 16:45 heat | 0.12 | 0.12 | 0.12 | 0.12 | 0.14 | 0.16 | 0.20 | 0.40 | 0.35 | 0.11 | 0.11
| $\Sigma\Delta f_{q}^{2}$ | 0.25 | 0.25 | 0.25 | 0.25 | 0.26 | 0.28 | 0.33 | 0.40 | 0.38 | 0.29 | 0.49
22 | 2011 50 18:45 heat | 0.13 | 0.13 | 0.13 | 0.13 | 0.16 | 0.16 | 0.20 | 0.37 | 0.45 | 0.15 | 0.11
| $\Sigma\Delta f_{q}^{2}$ | 0.19 | 0.19 | 0.19 | 0.18 | 0.20 | 0.19 | 0.28 | 0.39 | 0.35 | 0.20 | 0.30
25 | 2011 51 00:46 heat | 0.11 | 0.11 | 0.11 | 0.11 | 0.14 | 0.15 | 0.19 | 0.30 | 0.22 | 0.11 | 0.11
| $\Sigma\Delta f_{q}^{2}$ | 0.22 | 0.22 | 0.21 | 0.18 | 0.12 | 0.10 | 0.09 | 0.11 | 0.10 | 0.15 | 0.48
27 | 2011 51 04:46 heat | 0.10 | 0.10 | 0.10 | 0.10 | 0.11 | 0.12 | 0.15 | 0.25 | 0.19 | 0.12 | 0.11
| $\Sigma\Delta f_{q}^{2}$ | 0.29 | 0.29 | 0.30 | 0.26 | 0.25 | 0.23 | 0.20 | 0.19 | 0.19 | 0.35 | 0.85
28 | 2011 51 06:46 heat | 0.10 | 0.10 | 0.10 | 0.10 | 0.13 | 0.15 | 0.15 | 0.25 | 0.22 | 0.12 | 0.11
| $\Sigma\Delta f_{q}^{2}$ | 0.35 | 0.35 | 0.35 | 0.32 | 0.27 | 0.24 | 0.20 | 0.15 | 0.17 | 0.35 | 0.79
31 | 2011 51 12:47 heat | 0.11 | 0.10 | 0.10 | 0.10 | 0.13 | 0.12 | 0.15 | 0.30 | 0.22 | 0.12 | 0.11
| $\Sigma\Delta f_{q}^{2}$ | 0.31 | 0.31 | 0.30 | 0.30 | 0.21 | 0.24 | 0.18 | 0.19 | 0.18 | 0.23 | 0.56
32 | 2011 51 14:47 heat | 0.10 | 0.10 | 0.11 | 0.10 | 0.12 | 0.12 | 0.14 | 0.25 | 0.19 | 0.12 | 0.11
| $\Sigma\Delta f_{q}^{2}$ | 0.29 | 0.29 | 0.41 | 0.26 | 0.21 | 0.19 | 0.17 | 0.19 | 0.19 | 0.22 | 0.69

Results for epochs 10 - 18 are given in Table 3, for a variety of assumed
values of $\kappa$, ranging from $10^{4}$ (effectively a Maxwellian) down to
1.6, i.e. close to the lower limit given by Equation 8, as well as for a pure
Maxwellian ($\kappa\rightarrow\infty$). These epochs comprise the periods of
the ICME with significant Fe16+ (see Figure 3) and for which reasonable fits
to the data can be found. In all cases the lowest permissible $\kappa$’s in
the range 2 - 3 give the best model fit to the observed charge states. As
$\kappa$ increases towards infinity the fits deteriorate monatonically. The
heating required as a fraction of the increase in kinetic and gravitational
energy increases for decreasing $\kappa$, and is similar to that predicted by
Equation 4 for $S_{e}\sim 10^{4}$ for $\kappa=2-3$. The $\kappa$ distributions
are apparently less efficient at ionizing the CME ejecta, presumably because
so many electrons are placed at energies well above the ionization thresholds
where the ionization cross sections are decreasing with increasing electron
energy.

Figures 4 and 5 illustrate these calculations for epoch 16. Figure 4 shows the
evolution of the C, O, Mg, Si, and Fe charge state fractions as a function of
heliocentric radius, together with the evolution of the electron temperature
and density. The set of panels on the left show the results with a Maxwellian
electron distribution, while on the right the model with $\kappa=2$ is shown.
Figure 5 shows the final model charge state fractions for C, O, Mg, Si, and
Fe, as histograms, compared with in situ observations as black circles. The
difference betwen a Maxwellian and $\kappa$ distributions is most clearly seen
in the Fe and Si charge state distributions in Fig. 5, and also in the charge
state evolutions in Fig. 4. The $\kappa$ electron distribution produces a
wider range of charge states than the Maxwellian. This is understandable, and
to be expected, since the $\kappa$ distribution encompasses a wider range of
electron energies than does the Maxwellian. Excess population in lower charge
states than those modeled is visible for both Si and Fe. We attribute this to
mixing of the heated CME ejecta with either unheated ejecta or ambient solar
wind. Such mixing would also enhance lower charge states of C, O , and Mg, but
is less obvious here because these charge states are already populated anyway.

Figure 6: Left: Variation of ionization balance with heliocentric distance for
C, O, Mg, Si and Fe, and the electron temperature and density, for the 2011
February 15 CME ejecta epoch 25, assuming $\kappa=10^{6}$ electron
distributions. Right: The same but for $\kappa=3.5$ electron distributions.
The CME expansion velocity is superimposed on the top panel, to be read on the
right hand axis. Selected charge states are labeled. Others can be inferred by
comparsion with Fig. 7.

Figure 7: Left: Final charge state distributions for C, O, Mg, Si and Fe for
the 2011 February 15 CME ejecta epoch 25, assuming $\kappa=10^{6}$ electron
distributions. The model is shown as a histogram, and the data are shown as
filled black circles, with 2$\sigma$ uncertainties from the level 2 ACE data
given by the radius. Right: The same but for $\kappa=3.5$ electron
distributions.

###  5.2 2011 February 15 Event

This follows the approach taken above, but the CME and charge states
evolutions are quite different, and so this event offers an interesting
contrast with the analysis above. We start the Lagrangian plasma element at
1.07 R☉, with speed 75 km s-1, density $5\times 10^{9}$ cm-3 and temperature
$9\times 10^{6}$ K. This temperatures is higher than that assumed above, and
implies some pre-eruption CME heating. This presumably happens through
reconnection, in conditions (i.e. high Lundquist number) that yield much more
heat than kinetic energy. The higher pre-eruption temperature of $9\times
10^{6}$ K recalls the models of Gruesbeck et al. (2011). The erupting plasma
in this case is essentially recombining, and gives qualitatively different
charge state fractions patterns to continually heated ejecta.

The evolution of charge state distributions and the final charge state
distribution are shown for epoch 25 from Table 4, in a similar manner in
Figures 6 and 7, similarly to Figures 4 and 5. The rapid acceleration and high
density in this CME leads to strong cooling in the initial phase of the
evolution. This faster acceleration means that it reaches it final speed,
again about 1000 km s-1, much earlier than the 2010 April 03 event. Hence all
the CME heating happens early, and the charge states continue to evolve once
the heating has stopped, and the electron distribution relaxes back towards a
Maxwellian. Even so, because the initial heating takes the temperature up to
about $9\times 10^{6}$ K and being flare heating is assumed to heat the
electrons into a $\kappa$ distribution, this imprint remains in some of the
charge state distributions tabulated in Table 4, which shows, similarly to
Table 3, the results of simulations compared to data for a similar range of
$\kappa$ values. In Table 4, epochs 9 - 15 and 25 - 31 show a preference for
$\kappa\sim 2.6-4.0$, epochs 19-22 favor $\kappa>4.0$, while 12 and 18 show no
preference. Other epochs not shown typically also show no preference for
$\kappa$, and generally with worse fits than presented here.

Figure 8: Contours of electron velocity distribution function power law index
$\gamma=2\kappa$ in $\sigma-a$ space, for conductivity $\sigma$ and turbulent
spectrum $k_{||}^{-a}$. Lundquist number $S\sim 10^{4}$ is assumed on the
left, $S\sim 10^{8}$ on the right, and $L\sigma\sim S\times 10^{12}$ cm s-1.

##  6 Discussion

We are working within a picture where reconnection current sheets with
Lundquist number $S>S_{c}\sim 10^{4}$ break up into magnetic islands
(plasmoids). As these plasmoids merge, by further reconnection, and contract,
electrons are heated by a Fermi mechanism. This secondary reconnection,
although at lower Lundquist number, may also be plasmoid unstable if $S>S_{c}$
still holds, and this plasmoid production, merging and electron heating will
continue until $S<S_{c}$. Exactly what scale this occurs at is unclear. Song
et al. (2012) observe magnetic islands appearing and merging at heights 4.73 -
7.20 R☉ with size $\sim 10^{11}$ cm. Closer to the Sun, Liu et al. (2013) and
Lu et al. (2022) see magnetic islands forming 0.2 - 0.25 R☉ above the solar
surface with diameters $\sim 2\times 10^{9}$ cm during the 2012 July 19 flare.
Structures with these dimensions require an anomalous electrical conductivity
for a Lundquist number $S\sim 10^{4}$. Bemporad (2008) has previously argued
for anomalous conductivities of the order implied on the basis of observations
of the widths of current sheets. Tables 1 and 2 give value of the Lundquist
number based on such estimates of anomalous conductivity for the two CMEs of
interest here.

Our analysis allows for some quantitative statements of such things. The
“fitted” values of $\kappa=\gamma/2$ compare well with those determined in
flares by other methods. Analyses of hard X-ray and microwave emissions
associated with downgoing accelerated electrons in the 2017 September 10 flare
indicate $\kappa\simeq 2.5-4$ ($\gamma=5-8$ Fleishman et al., 2022; Li et al.,
2022) with harder spectra coming earlier in the flare when the magnetic field,
and presumably also the Lundquist number, were higher. That magnetic field
destruction powers the flare is confirmed by Fleishman et al. (2020), who
measure the evolution of the magnetic field during the event from microwave
emission, with electron jets being observed by Chen et al. (2018). This
observation, and the active region inferences of Del Zanna et al. (2022)
concern sunward directed electrons. For anti-sunward directed electrons, as
treated in this paper, Petersen et al. (2021) report similar spectra in flares
observed by Klassen et al. (2016) using the STEREO Solar Electron and Proton
Telescope. These various electron spectral indices can be compared with values
to be expected from equation 9. We assume the accelerated electrons are
scattered by parallel propagating waves with energy spectrum $\propto
k_{||}^{-a}$ where typically $1<a<2$. In equation 9 we write
$\nu=\Omega_{e}\pi/4\times\left(k_{||}/k_{m}\right)^{-a}$ (following Laming et
al., 2014, for scattering of ions) where $k_{m}\simeq 2S^{3/8}/L$ is the
wavevector where plasmoid growth is maximized (e.g. Loureiro et al., 2007),
and at maximum growth we assume $\delta B\sim B$. Then

$\nu=\frac{\pi}{4}\Omega_{e}\left(\frac{\Omega_{e}}{v_{||}}\frac{L}{2S^{3/8}}\right)^{-a}=\frac{\pi}{4}\Omega_{e}\left(\frac{\omega_{pe}}{c}\frac{V_{A}}{v_{||}}\sqrt{\frac{m_{i}}{m_{e}}}\frac{L}{2S^{3/8}}\right)^{-a}$
(12)

since $\Omega_{i}=\omega_{pi}V_{A}/c$ and
$\Omega_{e}=\omega_{pe}V_{A}/c\times\sqrt{m_{i}/m_{e}}$, and equation 8
becomes

$\displaystyle\gamma$ $\displaystyle=$
$\displaystyle\frac{3}{2}+\sqrt{\frac{9}{4}+\frac{15\pi}{2}\left(\frac{\omega_{pe}}{c}\sqrt{\frac{m_{i}}{m_{e}}}\right)^{1-a}\frac{V_{A}^{2}}{v_{in}^{2}}\left(\frac{v_{||}}{V_{A}}\right)^{a}\frac{\delta_{cs}^{2}}{L}\frac{S^{3a/8}}{\left(L/2\right)^{a}}}$
(13) $\displaystyle=$
$\displaystyle\frac{3}{2}+\sqrt{\frac{9}{4}+23.56\left(\frac{\sqrt{n_{e}}}{12422}\right)^{1-a}\left(\frac{v_{||}}{V_{A}}\right)^{a}\frac{S^{3a/8}}{\left(L/2\right)^{a-1}}}.$
(14)

where we have assumed $\left(V_{A}/v_{in}\right)^{2}\simeq S$ in the final
step. Figure 8 plots contours of $\gamma$ in the $a$-conductivity space
assuming $S\sim 10^{4}$ (left panel) and $S\sim 10^{8}$ (right panel). For a
given $S$, the product of loop length and conductivity remains constant. Our
inferred electron velocity distribution power law indices pick out the region
towards the bottom right hand corner of each contour plot. This matches with
Zank et al. (2022), who find $a\simeq 1.5$ in the sub Alfvénic solar wind
observed with Parker Solar Probe, and with the plasmoid dimensions referred to
above. Taking $a=1.5$ and $\gamma<8$ gives $L>10^{6}$ cm for $S=10^{4}$ and
$L>10^{10}$cm for $S=10^{8}$. These estimates become $L>10^{4}$cm and
$L>10^{7}$ cm respectively for $a=2$. For the case where $\gamma>8$ is
preferred, the reverse inequalities hold.

An independent estimate of the limiting plasmoid size comes from the
requirement that the electrons are collisionless at this scale to maintain a
$\kappa$ distribution. Uzdensky et al. (2010) argue that this is always true,
but assume that the electrical conductivity remains classical. We take the
electron-electron relaxation time $\tau_{ee}=1.5\times
10^{-20}v_{e}^{3}/n_{e}$ and demand that this be greater than the acceleration
time $\tau_{acc}\simeq d/v_{e}$ where $d$ is the magnetic island diameter.
These equations give

$d<1.85\times 10^{15}\frac{\left(E/100~{}{\rm eV}\right)^{2}}{n_{e}}$ (15)

which evaluates to $3\times 10^{5}-10^{6}$ cm for $n_{e}=2\times
10^{9}-5\times 10^{9}$ cm-3. This estimate favors $S=10^{4}$ for parallel and
perpendicular island dimensions of $l_{||}=\pi L/S^{3/8}=\pi\times
10^{3}S^{5/8}$ and $l_{\perp}=\delta$ in the range
$L/S^{1/2}-L/S^{1/3}=10^{3}S^{1/2}-10^{3}S^{2/3}$, where $L=10^{3}S$ from
results above. Similarly, the favored ratios of heating to kinetic energy
increase of order 1 in Table 3 also suggests $S\sim 10^{4}$. In Table 4, much
of the heating is already embodied in the initial temperature, so the heating
to KE ratio is less clear, but is likely much higher. Note that this limit
does not apply to epochs 19-22 in Table 4, where high $\kappa$ is favored by
the charge state analysis.

We interpret this as meaning that higher $S$ reconnection forms magnetic
islands that then merge in lower S reconnection. The process cascades until
the islands are small enough to allow electrons to be accelerated without
redistributing their energy through Coulomb collisions such that the non-
Maxwellian distribution survives. This condition appears to be reached once
$S$ reaches $10^{4}$, with $L\sim 10^{3}S$ and $\sigma\sim 10^{9}$ s-1 for
$a=1.5$. The last term in the square root in equation 13 is proportional to
$S^{1-5a/8}$ for $L\propto S$. For $a=1.5$, this goes as $S^{1/16}$ giving
harder electron spectra at lower $S$. For $a=2$, the opposite behavior
results, with this last term $\propto S^{-1/4}$ and harder electron spectra
resulting for increasing $S$. A wave spectrum with $a=2$ is suggested by
comparison with Arnold et al. (2021). They find $\gamma=3.3$ for $S=10^{8}$
and $L=10^{9}$ cm, which matches with $a=2$ in Figure 8 (right panel). They
also find harder electron spectra at higher Lundquist number. This difference
in behavior with $a$ can also be seen in Figure 8.

##  7 MHD simulations of Magnetic Reconnection

To show the formation of multiple magnetic islands in a large-scale
reconnection current sheet with a high S number and subsequent reconnection of
merging islands at smaller S numbers, we model a post-CME current sheet in the
solar corona in a compressible MHD simulation. We start from a 2.5D Harris
current sheet with initial conditions that characterize the solar corona’s
plasma and magnetic field and are identical to our previous work Provornikova
et al. (2016). This work examined the Sweet-Parker-like reconnection current
sheet formed for the global Lundquist numbers $S\lesssim 10^{4}$, which may
occur in the corona due to enhanced magnetic diffusivity $10^{11}-10^{12}$
cm2s-1 caused by turbulence and kinetic effects (see discussion in Sec. 2.3).
Here we model reconnection with magnetic islands, which occurs at S numbers
larger than the critical S number, typically $S_{crit}\sim 10^{4}$. We use the
GAMERA (Grid Agnostic MHD for Extended Research Application) MHD code (Zhang
et al., 2019) to solve compressible MHD equations in the Harris current sheet
setup. A predecessor of the GAMERA, the LFM (Lyon-Fedder-Mobarry) MHD code,
was used to simulate in a box the magnetic reconnection process in the Earth
magnetotail (Merkin & Sitnov, 2016).

Figure 9:  GAMERA MHD simulation of magnetic reconnection in Harris current
sheet representing post-CME current sheet in the solar corona. Panels (a-h)
show out-of-plane current $J_{z}$ in the simulation domain. Panels (i-k) show
out-of-plane current $J_{z}$ when two big islands form a secondary current
sheet where reconnection with formation of smaller islands occur (blue box
from panel (h)), panels (l-n) show $V_{y}$ velocity component indicating
reconnection outflow between merging and reconnecting islands. $J_{z}$ and
$V_{y}$ are shown in normalized units. Time is normalized to the Alfvén time.

The GAMERA code solves normalized MHD equations. We assume the following
(normalizing) plasma and magnetic field parameters in the coronal plasma
ambient to the CME current sheet within a few solar radii from the Sun. The
chosen parameters are informed by MHD simulations of the global solar corona
(Linker et al., 1999) available for the Carrington rotation 2095 (when the
eruption of April 3, 2010 CME occurred) by the Predictive Science group
111https://www.predsci.com/hmi/summary_plots.php. The plasma density is
assumed to be $n=10^{10}$ cm-3, magnetic field $B=10$ G, temperature
$T=10^{6}$ K, and characteristic length of the system $L=0.1R_{\sun}=70$ Mm.
The Alfvén speed in the reconnection inflow is $\sim 200$ km s-1 and plasma
beta parameter $\beta=p_{th}/p_{mag}=0.07$.

The initial conditions in the Harris current sheet are set by the magnetic
field profile $B_{x}=B_{0}tanh(y/\lambda)$ and the density
$n=n_{0}+(1/\beta)\cosh^{-2}(x/\lambda)$ in a simulation box $[-L_{x},L_{x}]$,
$[-L_{y},L_{y}]$, and $[-L_{z},L_{z}]$ with $L_{x}=2\pi$, $L_{y}=\pi$, and
$L_{z}=0.1$ (length scales are normalized to $L$). The resolution of the
spatial grid is $N_{x}=864$, $N_{y}=432$, and $N_{z}=8$. The thickness of the
current sheet $\lambda=0.2$. Magnetic reconnection is initiated by introducing
a small-amplitude disturbance of the magnetic potential
$\mathbf{B}=\mathbf{z}\times\nabla\psi$ in a form $\psi=0.1B_{0}\cos(2\pi
x/L_{x})\cos(\pi y/L_{y})$. Boundary conditions are set to be periodic at the
left/right boundaries perpendicular to the current sheet and perfectly
conducting in the top/bottom boundaries parallel to the current sheet. GAMERA
reconnection simulations were performed at CISL Cheyenne system (Cheyenne,
2023).

The disturbance initiates a magnetic reconnection process in the Harris
current sheet (Figure 9). The reconnection current sheet becomes thinner,
elongates (panel (b)), and breaks up into magnetic islands (panel (c)). The
islands grow and merge, forming larger islands (panels (d-g)). When two
islands are pushed to each other, the oppositely directed magnetic field with
a dominant $B_{y}$ component in adjacent islands forms a vertical current
sheet (panel (h)). We find from simulations that the ratio of a local
Lundquist number in such a “secondary” current sheet to a global Lundquist
number for the initial current sheet is $S_{local}/S_{global}\sim 0.01-0.1$
depending on the length of the secondary current sheet. This agrees well with
the ratio expected from the analytic theory, $S_{local}=\pi LV_{A}/\eta
S_{global}^{3/8}$, which is derived by replacing $L$ with $2\pi/k_{max}$ where
$k_{max}\simeq 2S^{3/8}/L$, the wavenumber of maximum growth.

If $S_{local}$ exceeds the critical S value, the reconnection process in a
local current sheet produces small-scale islands. Panels (i-n) in Figure 9
show a formation and ejection of a magnetic island in a vertical current
sheet. Since reconnection occurs at a lower S, fewer magnetic islands are
expected to form (Huang & Bhattacharjee, 2010) in secondary current sheets,
and these islands will be smaller, allowing non-Maxwellian electron
distributions a better chance of surviving. Such processes continue until the
S number for current sheets forming in this cascading process of island
merging reaches values below $S_{crit}$. In our cases above, only reconnection
just above $S_{crit}$ is sufficiently collisionless to allow the electron
$\kappa$ distribution to survive and ionize the ejecta, but this need not
always be the case. We speculate that more generally, ionization of the ejecta
should be characteristic of the hardest electron spectrum produced in
reconnection, and depending on the spectrum of turbulence, this can be either
at high or low $S$, (but still above $S_{crit}$).

Our MHD simulation of magnetic reconnection in a large-scale coronal current
sheet with a high S number shows that magnetic islands coalesce and form
smaller-scale secondary current sheets, which may further be disrupted
depending on the S number into the smaller-scale magnetic islands. Such system
evolution supports findings from the analysis of charge states that electrons
are accelerated in small-scale islands formed in lower S number current
sheets, as discussed in Sec. 6.

##  8 Conclusions

In this paper, we have attempted to model ion charge states in CME ejecta from
the initial reconnection event to their eventual detection by instrumentation
at 1 AU. This “end-to-end” analysis involves characterization of the
reconnection and the MHD expansion of the CME ejecta from observations with
STEREO A and B at near quadrature with the Earth and detection of the ejecta
particles themselves by ACE/SWICS near the L1 Lagrange point. We find electron
velocity distribution function, characterized by a kappa distribution with
values $\kappa=2.6-4.0$. This is more clearly seen for certain times during
the 2011 February 15 CME where the charge state evolution is dominated by
recombination. Our $\kappa$ value are close to the theoretical minimum of
$\kappa=1.5$, but this value is actually significantly disfavored. For
reference, Del Zanna et al. (2022) and Fleishman et al. (2022) also observe
$\kappa=3-5$ by different means, and Li et al. (2022) model $\kappa\simeq 3$
for the 2017 September 10 event, observed by Fleishman et al. (2022). Applying
these results to the theory of electron acceleration by merging magnetic
islands in the reconnection current sheet constrains the Lundquist number to
values close to its critical value of $S=10^{4}$. We speculate that no matter
what the initial “global” value of $S$, so long as $S>10^{4}$, the merging of
the islands so produced generates new current sheets at lower $S$ until they
become subcritical, as in Uzdensky et al. (2010) and Huang & Bhattacharjee
(2013). If any of these islands form and merge under conditions that render
the electrons collisionless, non-Maxwellian electron velocity distribution
functions, modeled here as $\kappa$ distributions, may result in subsequent
effects in the ionization of the CME ejecta and distortion of the charge state
distribution from that expected with Maxwellian electrons, subsequently
detected after freeze in at 1 AU.

This work has been supported by NASA HSR Grants NNH22OB102 and 80HQTR19T0064,
and by Basic Research Funds of the Office of Naval Research. E.P. acknowledges
support from the NASA Heliophysics Supporting Research program (contract
80NSSC19K0856). We acknowledge high-performance computing support from the
NSF-sponsored Cheyenne system provided by NCAR’s Computational and Information
Systems Laboratory (CISL). The authors thank Kareem Sorathia, Slava Merkin,
and Harry Arnold for helpful discussions.

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