the highest frequency over the nearly flat region of peak wave power,
henceforth simply denoted $f_{peak}$, for which the diffusion rate
$D_{\alpha\alpha}(E_{R,\min})\sim B_{w}^{2}(f)\cdot f/f_{cp}$ is maximized.

Next, we investigate the simplest explanation for the $\langle
j_{prec}/j_{trap}\rangle$ dependence on $E/E^{*}$, at $E<E^{*}$: cyclotron
resonance with progressively less intense EMIC waves at higher frequencies.
This is done in two steps: First, assuming that this precipitation is due to
quasi-parallel hydrogen band EMIC waves we use quasi-linear theory to infer
from the above fit to the ELFIN measurements the statistical EMIC wave-power
ratio $B_{w}^{2}(f)/B_{w}^{2}(f_{peak})$. And second, we compare this inferred
wave power ratio with published statistics of EMIC wave power $B_{w}^{2}(f)$
directly observed by the Van Allen Probes near the equator in 2012-2016 [Zhang
et al., 2016c]. As a reminder, nonlinear resonant scattering by an ensemble of
independent, short-range, and large amplitude wave packets that may partake in
the measured average wave spectral shape should produce average $\langle
j_{prec}/j_{trap}\rangle$ energy spectra similar to that caused by classical
diffusion by waves with the average wave power. In other words, the diffusive
formalism, even though borrowed from quasi-linear theory, also applies for
such nonlinear interactions if they participate statistically in the process.
Yet, since we focus now on the precipitation of sub-MeV electrons, at
$E<E^{*}$, which can reach cyclotron resonance only with high frequency waves,
at $f>f_{peak}$, of much lower amplitudes than at peak wave power, the
contribution from nonlinear interactions is likely much smaller than in the
case of multi-MeV electron precipitation.

Towards the first step, we start from the full quasi-linear expressions for
the precipitating to trapped flux ratio at ELFIN
$j_{prec}(\alpha)/j_{trap}(\alpha_{trap})$ ([Kennel & Petschek, 1966] and [Li
et al., 2013]). Pitch angles are referenced to the equator and $\alpha_{trap}$
is such that $\ln(\sin\alpha_{trap}/\sin\alpha_{LC})\sim 1/20$ (where
$\alpha_{LC}$ is the equatorial loss-cone angle). Averaging $j_{prec}$ over
equatorial pitch-angles $\alpha<\alpha_{LC}$, we get

$\frac{j_{prec}}{j_{trap}}\simeq\int_{0}^{1}\frac{I_{0}(z_{0}x)/I_{0}(z_{0})}{1+(z_{0}/20)I_{1}(z_{0})/I_{0}(z_{0})}dx,$

where $z_{0}=2\alpha_{LC}/\sqrt{D_{\alpha\alpha}\tau_{B}}$ [Li et al., 2013],
and $\tau_{B}\sim\gamma/(\gamma^{2}-1)^{1/2}$ is the bounce period [Schulz &
Lanzerotti, 1974]. This integral can be approximated as:
$j_{prec}/j_{trap}=0.9/z_{0}$ with less than 20% error for $z_{0}\in[0.9,8]$,
corresponding to $j_{prec}/j_{trap}\in[0.11,1]$. At low energies, $E<E^{*}$,
and for quasi-parallel left-hand-polarized hydrogen band EMIC waves [Kersten
et al., 2014] with a monotonically decreasing power $B_{w}^{2}(f)$ toward
higher frequencies $f>f_{peak}$ in Van Allen Probe statistics [Zhang et al.,
2016c], the most efficient wave-driven pitch-angle diffusion of electrons near
the loss-cone through cyclotron resonance should occur for $E(f)\sim
E_{R,\min}(f)$ at similar latitudes $\lambda_{R}$ close to the equator for all
$E$ in that range [Mourenas et al., 2016]. Bounce-averaging [Mourenas et al.,
2016] the full expression of the local pitch-angle diffusion rate in the cold
plasma approximation [Summers & Thorne, 2003; Su et al., 2012] then gives

$D_{\alpha\alpha}\approx\frac{B_{w}^{2}(f_{cp}/f)(1-f/f_{cp})^{3/2}}{(\gamma^{2}-1)^{1/2}\gamma\,(1-f/2f_{cp})}$

at an equatorial pitch-angle $\alpha\simeq\alpha_{LC}$.

Combining the theoretical scaling laws for $j_{prec}/j_{trap}$ and
$D_{\alpha\alpha}$ and the fit to ELFIN observations
$j_{prec}/j_{trap}\sim\gamma^{2}$ in Figure 18, gives us the estimate of the
wave-power spectrum consistent with these observations:

$\frac{B_{w}^{2}(E)}{B_{w}^{2}(E^{*})}=\frac{(1+2E)^{4}E(E+1)}{(1+2E^{*})^{4}E^{*}(E^{*}+1)}\cdot\frac{f(1-f_{peak}/f_{cp})^{3/2}(1-f/2f_{cp})}{f_{peak}(1-f/f_{cp})^{3/2}(1-f_{peak}/2f_{cp})}$

To obtain the mapping between energy and frequency, we utilize the cyclotron
resonance condition coupled to the cold plasma dispersion relation. Its
expression for $E<E^{*}$ and $f>f_{peak}$ is combined with its expression for
$E^{*}$ and $f_{peak}$, the (highest-)frequency of peak EMIC wave-power
($f_{peak}/f_{cp}\sim 0.37-0.41$, as seen in statistical wave observations
from the Van Allen Probes [Zhang et al., 2016c]). We also assume an ion
composition with $>$94% protons, as appropriate for when hydrogen band waves
are present [Kersten et al., 2014; Ross et al., 2022]. This yields a second
order equation for $f/f_{cp}$, with solution $f/f_{cp}\simeq
2/(1+\sqrt{1+4C})$, where
$C=(f_{cp}/f_{peak})(f_{cp}/f_{peak}-1)E(1+E)/\left(E^{*}(1+E^{*})\right)$.

The resonant frequency $f(E)/f_{cp}$, expressed as a function of the resonance
energy in the above equation, allows us, by substitution in the previous
equation, to obtain the EMIC wave-power ratio
$B_{w}^{2}(f)/B_{w}^{2}(f_{peak})=B_{w}^{2}(E)/B_{w}^{2}(E^{*})$ inferred from
ELFIN, with an error smaller than 40% for $j_{prec}/j_{trap}\in[0.11,1]$. Note
that the above normalizations of $j_{prec}/j_{trap}$ to its level at $E^{*}$
and of EMIC wave power to its level at $f_{peak}/f_{cp}$ allow us to eliminate
the dependencies of $j_{prec}/j_{trap}$ and $D_{\alpha\alpha}$ on
$f_{pe}/f_{ce}$ and on the non-normalized EMIC wave power at the equator
conjugate to ELFIN, which are unknown. Here we only had to assume that the
ensemble of $j_{prec}/j_{trap}$ measurements from ELFIN are statistically
compliant to an ensemble of EMIC wave power observations on another,
equatorial, platform in order to convert the energy spectrum of the
precipitation to a frequency spectrum of wave power that should be consistent
with that precipitation.

We proceed, now, to the second step in our consistency check between the ELFIN
observations of electron precipitation and independently collected EMIC wave-
power spectra, in order to assess the validity of quasi-linear diffusion in
describing the precipitation down to low energies. Figure 19(a) compares the
wave power ratio $B_{w}^{2}(f)/B_{w}^{2}(f_{peak})$ inferred above from ELFIN
statistics of $j_{prec}/j_{trap}$ (solid blue curve) with the measured
hydrogen band EMIC wave power ratio from Van Allen Probes statistics when
$f_{pe}/f_{ce}>15$ [Zhang et al., 2016c] in different MLT sectors (black,
green, magenta, and red curves). In the high density plasmaspheric plume or at
the plasmapause, as implied by the $f_{pe}/f_{ce}$ ratio criterion used in
this subset of wave data, the cyclotron resonance condition can indeed be
satisfied for the optimum parameters $E^{*}\sim 1.45$ MeV in Figures 18(a,b)
and $f_{peak}/f_{cp}\sim 0.37$ from the above wave statistics. Figure 19(c)
indicates that $200$ keV electrons are then in resonance with waves at
$f/f_{cp}\sim 0.8$ near the equator. We see in Figure 19(a) that the observed
statistical EMIC wave power ratios in the 12-22 MLT sector (magenta and red
curves) agree quite well with the power ratios inferred from ELFIN data (blue
curve with dashed lines indicating uncertainties) over two decades in power,
in the frequency range $f/f_{cp}\sim 0.37$ up to $f/f_{cp}\sim 0.8-0.95$.
Therefore, in high density regions (those with $f_{pe}/f_{ce}>15$), the
observed EMIC wave power ratio is consistent with that inferred from ELFIN
precipitation of $j_{prec}/j_{trap}$ from $\sim 1.45$ MeV down to $200$ keV,
sometimes even down to $100$ keV.

In regions of lower plasma density, with $5<f_{pe}/f_{ce}<15$, EMIC wave
statistics from the Van Allen Probes show that $f_{peak}/f_{cp}\sim 0.41$
[Zhang et al., 2016c], corresponding to a higher $E^{*}\sim 2.5$ MeV for
$f_{pe}/f_{ce}\sim 13-15$. For these parameters, Figure 19(b) shows that in
the 12-22 MLT sector the observed EMIC wave power ratios in the range
$f/f_{cp}\sim 0.41$ to $f/f_{cp}\sim 0.90-0.95$ [Zhang et al., 2016c] at Van
Allen Probes (magenta and red curves) are consistent with those inferred from
ELFIN statistics (blue curves). In the above $f_{pe}/f_{ce}$ range, the
frequencies where agreement prevails correspond to $j_{prec}/j_{trap}$ from
$\sim 2.5$ MeV down to $\sim 100-200$ keV.

It is evident from Figures 19(a,b) that the inferred and observed EMIC wave
power ratios agree well, and up to high frequencies (low resonance energies),
only in the 12-22 MLT sector, i.e., near dusk. This is quite consistent with
the fact that most of the EMIC wave-driven events at $L<7$ in our database
occurred near dusk (Figure 17(b)). It is also consistent with the predominance
of similar events at dusk in Firebird-II statistics [Capannolo et al., 2021].
In the 0-3 MLT sector (black lines in Figures 19(a,b)) the observed EMIC wave
power is consistent with EMIC wave-driven electron precipitation down to $\sim
300$ keV only when $f_{pe}/f_{ce}>15$. Few events exist in this sector in our
database. In the 4-12 MLT sector (green lines) the observed EMIC wave power is
too weak to drive significant electron precipitation below $\sim 1$ MeV,
consistent with the absence of precipitation events in that sector in our
database. (Note: Since EMIC wave power spectra depend weakly on $L$ in Van
Allen Probes statistics [Zhang et al., 2016c; Ross et al., 2021], the present
results should hold for $L\in[4,7]$.)

Therefore, we see that the observed, moderately efficient electron
precipitation at energies as low as $\sim 200-300$ keV could simply be due to
quasi-linear scattering by moderate intensity EMIC waves at high frequencies,
up to $f/f_{cp}\sim 0.8-0.9$, provided that such waves are present during a
majority of these events with a similar average intensity as in statistical
empirical wave models. Such waves may still be sufficiently below the proton
gyrofrequency to evade hot plasma effects, at least to first order [Chen et
al., 2011; Ross et al., 2021]. Assuming a maximum wavenumber that can be
attained in the presence of hot plasma effects given by $kc/\Omega_{pi}\sim 2$
(corresponding to a typical cold ion temperature of $\sim 10$ eV in a
plasmaspheric plume or just within the plasmasphere) for left-hand-polarized
hydrogen band waves in a plasma with more than 94% protons, the minimum
electron energy for cyclotron resonance can indeed be as low as $\sim 150$ keV
to $350$ keV for $f_{pe}/f_{ce}\sim 15$ to $30$, when a sufficient transient
hot $H^{+}$ temperature anisotropy $A>2.3$ generates such high-frequency waves
[Chen et al., 2011, 2013]. A typical precipitation event of this kind has been
analyzed in Section 4.2.

Another important point is that the measured ratios $j_{prec}/j_{trap}\sim
1/8-1$ from low energy, $E\ll E^{*}$, to $E^{*}$ actually correspond to a
quasi-linear diffusion close to the strong diffusion regime with
characteristic time scales for partial or full loss-cone filling on the order
of a quarter of a bounce period $\tau_{B}/4\sim 0.25$ s [Kennel & Petschek,
1966]. Such fast diffusive time scales are therefore still consistent with the
time scales of the observed electron precipitation at ELFIN. Thus, even in the
quasi-linear regime it is possible for temporal or spatial wave power
variations encountered by drifting and bouncing electrons to explain the
short-lived (sub-spin), bursty nature of the precipitation at ELFIN.

The present results agree with test particle hybrid simulations suggesting
that low-amplitude, high-frequency hydrogen band EMIC waves could be the main
cause of 0.55 MeV electron precipitation [Denton et al., 2019]. Several
previous studies have also provided hints of a prevalence of hydrogen band
waves in driving sub-MeV relativistic electron precipitation [Chen et al.,
2013; Qin et al., 2018; Zhang et al., 2021]. Even when such high-frequency
waves are not evident at apparently conjugate measurements at the equator,
this may be due to mapping uncertainties - the waves could still be present
$\gtrsim$ 0.5 Earth radii away where the ionospheric electron precipitation
measurements actually map to. Further analysis of EMIC wave data and sub-MeV
precipitation with ELFIN’s twin spacecraft, providing higher spatio-temporal
resolution of these phenomena, will be able to further study these points in
greater detail. In addition, one cannot totally rule out a possible role of
helium band EMIC waves in driving some sub-MeV electron precipitation, in
spite of the expected strong cyclotron damping very close to the helium
gyrofrequency [Chen et al., 2013; Cao et al., 2017a]. A similar study as in
Figure 19, but focusing on helium band waves, could be performed to check this
point. This would require extensive numerical calculations including hot
plasma effects in the dispersion relation [Chen et al., 2013].

As a final note, we comment on the global contribution of EMIC waves to sub-
MeV electron scattering. Although bursts of EMIC waves can drive relatively
intense sub-MeV electron precipitation in a narrow MLT sector, their very weak
time- and MLT-averaged power at high frequencies should prevent them from
contributing significantly to the global loss rates of sub-MeV electrons up to
high equatorial pitch-angles $\alpha\sim 90^{\circ}$, above and beyond what
can already be done by typical chorus waves in the dawnside trough [Mourenas
et al., 2016; Boynton et al., 2017; Agapitov et al., 2018; Drozdov et al.,
2019, 2020; Miyoshi et al., 2020; Ross et al., 2021; Zhang et al., 2022]. This
is because at a fixed energy $E\sim E^{*}$, cyclotron resonance between
electrons of equatorial pitch-angle $\alpha$ and hydrogen band EMIC waves at a
frequency $f$ indeed corresponds to a scaling
$\cos^{2}\alpha/\cos^{2}\alpha_{LC}\sim(f_{cp}/f-1)/(f_{cp}/f_{peak}-1)f_{peak}/f$.
This implies that electrons of higher $\alpha$ reach resonance with waves of
higher frequency $f>f_{peak}$ (e.g. see first standalone equation in Mourenas
et al. [2016]). Therefore, the steep decrease of EMIC wave power from
$f/f_{cp}\sim 0.4$ to $\sim 0.9$, by two orders of magnitude in Figure 19,
implies a rapid decrease of the quasi-linear pitch angle diffusion rate
$D_{\alpha\alpha}(\alpha)$ away from the loss-cone [Summers & Thorne, 2003;
Mourenas et al., 2016]. This is consistent with pitch angle bite-out
signatures (stronger electron loss near the loss-cone than at higher pitch
angles) produced by the most intense EMIC waves [Usanova et al., 2014; Adair
et al., 2022] at energies of $\sim 1-4$ MeV. Such signatures demonstrate the
inefficacy of EMIC waves in depleting the relativistic electron flux at
equatorial pitch angles far from the loss cone, absent intense chorus waves
[Mourenas et al., 2016; Zhang et al., 2017]. The same arguments hold at sub-
MeV energies, except that cyclotron resonant EMIC waves have higher
frequencies than at $\sim 1-4$ MeV and, therefore, much lower amplitudes,
making them even less efficient than typical intense chorus waves in the
dawnside trough in driving the precipitation of electrons near the loss-cone.

##  6 Summary and discussion

Relativistic electrons in the near-Earth environment are an important
contributor to space weather and may play a significant role in charged
particle energy input to the atmosphere from space. The question of how such
electrons, after being accelerated in near-Earth space by waves or transported
into it by, say, radial diffusion, are lost remains open. This question is
particularly vexing for such electron loss from the outer radiation belt, at
(L$<$7), which has the highest fluxes of such electrons and is thus most
important for space weather. Aside from magnetopause shadowing (the result of
magnetospheric compressions changing electron drift paths from trapped to
open), and field-line curvature scattering precipitation at the isotropy
boundary, both affecting L-shells beyond the outer radiation belt, and
whistler-mode wave-driven precipitation which mostly affects sub-MeV
electrons, most attention on relativistic electron loss has been placed on
these electrons’ interaction with EMIC waves. This is because EMIC waves can
resonate with electrons of highly relativistic energies $>1$ MeV under high-
density, low magnetic field plasma conditions that can be realized in the
outer edge of the plasmasphere, well inside the outer radiation belt at active
times. As they can act on trapped, outer radiation belt electrons, such waves
can change significantly the outer radiation belt fluxes and therefore are an
important process to include in space weather models. The recent launch of the
ELFIN mission, with a goal to determine whether EMIC waves are predominantly
responsible for this loss or whether other wave processes are implicated,
provided a unique dataset of 50 - 5000 keV electrons obtained on a polar, low-
altitude orbit, with which we can address this question comprehensively and
for the first time. In this paper, we primarily focused on the question of
whether EMIC waves can be definitively shown to be responsible for the
observed scattering of relativistic electrons, and whether whistler-mode
chorus waves (the other candidate for pitch-angle scattering of sub-MeV and up
to MeV electrons into the loss cone) simultaneously might be implicated.

After a short review of EMIC wave generation and its interaction with
energetic electrons in Section 2, we presented in Section 3 the first
comprehensive examples of typical EMIC wave-driven precipitation of such
electrons from ELFIN. The high energy and pitch-angle resolution of ELFIN
allowed us to identify and quantitatively investigate the energy range of such
precipitation and study its properties using the precipitating-to-trapped flux
ratio, $R$, measured by a single detector on a spinning platform, thus
avoiding problems with multiple detector inter-calibration. This ratio
increasing with energy and peaking in the range $>$0.5 MeV is a tell-tale
signature of EMIC wave-driven precipitation. In Section 3.1 we showed how this
spectral signature is differentiated quite well from that of whistler-mode
chorus driven precipitation which is a decreasing ratio with energy from as
low as 50 keV (the lower energy limit of the ELFIN detector) to $>$0.5 MeV.
(Both spectral types can be observed on occasion simultaneously, as expected,
since whistler-mode chorus and EMIC waves can co-exist – in that case, the two
spectral shapes can still be well separated.)

Next, in Section 3.2 we used a case study of typical EMIC wave-driven
emissions, accompanied by equatorial measurements of EMIC waves on THEMIS to
demonstrate that the bursty nature of the precipitation at ELFIN (lasting only
7-14s, or 2.5-5 ELFIN spin periods) is consistent with the spatial extent of
the EMIC wave region at the equator (lasting 6min). In this case, the
equatorial extent of the EMIC region was $\Delta L\approx$0.23 which is
consistent with published statistical averages of $\Delta L\approx$0.5 [Blum
et al., 2016, 2017]. Thus, the burstiness of EMIC wave-driven precipitation at
polar orbiting, low-altitude spacecraft like ELFIN is likely typical. We have
also shown that the burstiness can extend to sub-spin time-scales, 1-5 spin
sectors, at least in part due to the spatial variability of the wave-power at
the equator. This can cause the precipitating-to-trapped flux ratio to be, on
occasion, aliased and exceed unity. While there are reported cases when this
flux ratio on ELFIN exceeds unity due to nonlinear scattering, an assertion
supported by fortuitous measurements of the EMIC wave amplitudes at conjugate
observatories [Grach et al., 2022], conclusions on the preponderance and
significance of nonlinear effects cannot be drawn from the ratio alone without
further statistical or detailed case study analyses, both left for the future.
However, we have argued that nonlinear effects (from presumably mainly short
wave packets – e.g., see various examples of such short packets in [Usanova et
al., 2010] and [An et al., 2022]) are statistically incorporated into the
spectral shapes of precipitation at ELFIN, just as these high-amplitude EMIC
wave packets are incorporated into the statistical averages of equatorial wave
spectra, so that a direct comparison using a diffusive formalism should be
possible.

In Section 3.3 we showed that consecutive ELFIN passes over the same MLT
region can reveal the EMIC wave-driven precipitation’s spatio-temporal
evolution in location, extent (in $L$ and $\Delta L$), and intensity. This
allows us to study the EMIC wave-generation region in a geophysical context
(e.g., during substorms and storms) as well as the potential of the waves for
reducing the outer radiation belt flux.

To evaluate the consistency of the observed energy of moderate and strong
precipitation at ELFIN (ratios $R\in$[0.3,0.5] and $R>$0.5, respectively) with
the resonance energy expected for EMIC wave-driven precipitation, we utilized
in Section 4 conjunctions with ground based or equatorial assets. Using such
data-informed estimates of $f_{pe}/f_{ce}$ and of $f_{\max}/f_{ci}$, with only
occasional support from models, we examined the resonance energy that would be
consistent with these estimates. We found it to be in agreement with the
observed energies of moderate and strong precipitation seen at ELFIN. This
demonstrates that the observed precipitation as identified by the ratio $R$ is
indeed consistent with theoretical expectation from resonant interactions.

We proceeded, in Section 5, to study statistically an ensemble of EMIC wave-
driven events observed by ELFIN. These were identified in two years of ELFIN
data (2019-2021) as enhancements in the precipitating-to-trapped flux ratio
$R$, peaking at $>0.5$MeV. The average event duration on ELFIN, 17s (6 spins),
or $\Delta L\sim$ 0.56, is consistent with published reports of the typical
radial extent of EMIC waves in the magnetosphere, $\Delta L\sim$ 0.5 [Blum et
al., 2016, 2017], validating our assertion that the bursty nature of the
precipitation is due to the spatial localization of the EMIC wave interaction
region in the magnetosphere. The most populous category of events is those
occurring at pre-midnight and dusk, at $L\sim$5-7. A second class of events
was found at midnight, post-midnight and dawn at $L\sim$8-12. These two
categories are roughly co-located with the two main populations of EMIC waves
in the magnetosphere [Min et al., 2012], further solidifying our assertion
that our observations of relativistic electron precipitation are caused by
EMIC waves.

The peak precipitating-to-trapped flux ratio exceeds one for many of the
events studied. Considering that such a ratio, if scrutinized and confirmed,
could signify nonlinear interactions, the only means of exceeding the quasi-
linear strong diffusion limit (e.g., Kubota et al. [2015]; Grach & Demekhov
[2020]), we examined further how it may arise. We found that on many occasions
extreme burstiness of the precipitation, lasting 1-5 spin phase sectors was
evidenced in the data. (Each ELFIN spin period has 16 sectors and the loss
cone most often contains 6 such sectors.) Examination of EMIC waves at THEMIS
E on one event studied revealed that this burstiness is, in fact, to be
expected at ELFIN due to the spatial localization of the EMIC wave coherence
and amplitude arising from interference of wave packets in the presence of
extreme density gradients. This burstiness in precipitation can result in
temporal aliasing of ELFIN’s precipitating-to-trapped flux ratio, $R$,
resulting in values that can exceed 1 or be well below 1. Careful analysis of
individual events together with modeling of measured wave spectra from
ancillary datasets can, at times, confirm the nonlinear nature of the
scattering (Grach et al. [2021]). Either future extensive case-by-case
analysis or careful statistical analysis to identify and remove events subject
to aliasing is required to establish the preponderance of nonlinear effects.
However, we argued that by analogy with similar effects for whistler-mode
chorus waves, it is still quite likely that precipitation due to nonlinear
EMIC wave scattering can conform to a statistical analysis using a diffusive
treatment, because precipitation bursts contribute to the average flux (and
flux ratio) at a level commensurate to the contribution of nonlinear wave
power bursts in statistical averages of equatorial wave power. We proceeded
with such a statistical analysis even though we recognize that in small or
large part the precipitation and the associated wave amplitudes considered may
incorporate nonlinear effects.

Using this database we have shown (Section 5.1) that:

  * •

The typical energy of the peak precipitating-to-trapped flux ratio, $E^{*}$, a
measured proxy for the resonance energy at the frequency of peak wave power,
$E_{peak}$, is in the range $\sim 1-3$ MeV. This is consistent with
expectation for EMIC wave-electron resonant interactions, based on the cold
plasma dispersion relation and prior statistics of wave-power near peak wave-
power, $f_{peak}$, at the equator.

  * •

The above measured energy $E^{*}$ decreases with $L$-shell as $\sim L^{-0.5\pm
0.1}$. This dependence is also in good agreement with theoretical
expectations.

  * •

The typical energy of the half-peak of the precipitating-to-trapped flux
ratio, $E^{*}_{\min}$, a proxy for the minimum resonance energy $E_{R,\min}$
corresponding to significant wave-driven electron scattering toward the loss-
cone, is in the range $\sim 0.5-1.5$ MeV and falls off with $L$-shell at the
same rate as $E^{*}$. This too is consistent with theoretical expectations
based on equatorial wave-power estimates at the maximum frequency below the
ion gyrofrequency where wave-power remains significant. This shows that sub-
MeV electron precipitation accompanying multi-MeV electron precipitation, is
also likely driven by EMIC waves.

We next examined (Section 5.2) the properties of the most intense ($R>$1/2),
highly relativistic (having $E^{*}>1$ MeV), EMIC wave-driven electron
precipitation events, of the most populous category in our database (L$<$7).
We consider these to be the most efficient EMIC wave-driven events in our
database. Such events are highly correlated with geomagnetic activity. And, as
expected from the statistics of all events, they are predominantly seen near
dusk (MLT $\sim$ 18) with their occurrence rates dropping precipitously
(though still finite) at noon and post-midnight. We found that:

  * •

About 35% of the time, the most efficient EMIC wave-driven events are still
associated with moderate or strong ($R>$1/3) precipitation of electrons with
energies as low as 100-300 keV. Only 6.5% of the time do such events have a
definitively negative slope of $R$ versus E, at that low energy range. For
most events (at least 78% and potentially as many as 93.5%) that slope is
positive. This suggests that simultaneous energetic electron scattering by
whistler-mode chorus and EMIC waves (as was presented in Figure 1, right
column, and discussed in Section 3.1) are a minority, occurring for only one
in every 16 of the highly relativistic, strong EMIC wave-driven events.

  * •

The average $R$ versus $E$ spectrum of the most efficient EMIC wave-driven
events exhibits an exponential increase with energy (more precisely with the
square of the Lorentz factor, $R\sim\gamma^{2}$) up to its peak value $R\sim
1$ at $E\sim E^{*}$ (whose average value is $\langle E^{*}\rangle\sim 1.45$
MeV). Examined versus energy normalized to the peak-precipitation energy,
$E/E^{*}$, $R$ exhibits an exponential decay with normalized energy away from
(on both sides of) unity: ($R\sim(E/E^{*})^{-1}$ above and
$R\sim(E/E^{*})^{+1}$ below $E/E^{*}=1$). This decay is in agreement with
quasi-linear diffusion theory for resonant scattering by a typical equatorial
EMIC wave power spectrum that falls off away from its peak wave power, which
is consistent with observations.

  * •

Based on both points above, the majority of the weaker precipitation below $1$
MeV is thus probably due to electron resonant interaction with EMIC waves at
frequencies $f>f_{peak}$, i.e., above the peak power frequency of hydrogen
band EMIC waves (as was also inferred by the $E^{*}_{\min}$ decrease as a
function of L-shell in Section 5.1 and summarized above). Even though this
wave power is much smaller than at $f_{peak}$, it is still finite [Denton et
al., 2019; Zhang et al., 2016c, 2021]. But because of the significantly higher
trapped electron flux at lower energies, even waves of such small power may
still produce finite precipitating fluxes at $\sim 200-500$ keV, as observed.

  * •

Very low energy ($\sim 100$ keV) electron precipitation at times of strong,
highly relativistic precipitation, could still be predominantly due to
interactions with very low-amplitude high-frequency hydrogen band EMIC waves,
excited very near the equatorial gyrofrequency, $f_{cp}$, by $10-100$ eV
anisotropic ion populations [Teng et al., 2019; Zhang et al., 2021]. Such
waves can more easily experience strong cyclotron damping by low energy ions,
likely causing them to be transient and rendering their equatorial
observations sparse. Since $\sim 100$ keV electrons cannot reach cyclotron
resonance with the most frequent hydrogen band EMIC waves [Zhang et al.,
2016c], other precipitation mechanisms should also be examined and quantified.
Those include nonresonant scattering by EMIC wave-packets with sharp edges
[Chen et al., 2016; An et al., 2022], which can scatter electrons well below
the minimum resonance energy while still exhibiting a rising $R$ versus $E$
spectrum. However, simultaneous whistler-mode wave-driven precipitation which
is very efficient at such low energies, is clearly contributing to this
precipitation (as discussed above), even though it is likely a minority, and
should also be further considered [Artemyev et al., 2016; Ma et al., 2016b].

Our results confirm the crucial role played by EMIC waves in relativistic
electron losses in the Earth’s outer radiation belt (at $L$-shells between 5
and 7, where ELFIN detected most EMIC wave-driven precipitation events). We
find that the main predictions of the classical resonant scattering models of
EMIC wave-driven precipitation (e.g., typical energy of precipitation peaks,
the energy spectra and the $L$-shell dependence of minimum and peak-power
resonance energy) are consistent with ELFIN case studies and statistical
results presented herein. However, ELFIN’s observed fine structure of the
precipitation in energy, pitch-angle and time reveal interesting details
pertaining to the nature of low-energy precipitation concurrent with strong,
highly relativistic electron scattering, and the origin of precipitation
exceeding the strong diffusion limit, that deserve further attention.

Towards that end, significant new information on EMIC wave-driven
precipitation can be obtained by further combining ELFIN electron measurements
and theoretical modelling of wave-particle interactions with several ancillary
datasets: First, ELFIN conjunctions with ground-based and equatorial
spacecraft such as Van Allen Probes, THEMIS, ERG and MMS exist, can provide
important information on the EMIC waves implicated in the scattering. The
equatorial spacecraft can also provide local density and magnetic field
information, thus constraining the $f_{pe}/f_{ce}$ ratio. The large dataset of
ELFIN (spanning four years of operation, from 2018 to 2022) provides ample
opportunities for studies using such conjugate observations. These also
present good opportunities to investigate the relationship between the fine
structure of EMIC waves at the equator (wave packet coherence [Blum et al.,
2016, 2017], frequency ranges [Kersten et al., 2014; Zhang et al., 2016c], and
hot plasma effects [Cao et al., 2017a; Chen et al., 2019]) and the energy
distribution of precipitating electrons at ELFIN. Second, ELFIN’s energetic
particle detector for ions (EPDI), which measures total ions over the same
energy range as electrons (50 - 5000 keV), has been calibrated and has
collected data for approximately three months prior to the end of the mission
on both satellites. And finally, ELFIN’s fluxgate magnetometer (FGM)
instrument, which provides field-aligned current information and has the
capability of measuring waves up to a Nyquist frequency of 5Hz, has been
operating well and has provided data on both satellites for more than a year
towards the end of the mission (note that sub-spin resolution calibration to
reveal EMIC waves is still on-going). These datasets can provide information
on energetic ion scattering and precipitation by the same EMIC waves that
scatter energetic electrons in the inner magnetosphere (when the ion resonance
energy is sufficiently high at appropriate $f_{pe}/f_{ce}$ ratios), on the
location of the EMIC wave excitation relative to large scale magnetospheric
current system sources, and on EMIC waves potentially seen at ELFIN at the
same time as electron precipitation. Critical open questions are (i) What is
the overall contribution of nonlinear interactions and what can we learn about
the physics of such interactions from ELFIN’s precipitation energy and pitch-
angle spectrum? (ii) What effects are responsible for precipitation at sub-MeV
energies? (iii) What is the relative contribution of EMIC wave-driven
precipitation to the total loss of energetic electrons in the inner
magnetosphere and to the total atmospheric energy deposition by energetic
particles? There is already good theoretical background for further model
development of nonlinear resonant interactions [Albert & Bortnik, 2009; Kubota
et al., 2015; Kubota & Omura, 2017; Grach & Demekhov, 2020; Grach et al.,
2021] and nonresonant interactions [Chen et al., 2016; An et al., 2022]. Such
development can proceed on solid grounds only if guided by statistically-
derived properties of the observed electron precipitation and ancillary
measurements afforded by the above datasets.

###### Acknowledgements.

At UCLA, we acknowledge support by NASA awards NNX14AN68G (5/22/2014 –
7/15/2022) and 80NSSC22K1005 (7/16/2022 - present), NSF award AGS-1242918
(9/24/2014-7/31/2019) and AGS-2019950 (May 2020 – present). We are grateful to
NASA’s CubeSat Launch Initiative program for successfully launching the ELFIN
satellites in the desired orbits under ELaNa XVIII. We thank the AFOSR for
their early support of the ELFIN program under its University Nanosat Program,
UNP-8 project, contract FA9453-12-D-0285 (02/01/2013-01/31/2015). We also
thank the California Space Grant program for student support during the
project’s inception (2010-2014). The work at UCLA was also supported by NASA
contract NAS5-02099 for data analysis from the THEMIS mission. We specifically
thank: J. W. Bonnell and F. S. Mozer for the use of EFI data in determining
spacecraft potential derived density from THEMIS E, J. P. McFadden for use of
ESA data in obtaining electron temperature measurements to incorporate into
the procedure for electron density determination from the spacecraft
potential, and K. H. Glassmeier, U. Auster and W. Baumjohann for the use of
FGM data provided under the lead of the Technical University of Braunschweig
and with financial support through the German Ministry for Economy and
Technology and the German Center for Aviation and Space (DLR) under contract
50 OC 0302. Additionally, we thank the operators of the Magnetic Induction
Coil Array, in particular Dr. Marc Lessard of UNH and the operators of the
South Pole, PG3, and PG4 sites at NJIT, UNH, and Virginia Tech; and the
Sodankyla Geophysical Observatory for providing the Finnish pulsation
magnetometer network data. The induction coil magnetometer at South Pole is
operated and managed by New Jersey Institute of Technology (NJIT) under NSF
grant OPP-1643700. X.-J.Z. and A.A. acknowledge NASA awards 80NSSC20K1270,
80NSSC22K0517; X.-J.Z. also acknowledges NSF award 2021749; X.A. acknowledges
NSF grant NO. 2108582; W. Li acknowledges NSF award AGS-2019950 and NASA award
80NSSC20K1270; M.D.H. was supported by NSF AGS-2027210; X. M. acknowledges
NASA contract 80NM0018D0004 to the Jet Propulsion Laboratory, California
Institute of Technology; S.R.S. thanks Dasha Gloutak of CU Boulder and Samuel
Rickert of UCLA. We acknowledge the critical contributions and talent of
numerous volunteer team members (more than 300 students contributed to this
program since its inception) who made this challenging program both successful
and fun. ELFIN would not have been possible without the advice and generous
time contributions from numerous members of the science and technology
community who served as reviewers, became unofficial mentors, or were the
bouncing board of ideas during trade studies or tiger team deliberations.
Special thanks go to UCLA’s: Steve Joy and Joe Mafi; The Aerospace
Corporation’s: David Hinkley, Eddson Alcid, David Arndt, Jim Clemmons (now at
UNH), Chris Coffman, Joe Fennel, Michael Forney, Jerry Fuller, Brian Hardy,
Petras Karuza, Christopher Konichi, Justin Lee, Pierce Martin, Leslie
Peterson, David Ping, Dan Rumsey and Darren Rowen; NASA GSFC’s: Gary Crum,
Thomas Johnson, Nick Paschalidis, David L. Pierce, Luis Santos and Rick
Schnurr; NASA JPL’s: Matt Bennett, Olivia Dawson, Nick Emis, Justin Foley,
Travis Imken, Allen Kummer, Andrew Lamborn, Marc Lane, Neil Murphy, Keith
Novac, David R. Pierce, Sara Spangelo, and Scott Trip; AFRL’s: Travis Willett,
David Voss and Kyle Kemble; UCB’s: Peter Berg, Manfred Bester, Dan Cosgrove,
Greg Dalton, David Glaser, Jim Lewis, Michael Ludlam, Chris Smith, Rick
Sterling, and Ellen Taylor; Tyvak’s: Bill McCown, Robert Erwin, Nathan Fite,
Ehson Mosleh, Anthony Ortega, Jacob Portucalian, and Austin Williams; Cal
State University Northridge’s: James Flynn and Sharlene Katz; JHU/APL’s: Jeff
Asher and Edward Russell; Montana State University’s: Adam Gunderson;
Space-X’s: Michael Sholl; Planet Labs’: Bryan Klofas; and LoadPath’s: Jerry
Footdale.

## Conflict of Interest

The authors have no relevant financial or non-financial interests to disclose.

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Figure 1: Overview of two consecutive ELFIN A science zone crossings: one at
the nightside/dawn sector (left) and the other at the dayside/dusk sector
(right, primed panel letters). From top to bottom shown are 3 energy-
spectrograms (a-c), 2 pitch-angle spectrograms (d-e) and the satellite’s
L-shell and magnetic latitude (f) computed using the international geophysical
reference field (IGRF) model. All spectrograms show products derived from the
number-flux of electrons (measured in individual sectors, in units of $1/{\rm
cm}^{2}/{\rm s}/{\rm sr}/{\rm MeV}$) averaged over the selected pitch-angle
and energy range as follows: The energy spectrograms in (a) and (a′) are for
locally trapped electrons (only pitch angles outside the loss cone and anti-
loss cone, near perpendicular to the local field line direction, were
included); those in (b) and (b′) are for precipitating electrons (with pitches
in the loss cone); and those in (c) and (c′) are precipitating-to-trapped flux
spectra ratios formed from the panels right above. The pitch-angle
spectrograms in (d-e) and (d′-e′) are average fluxes in two broad energy
ranges: a low energy range, 80-270 keV, and a high energy range, 0.4 - 3.4
MeV. The horizontal lines demarcate 90$\deg$ (vertically centered solid line),
the loss cone (the other solid line) and the anti-loss cone (the dashed line).
Horizontal color bars above Panel (a) represent magnetospheric regions
identified based on the data and discussed in the main text. Arrows in Panels
(c′) and (e′) represent spectral features also discussed in the main text.
Figure 2: Overview of ELFIN A observations for a science zone crossing,
exhibiting a typical EMIC wave-driven precipitation signature. Format is
identical to that of Figure 1 Figure 3: Observations from THEMIS E at the
equator at an MLT and UT near those of the science zone crossing by ELFIN A
depicted in Figure 2. Panels (a-d) show about an hour of data centered around
an EMIC wave emission: power spectral density of the magnetic field measured
by the fluxgate magnetometer, FGM, instrument (a), wave normal angle (b),
ellipticity (c), electron density inferred from the spacecraft potential
computed on-board by the electric field instrument, EFI, and processed on the
ground using the measured electron temperature by the electrostatic analyzer,
ESA, instrument (d). The black dashed line is the $He^{+}$ gyrofrequency.
Panel (e) shows $\sim$2.5 min of data from a single magnetic field component
in a field-aligned-coordinate system (FAC), $B_{X,FAC}$. It is oriented
perpendicular to the average magnetic field direction (hence, near the plane
of polarization) and lies on a plane also containing the sunward direction.
Panel (f) is an expanded, $\sim$1 min long, view of the same quantity as above
it. Arrows in Panels (e) and (f) are discussed in the main text. Figure 4:
Average spectra of precipitating (red) and trapped (blue) electrons at the
moment of the strongest precipitation (04:10:42-04:10:48 UT) at ELFIN A for
the event of Figure 2. Detrended fluxes (solid, thick lines) are measured
averages (dotted, thin lines) minus the trends (dashed, thin lines). Trends
are average fluxes from 6s immediately before and 6s immediately after the
strongest precipitation interval.  Figure 5: An example of prolonged
relativistic electron precipitation, presumably due to the long-lasting
presence of EMIC waves in the equatorial magnetosphere. Three consecutive
northern hemisphere, equatorward science zone crossings by ELFIN A at post-
noon/dusk (MLT $\sim$16.75) are depicted in three panels per crossing (a-c;
d-f; g-i), arranged in time to have a common L-shell and magnetic latitude,
shown in Panel (j). Each crossing’s three panels have the same format as the
top three panels in Figures 1 and 2, i.e., they are energy spectrograms of
trapped fluxes, precipitating fluxes and precipitating-to-trapped flux ratio.
Figure 6: Overview of ELFIN A observations during a $He^{+}$ band EMIC wave-
driven event, on 02 Nov 2020, in a format similar to that of Figure 1. Figure
7: ELFIN A projections to the ionosphere in the north and south for the event
of Fig. 6. Diamonds and asterisks mark the start and end times of the
trajectories; crosses are 1 min tickmarks; thick traces denote times of
intense relativistic electron precipitation identified from Fig. 6(c) as a
putative EMIC wave-driven precipitation event. Figure 8: Magnetic field
spectra from the ground-based station at SPA in the Antarctic. As shown in
Fig. 7, SPA is in close conjunction with ELFIN-A during this event.
Superimposed in the spectra are $He^{+}$ and $O^{+}$ equatorial
gyrofrequencies (horizontal lines) using the magnetic field at their
equatorial projection, inferred from the T89 model. The vertical magenta lines
mark the time interval of ELFIN A’ science zone crossing during this event.
Figure 9: Theoretical estimates of minimum resonance energy and comparison
with observations for the event of Fig. 6. From top to bottom: average
precipitating-to-trapped flux ratio during the time of peak precipitation
(15:18:51-15:19:03UT); contour plot of minimum resonance energy (in MeV) as a
function of the maximum unstable frequency $f_{\max}$ (normalized to the
relevant ion cyclotron gyrofrequency $f_{cHe}$) and of the $f_{pe}/f_{ce}$
ratio for a $He^{+}$ ion concentration of 5$\%$; and same as the panel above,
but for a 10$\%$ $He^{+}$ concentration. Red and yellow colors depict the
electron energy ranges for which ELFIN A measured strong precipitation and
moderate precipitation (R$>$0.5 and 0.5$>$R$>$0.3, respectively). The plot
shows that resonance energies exhibiting moderate and strong precipitation at
ELFIN are consistent with the range of parameters $f_{\max}/f_{cHe}$,
$f_{pe}/f_{ce}$ inferred from in-situ measurements at conjugate platforms (at
the intersection of the corresponding grayed areas), for a reasonable range of
$He^{+}$ concentrations. Figure 10: Overview of ELFIN A observations during a
$H^{+}$ band EMIC wave-driven event, on 6 December 2020, in a format similar
to that of Figure 1. Figure 11: ELFIN A projections to the ionosphere in the
north and south for the event of Fig. 10, in a format similar to that of
Figure 7. Figure 12: Magnetic field power spectral density (arbitrary units)
from the Finland ground-based station at SOD, located as shown in Fig. 11.
Superimposed $H^{+}$, $He^{+}$ and $O^{+}$ equatorial gyrofrequencies
(horizontal lines) using the equatorial magnetic field conjugate to these
stations. The magenta vertical lines bracket the time interval of ELFIN A’s
science zone crossing during this event. Figure 13: Theoretical estimates of
minimum resonance energy and comparison with observations for the event of
Fig. 10. From top to bottom: average precipitating-to-trapped flux ratio
during the time of peak precipitation (20:20:33 - 20:20:48 UT); contour plot
of minimum resonance energy (in MeV) as a function of the maximum unstable
frequency $f_{\max}$ (normalized to the relevant ion cyclotron frequency,
$f_{cp}$), and of the $f_{pe}/f_{ce}$ ratio for $0\%$ helium concentration;
and same as the panel above but for a $2.5\%He^{+}$ concentration. Red and
yellow colors depict the electron energy ranges for which ELFIN A measured
strong and moderate precipitation (R$>$0.5 and 0.5$>$R$>$0.3, respectively).
The plot shows that resonance energies exhibiting moderate and strong
precipitation at ELFIN are consistent with the range of parameters
$f_{\max}$/$f_{cp}$ and $f_{pe}/f_{ce}$ inferred from in-situ observations at
conjugate platforms ($f_{\max}/f_{cp}\sim$0.6-0.9 and
$f_{pe}/f_{ce}\sim$10-12) for a reasonable range of $He^{+}$ concentrations.
Figure 14: Overview of ELFIN A observations on 10 September 2020, which is
being examined together with concurrent TEC observations. Two consecutive
northern hemisphere, equatorward science zone crossings at midnight/pre-
midnight (MLT $\sim$0.75) are depicted in three panels per crossing (a-c; and
f-g), arranged in time to have a common L-shell and magnetic latitude, shown
in Panel (e). Each crossing’s three panels have the same format as the top
three panels in Figures 1 and 2, i.e., they show energy spectrograms of
trapped fluxes, precipitating fluxes and precipitating-to-trapped flux ratio.
A fourth panel for each crossing (Panels (d) and (h), respectively) show the
TEC values at the satellite ionospheric projection (after averaging over 3
degrees each in latitude and longitude, and over 20min in time). Figure 15:
Projections of ELFIN A science zone crossings of Figure 14 onto the Madrigal
TEC maps at 11:30:00 UT (top panel) and 13:00:00 UT (bottom panel) in the
northern hemisphere: start and end times of ELFIN A trajectories are marked by
a diamond and an asterisk, respectively; crosses are 1 min marks. Regions of
interest along the ELFIN trajectory are denoted with same colors as at the top
of Figure 14 (blue: plasma sheet, magenta: outer radiation belt, black:
plasmasphere). The thick magenta line sections depict the times of
relativistic electron precipitation related to EMIC wave scattering identified
from Figure 14, Panels (c) and (c$*\prime$), respectively. Numbered black
traces in the TEC maps show contours of TEC. Figure 16: Properties of
relativistic electron precipitation inferred from $\sim 50$ EMIC wave-driven
events (comprising $\sim 310$ individual spin samples) measured by ELFIN A&B.
(a) Ensemble spatial distribution versus $L$-shell. (b) Occurrence rate of
events (total event duration normalized by the ELFIN residence time in each
bin) in $(L,MLT)$ space. (c) Peak precipitating-to-trapped flux ratio at each
spin sample (circle) and average of that ratio within each $L$-shell bin
(solid curve), both shown as a function of $L$-shell. (d) Energy of the peak
precipitating-to-trapped electron flux ratio, $E^{*}$, at each spin sample as
a function of $L$. (e) Energy of half-peak (below the peak) of the
precipitating-to-trapped electron flux ratio, denoted by $E^{*}_{min}$ at each
spin sample, shown as a function of $L$. Red dashed lines in Panels (d) and
(e) depict theoretical estimates of the respective quantities based on
statistical wave spectra (derived as discussed in text). Dashed and dotted
blue lines in Panel (e) depict theoretical lower limits of the resonance
energy estimated based on wave cyclotron damping at $kc/\Omega_{pi}\sim 1$ and
$kc/\Omega_{pi}\sim 2$, respectively. Figure 17: Properties of the most
efficient EMIC wave-driven precipitation events, those exhibiting highly
relativistic (E${}^{*}>$1 MeV), strong (R = jprec/jtrap $>$ 1/2) electron
precipitation, observed at L $<$ 7\. (a) Fraction of these most efficient
events in each $AE^{*}$ bin divided by total number of most efficient events
($AE^{*}$ is the maximum AE in the preceeding 3 hours). (b) Same as (a) but as
a function of MLT. (c) Fraction of most efficient events that also have
$0\leqq R<1/3$) in the three low-energy range categories listed in the
abscissa. The rest (those with $R>1/3$), representing concurrent low-energy
moderate or strong precipitation, can still be a significant fraction of the
most efficient EMIC wave-driven precipitation events (15 - 35%, depending on
energy range category). (d) Fraction of most efficient EMIC wave-driven
precipitation events satisfying two different criteria for
$R=j_{prec}/j_{trap}$ at lower energy as defined in annotations. Implications
are discussed in the main text. Figure 18: Spectral properties of EMIC wave-
driven electron precipitation. (a) Average spectrum of trapped fluxes (black
curve) and of precipitating-to-trapped flux ratio (red curve) for events with
$j_{prec}/j_{trap}>1/2$ at $E^{*}>1$ MeV. The dotted blue curve shows a best
least-squares fit to the average flux ratio at low energy. (b) Average
precipitating-to-trapped flux ratio for the events of Panel (a), plotted as a
function of the normalized energy $E/E^{*}$, where $E^{*}$ is the peak
precipitating-to-trapped flux ratio energy for each event (a proxy for the
minimum resonance energy $E_{R,\min}$ with the most intense waves). Best
least-squares fits to the average flux ratio are shown below the peak,
$\langle j_{prec}/j_{trap}\rangle\approx\gamma^{2}(E)/\gamma^{2}(E^{*})\sim
0.065\,\gamma^{2}(E)$ for $E^{*}=\langle E^{*}\rangle\sim 1.45$ MeV (dashed
blue curve), and above the peak, $\langle j_{prec}/j_{trap}\rangle\approx
1.01\cdot(E/E^{*})^{-1.01}$ (dashed red curve). Note that if the last point
($E/E^{*}\approx 4.84$) with a small number of measurements was discarded, the
best fit would become $\langle j_{prec}/j_{trap}\rangle\approx
0.92\cdot(E/E^{*})^{-0.8}$ (dotted red curve).  Figure 19: (a) EMIC wave power
ratio $B_{w}^{2}(f)/B_{w}^{2}(f_{peak})$ inferred, using quasi-linear theory,
from ELFIN statistics of precipitating-to-trapped electron flux ratio
$j_{prec}/j_{trap}$ (solid blue curve), and compared with statistical EMIC
wave power ratios obtained from Van Allen Probes 2012-2016 observations in
four different MLT sectors when $f_{pe}/f_{ce}>15$ (black, green, magenta, and
red curves). Specifically, the solid blue curve was derived from the best fit
to $\langle j_{prec}/j_{trap}\rangle$ in Figure 18(a) and the dashed blue
curves show its uncertainty range, corresponding to the uncertainty in $E^{*}$
from the measurements, also depicted in Panel (c), below. (b) Same as (a) but
for $f_{pe}/f_{ce}\in[5,15]$. (c) Energy $E$ of electrons near the loss cone
in cyclotron resonance with EMIC waves at $f/f_{cp}$ in Panel (a), assuming
waves at $f_{peak}/f_{cp}\sim 0.37$ in resonance with $E^{*}\sim 1.45$ MeV
electrons for $f_{pe}/f_{ce}>15$. Dashed curves are based on the measurement
uncertainty range in $E^{*}$. (d) Same as (c) for $f_{peak}/f_{cp}\sim 0.41$
and $E^{*}\sim 2.5$ MeV for $f_{pe}/f_{ce}\in[5,15]$, corresponding to wave
power ratios in Panel (b).