11institutetext: Department of Physics and Astronomy, University of Turku,
FI-20500 Turku, Finland
11email<EMAIL_ADDRESS>22institutetext: LPC2E/CNRS, UMR 7328,
3A Avenue de la Recherche Scientifique, Orleans, France 33institutetext: Space
Sciences Laboratory, University of California, Berkeley, CA, USA
44institutetext: The Blackett Laboratory, Department of Physics, Imperial
College London, London, UK 55institutetext: Center for mathematical Plasma
Astrophysics, Department of Mathematics, KU Leuven, Celestijnenlaan 200B,
B-3001 Leuven, Belgium 66institutetext: Queen Mary University of London,
School of Physics and Astronomy, London, UK 77institutetext: The Johns Hopkins
University Applied Physics Laboratory, Laurel, MD 20723, USA 88institutetext:
Swedish Institute of Space Physics, P.O. Box 537, SE-751 21 Uppsala, Sweden
99institutetext: Institute of Experimental & applied Physics, Kiel University,
24118 Kiel, Germany 1010institutetext: Heliophysics Science Division, NASA
Goddard Space Flight Center, Greenbelt, MD 20771, USA

# Relativistic electron beams accelerated by an interplanetary shock

Immanuel C. Jebaraj 11 N. Dresing 11 V. Krasnoselskikh 22 3 3 O. V. Agapitov
33 J. Gieseler 11 D. Trotta 44 N. Wijsen 55 A. Larosa 66 A. Kouloumvakos 77 C.
Palmroos 11 A. Dimmock 88 A. Kolhoff 99 P. Kühl 99 S. Fleth 99 A. Fedeli 11 S.
Valkila 11 D. Lario 1010 Yu. V. Khotyaintsev 88 R. Vainio 11

###### Abstract

Context. Collisionless shock waves have long been considered amongst the most
prolific particle accelerators in the universe. Shocks alter the plasma they
propagate through and often exhibit complex evolution across multiple scales.
Interplanetary (IP) traveling shocks have been recorded in-situ for over half
a century and act as a natural laboratory for experimentally verifying various
aspects of large-scale collisionless shocks. A fundamentally interesting
problem in both helio and astrophysics is the acceleration of electrons to
relativistic energies ($>300$ keV) by traveling shocks.

Aims. The reason for an incomplete understanding of electron acceleration at
IP shocks is due to scale-related challenges and a lack of instrumental
capabilities. This letter presents first observations of field-aligned beams
of relativistic electrons upstream of an IP shock observed thanks to the
instrumental capabilities of Solar Orbiter. This study aims to present the
characteristics of the electron beams close to the source and contribute
towards understanding their acceleration mechanism.

Methods. On 25 July 2022, Solar Orbiter encountered an IP shock at 0.98 AU.
The shock was associated with an energetic storm particle event which also
featured upstream field-aligned relativistic electron beams observed 14
minutes prior to the actual shock crossing. The distance of the beam’s origin
was investigated using a velocity dispersion analysis (VDA). Peak-intensity
energy spectra were anaylzed and compared with those obtained from a semi-
analytical fast-Fermi acceleration model.

Results. By leveraging Solar Orbiter’s high-time resolution Energetic Particle
Detector (EPD), we have successfully showcased an IP shock’s ability to
accelerate relativistic electron beams. Our proposed acceleration mechanism
offers an explanation for the observed electron beam and its characteristics,
while we also explore the potential contributions of more complex mechanisms.

###### Key Words.:

collisionless shocks – electron acceleration

##  1 Introduction

Shock waves are ubiquitous in space plasmas and are the most prolific particle
accelerators in most systems. They can be directly probed in the heliosphere
due to the presence of planetary bow shocks (created due to the planetary
obstacles in the solar wind flow), and interplanetary shocks which are driven
by solar activity such as coronal mass ejections (CMEs). CMEs are large-scale
eruptions of plasma and magnetic fields which travel away from the Sun that
when their speed exceeds the information speed of the medium (i.e. the fast-
magnetosonic speed), shock waves are generated. While ion acceleration in such
shocks has been extensively studied (Lee et al., 2012, and references
therein), electron acceleration is challenging due to the large scale
separation111Ratio of gyro-radii ($r_{L}$)
$\sim\sqrt{\frac{m_{i}}{m_{e}}}=42.8$, where $m_{i}$ and $m_{e}$ are the mass
of protons and electrons, respectively. and physical limitations such as their
retention at the shock and magnetization (see, Lembege et al., 2004, for
review). However, observational surveys have found a small number of
interplanetary (IP) shocks that were associated with a significant increase in
electron fluxes from sub-relativistic to relativistic energies (e.g., Sarris &
Krimigis, 1985; Lopate, 1989; Dresing et al., 2016; Mitchell et al., 2021;
Talebpour Sheshvan et al., 2023). Furthermore, a strong correlation was found
between the characteristics of relativistic ions and electrons, suggesting
potential similarities in their acceleration mechanisms (Posner, 2007; Dresing
et al., 2022). While acceleration of sub-relativistic electrons ($<10$ keV) by
IP shocks is widely acknowledged due to the common occurrence of type II radio
emissions (Krasnoselskikh et al., 1985; Bale et al., 1999; Magdalenić et al.,
2014; Jebaraj et al., 2020; Kouloumvakos et al., 2021; Jebaraj et al., 2021),
acceleration of relativistic electrons ($>300$ keV) by IP shocks remains an
open question.

The characteristics and dynamics of collisionless shock waves are complex and
are defined across multiple scales (Galeev & Karpman, 1963; Karpman, 1964;
Sagdeev, 1966; Kennel et al., 1985; Krasnoselskikh et al., 2013). These
complexities are further enhanced when addressing the behavior of electrons at
shock fronts (Balikhin et al., 1989; Trotta & Burgess, 2019; Agapitov et al.,
2023). An important property of shocks is their geometry, that is the angle
between the upstream magnetic field (B) and the shock normal
($\hat{\textbf{n}}$), $\theta_{Bn}$. Quasi-perpendicular shocks
($\theta_{Bn}>45^{\circ}$) are often considered more efficient in accelerating
electrons to high energies (Schatzman, 1963; Bulanov et al., 1990; Mann et
al., 2009). The typical width of these shocks are larger than the electron
gyro-radius (Walker et al., 1999; Hobara et al., 2010) and therefore their
rapid acceleration can be explained via two main mechanisms. The first is the
“fast-Fermi” process, where acceleration happens via magnetic mirroring (due
to the steep magnetic gradient at quasi-perpendicular shocks, Leroy &
Mangeney, 1984; Wu, 1984). This mechanism has been exploited for almost four
decades when discussing the so-called “field-aligned beams” (FABs hereon) of
energetic particles (e.g., Pulupa & Bale, 2008). The second process is a
gradient drift mechanism commonly known as “shock-drift acceleration” (SDA;
Hudson, 1965; Ball & Melrose, 2001) which is similar to fast-Fermi
acceleration, but deviates due to the fact that the electron drift along the
small-scale electric field gradients within the shock ramp (Krasnoselskikh et
al., 2002; Vasko et al., 2018; Dimmock et al., 2019; Hanson et al., 2020).

Spacecraft upstream of the terrestrial bow shock have routinely measured
$10-100$ keV (near-relativistic electrons) (e.g., Wilson et al., 2016) and far
higher energy electrons at other planetary bow shocks such as those of Jupiter
and Saturn (e.g., Masters et al., 2017). The most frequently observed
similarity in conditions across all these planetary bow shock observations is
the presence of a shock that has exceeded a certain critical point beyond
which the downstream sound speed ($c_{s}$) is greater than the flow speed.
Such shocks are termed, “super-critical” and behave fundamentally differently
from those where $c_{s}$ is slower than the flow speed downstream (sub-
critical). This is characterized best by the ratio between the shock speed and
the fast-magnetosonic speed known as the Mach number $M$. Super-critical
shocks are usually strong shocks, while sub-critical are weak shocks.
Depending on the shock geometry, super-critical shocks can generate ion-
kinetic structures in the region ahead of the shock (foreshock), which trap
and energize particles (Kennel, 1981; Kis et al., 2013). However, observations
of electron trapping and energization are rare in IP shock waves, which are
generally weak shocks (Armstrong & Krimigis, 1976; Sarris & Krimigis, 1985;
Shimada et al., 1999). Super-critical shocks may also be crucial for ion
acceleration and may be the reason for the good correlations between
relativistic ions and electrons at strong coronal shocks (Kouloumvakos et al.,
2019; Dresing et al., 2022).

In this study, we present measurements of relativistic electron beams found
upstream of a travelling shock wave driven by an IP coronal mass ejection
(ICME). To our knowledge, this is the first such study made possible thanks to
the enhanced time resolution (1 second) of the Energetic Particle Detector
(EPD; Rodríguez-Pacheco et al., 2020) instrument suite onboard Solar Orbiter
(SolO; Müller et al., 2013). We utilized the Electron Proton Telescope (EPT)
which covers electron energy ranges from $\approx 30$ keV – $300$ keV, and the
High Energy Telescope (HET) which covers $>400$ keV. The letter is structured
as follows; An overview of the event is presented in Sect. 2, experimental
details of the FABs and a demonstration of their origin from a remote location
of the shock are presented in Sect. 3.1, and a robust peak-intensity spectral
analysis in Sect. 3.2. Finally, a simple shock acceleration model for FABs is
presented in Sect. 4, before discussing the results and its implications in
Sect. 5

##  2 Overview of the insitu shock and associated energetic electrons

Figure 1: Overview of the in-situ shock and energetic electrons observed by
Solar Orbiter on 25 July 2022. The arrival of the shock at 06:22:47 is marked
by the vertical, gray dashed line in all panels. The first panel shows the
differential intensities of energetic electrons observed by the south
telescopes of EPT and HET (covering a stable pitch angle of
$\leavevmode\nobreak\ 180^{\circ}$ throughout the shown period). The upstream
FABs are observed between 06:06:00–06:08:30 UT and indicated by the gray
shaded region. Fig. 3 shows the differential intensities across all viewing
directions. The second panel presents the pitch-angle coverages for the
different viewing directions of EPT and HET, where the gradients in color
indicate the different telescope openings of EPD/EPT (smaller) and EPD/HET
(larger). The third & fourth panels show the components of the magnetic field,
and bulk flow velocity, while the fifth panel presents the ion number density
and proton temperature.

On 25th July 2022, at 06:22:45 UT, the SolO spacecraft encountered an IP fast-
forward shock wave as seen in the last three panels of Fig. 1. It was linked
to a weak C1 solar flare and CME on 23rd July 2022 at about 18:30 UT. A
remarkable aspect related with the in-situ shock crossing, as shown in the
first two panels of Fig. 1 (and Fig. 3), is the presence of upstream field-
aligned beams of relativistic electrons (up to $\sim 1$ MeV) marked by the
gray shaded region. The shock was also associated with an energetic storm
particle (ESP) event for electrons, the features of which are outside the
scope of this letter, but a short summary is provided in Appendix A.

The shock arrived at SolO approximately 36 hours after the solar event, which
results in a average transit speed of 1100 km s-1. In order to obtain the in-
situ characteristics of the shock wave, such as its speed and strength, a more
robust physical approach involving multiple steps was adopted.

Firstly, the shock normal ($\hat{\textbf{n}}$) was estimated using a
combination of the mixed-mode method, magnetic and velocity coplanarity
(Trotta et al., 2022, and references therein) and minimum variance analysis
(MVA; Sonnerup & Scheible, 1998) to be $\hat{\textbf{n}}=[0.93\pm
0.05,0.018\pm 0.12,-0.16\pm 0.25]$. Next, the shock’s obliquity was estimated
to be $\theta_{Bn}\sim 64^{\circ}\pm 6^{\circ}$ indicating a quasi-
perpendicular geometry. The upstream and downstream bulk flow speeds were then
estimated in the shock rest frame, $V_{u}^{sh}\sim 350\pm 50$ km s-1 and
$V_{d}^{sh}\sim 125\pm 50$ km s-1, which results in a shock speed, $V_{sh}\sim
775\pm 50$ km s-1 in the observers frame. The Alfvén Mach number ($M_{A}$) was
then estimated using the upstream Alfvén speed,
$c_{A}=B_{up}/\sqrt{\mu_{0}n_{i}m_{i}}$ (assuming $Z=1$, $m_{i}$ and $n_{i}$
are proton mass and number density), and proxies (for $M_{A}$) established in
Gedalin et al. (2021) to be $M_{A}\sim 3.3\pm 0.7$. Single spacecraft
techniques to obtain shock parameters are cumbersome and may possess large
uncertainties and proxies act as constraints. Another indicator of the shock’s
strength is its compression ratios: $r_{B}\sim$3 for magnetic compression
($B_{down}/B_{up}$) and $r_{gas}$ of at least $\sim$2.1 for gas compression
($n_{down}/n_{up}$). The estimated $r_{gas}$ may be influenced by data
uncertainties and therefore for simplicity, $r_{gas}$ is “at least” $\sim 2.1$
as for a quasi-perpendicular shock, $r_{gas}$ is expected to be similar to
$r_{B}$. This uncertainty was also considered when estimating $V_{sh}$ and
$M_{A}$.

##  3 Upstream field-aligned electron beams

###  3.1 Observational details and the properties of the electron beams

During the period of shock arrival (c.f. Fig. 1), the pitch-angle (PA)
coverage of EPD was unusually stable due to the very stable magnetic field
vector. The orientation of the magnetic field was north-south, resulting in
field-aligned anti-sunward propagating particles being detected in the south
telescopes of SolO/EPD instruments with PA close to 180∘. Consequently, the
north telescope measured particles propagating toward the Sun, while the sun
and anti-sun telescopes covered particle PA perpendicular to the magnetic
field ($\sim 90^{\circ}$). Taking this into account, electron observations
from only the south telescope are presented in the main text. Detailed
observations from all telescopes can be found in Appendix A. Since the
measured electron intensities may be subject to ion contamination, robust
contamination corrections were applied using different techniques for both EPT
(Appendix A.1) and HET (Appendix A.2) measurements.

The first panel in Fig. 1 presents the differential electron intensities
observed by the south telescopes of EPT and HET post-correction, which show a
clear increase in the electrons above the background. Thanks to the high time
resolution provided by EPT and HET (full resolution is 1 second), it was
possible to identify, for the first time, fine features, particularly,
electron FABs upstream of the shock between 06:06:00 and 06:08:30 UT ($\sim$
14 minutes prior to shock arrival) in the south facing telescopes (indicated
by the gray shaded region in Fig. 1). Such FABs have never been clearly
measured upstream of interplanetary shocks before, possibly due to the lack of
sufficient time resolution of the particle measurements. On closer inspection,
this particular FAB had two peaks within it which are distinguishable over the
same range of energies. Since the FABs were only observed by the south
telescopes, and the lower-energy STEP telescope only has a field of view in
the sunward direction, the FABs could not be observed in lower energies (sub-
relativistic, i.e. $<10$ keV). However, Langmuir wave packets were registered
by the Time Domain Sampler (TDS) instrument, which is a part of the Radio
Plasma Waves instrument suite (RPW; Maksimovic et al., 2020). Two wave packets
(not shown), first at 06:10:05 UT and then 06:11:59 UT, are likely evidences
of the FABs extending to sub-relativistic energies. Other than small amplitude
Langmuir waves, no strong wave activity was measured during this time, making
it improbable for the FABs to be generated locally. The presence of Langmuir
waves also indicates the remote origin of the beam, as it must have propagated
a distance, at the very least, of the order of its relaxation length (Ryutov,
1969; Voshchepynets et al., 2015; Jebaraj et al., 2023b).

Under the assumption that electrons of all energies were injected at the same
time (e.g., Vainio et al., 2013), a velocity dispersion of the FABs allows to
estimate the injection time and the propagated path length of the FABs. The
details of this analysis can be found in Appendix A.3. The results of the
analysis shown in Fig. 7 suggest that the electrons were accelerated in a
region $\sim$ 13 $R_{\odot}$ away from the observer and were injected at
06:05:46 UT.

###  3.2 Assessing the characteristics of electron acceleration using peak
energy spectra

Figure 2: Observation (top) vs model (bottom) comparison of the peak-intensity
electron spectra. In the upper two panels, points in aqua signify the first
peak, while points in blue denote the second peak. The corresponding fits are
indicated by orange for the first peak and blue for the second. The background
that has been subtracted from each energy channel is represented by points in
gray.(top-left) Broken power-law fits applied to both peaks. (top-right)
Broken power-law fits with an exponential roll-over (or cutoff) for the two
peaks. (bottom-left) Ensemble of runs indicated by the gray shaded region with
varying $\kappa\sim 3-4$ and $\theta_{Bn}\sim 87^{\circ}-89.5^{\circ}$. A
single power-law fit and exponential roll-over (blue dashed line), and a
broken power-law fit (red dashed line) are shown for the modeled mean. The fit
for the first peak from top-left panel is illustrated by the olive-black dash-
dot line. The mean roll-over ($E_{c}$) and break energies ($E_{b}$) of the
ensemble and its standard deviation are also shown. (bottom-right) Modeled
results using $\kappa\sim 3.5$ for different $\theta_{Bn}$
($88^{\circ}-89.5^{\circ}$). The corresponding break energies are marked by
the vertical dashed lines.

The generation and propagation of the FABs can be understood better by
constructing a peak-intensity spectrum across all observed energy channels
(e.g., Dresing et al., 2020). The two top panels of Fig. 2 present background-
subtracted and contamination corrected electron peak-intensity spectra of the
two well-defined peaks within the FAB, using measurements from both EPT and
HET south-facing telescopes and a time resolution of 15 seconds. Light gray
markers represent the subtracted background, while the colored markers
indicate the peak electron fluxes.

The method for spectral fitting outlined in Dresing et al. (2020); Strauss et
al. (2020) allows for different fit models. For the FABs, we use a broken
power law defined by two different spectral indices $\delta_{1}$ and
$\delta_{2}$ separated at a break-energy $E_{b}$, a single power law
($\delta_{1}$) with an exponential roll-over or cutoff energy $E_{c}$, and a
broken power law with an exponential roll-over at higher energies. Depending
on the respective peak and the energy range taken into account in the fitting
procedure, different models that fit the data were obtained. The top-right
panel of Fig. 2 shows the fit applied to both peaks over all available EPT
channels and therefore fitting also the “ankle-like” feature below $100$ keV,
which is more pronounced in the spectrum of the second peak than the first.
The resulting fits both yield a similar exponential roll-over at higher
energies. If the ankle is excluded from the fit as done in Fig. 2 (top-left),
the fit results in a broken power law for both peaks. Although these fits seem
to yield a better agreement with HET points, it is hard to conclude what type
of fit, namely, a broken power law or an exponential roll-over is the best
representation of the spectrum. This is on the one hand due to the HET
electron peak fluxes potentially being subject to a small amount of proton
contamination, which would lead to slightly increased fluxes, and on the other
hand because of the ion-contamination correction, which was applied to the EPT
data points, not being perfect. The presence of a break or spectral roll-over
is without doubt, as it is already evident in the uncorrected EPT data (not
shown).

The cutoff energies, $E_{c}$ for both the fits on the right panel are similar,
$\sim 200$ keV. However, the break energies, $E_{b}$ for the fits in the left
panel are both found above $\sim 200$ keV, but are slightly different. The
most significant difference in the two spectra was at lower energies, the
second peak exhibits a more pronounced ankle at approximately $\sim$ 70 keV.
The lower spectral index of the first peak in the right panel is
$\delta=-1.19\pm 0.14$, while that of the second peak is $\delta=-1.96\pm
0.54$. For comparison, the spectral indices for the first peak in the left
panel are, $\delta_{1}=-1.66\pm 0.08$ and $\delta_{2}=-5.69\pm 2.09$, while
that of the second peak are $\delta_{1}=-0.94\pm 0.06$ and
$\delta_{2}=-5.02\pm 0.79$, respectively. We also fit an averaged spectrum of
a time period of 2.5 min containing both peaks, which results in a broken
power law with $\delta_{1}=-1.4\pm 0.04$, $\delta_{2}=-5.2\pm 2.58$ and
$E_{b}=261.8\pm 47$ keV (not shown).

##  4 A model for electron acceleration at IP shocks

From the observations presented in this letter, it is understood that the FABs
were accelerated at a remote location on the shock similar to how they are
accelerated at the near-perpendicular terrestrial bow shock. If so, then the
curvature of the shock (or large scale deformations for IP shock at 1 AU) and
the time-varying properties of the upstream magnetic field become crucial.
Acceleration happens under the exclusive assumption that there exists a region
on the shock wave separated by $\sim 13R_{\odot}$ that is near-perpendicular
($89.9^{\circ}>\theta_{Bn}>85^{\circ}$) and super-critical 222When the
downstream medium is subsonic, i.e. the sound speed $c_{s}>V_{d}^{sh}$, the
shock undergoes fundamental changes in its evolution and structure
(Krasnosel’skikh, 1985) (Appendix B). When these conditions are met,
acceleration happens based on the conservation of magnetic moment (fast-Fermi
mechanism, Sonnerup, 1969; Wu, 1984; Leroy & Mangeney, 1984) through which
electrons can gain a significant amount of energy in a single encounter with a
near-perpendicular shock.

A primary aspect of super-critical shocks is that in order to dissipate excess
energy, they reflect a portion of upstream ions forming an overshoot magnetic
field (Krasnoselskikh et al., 2013). The steep magnetic gradient of the
overshoot enables the near-relativistic and relativistic electrons to conserve
their magnetic moment and as such, a relativistic approach to the fast-Fermi
mechanism must be adopted (Appendix B). It is noteworthy to mention that a
sub-critical shock is also capable of reflecting electrons but to a far lesser
degree as far less particles would participate in such a process. The second
important assumption is that in a near-perpendicular geometry, the shock speed
along the magnetic field line scales as a function of geometry. This limits
acceleration to only those particles that exceed it, resulting in dilute
beams.

We perform semi-analytical modeling based on the relativistic form of the
reflection conditions provided in Leroy & Mangeney (1984). When
$\theta_{Bn}>89.5^{\circ}$, the shock speed must also be Lorentz transformed
as it is relativistic. The upstream “seed” electron distribution function was
assumed to be a kappa-distribution (Appendix B) with $\kappa\sim 3-4$
(considered realistic, Krauss-Varban & Burgess, 1991). For the parameters of
the shock, an upstream magnetic field similar to the one measured in-situ by
SolO was used, i.e. $\sim$10 nT. The magnetic compression including the
overshoot magnetic field is considered to be $B_{overshoot}/B=7$ (based on,
Mellott & Livesey, 1987) yielding a maximum $B$ of 70 nT. Lastly, the effect
of the electrostatic potential cannot be ignored for electrons and as such,
100 eV was used (obtained through proxies, Appendix B).

The modeling results are presented in the bottom two panels of Fig. 2. An
ensemble of runs was performed using kappa values between $\kappa\sim 3-4$,
and shock obliquity between $\theta_{Bn}\sim 87^{\circ}-89.5^{\circ}$. The
gray shaded region in the bottom-left panel of Fig.2 shows the results and two
different fits to the mean of all runs, namely, a single power-law with a
roll-over (blue dashed line), and a broken power-law (red dashed line). The
first has a power-law with spectral index $\delta\sim-2.4$ which is followed
by an exponential roll-over, while the second has two power-laws with spectral
indices $\delta_{1}\sim-2.5$ and $\delta_{2}\sim-6.6$. Both fits are used, but
the $\chi^{2}$ of the first fit is lower than that of the second. The mean
cutoff energy ($E_{c}$) for the first fit was $\sim 290$ keV and the mean
break-energy ($E_{b}$) of the second fit was $\sim 330$ keV. The vertical red
shaded region shows the $2\sigma$ standard deviation of both fits which are
similar. For comparison, the broken power-law of the first peak (i.e.,
$\delta_{1}\sim-1.66$, and $\delta_{2}\sim 5.7$, marked by the orange fit in
top-left panel of Fig. 2) from observations is represented in black. Both the
modeled spectral indices and the break-energy range are comparable to the
observations.

In a general sense, fast-Fermi acceleration simply shifts the seed
distribution minus the transmitted electrons to higher energies (see, Appendix
B). This is also corroborated by the fact that a single power-law with an
exponential roll-over produces a better $\chi^{2}$ than a broken power-law
fit. The mechanism does not change the shape of the source
$\kappa$-distribution function. The shift (i.e. energy gain) itself is highly
dependent on the shock obliquity, and this is clearly shown in the bottom-
right panel of Fig.2. When $\kappa=3.5$ is a fixed value and the obliquity is
varied between $\theta_{Bn}=88^{\circ}-89.5^{\circ}$, the resulting energy
spectra are significantly different for energies above 100 keV. The difference
is quantified by the dotted vertical lines representing the break energies
($E_{b}$) for each $\theta_{Bn}$. The effect of a changing $\kappa$ index
would then be responsible for the steepness of the spectrum since it
determines the number of electrons in the tail. A $\kappa=2$ can be
characterized with a power-law index of $\delta\sim-1.5$ at suprathermal
energies and an exponential roll-over above a certain energy range (Oka et
al., 2018). Then, the steepest result possible is when the distribution tends
towards a Maxwellian distribution as $\kappa\rightarrow\infty$.

Finally, it is worth mentioning that the model used here makes two main
simplifications, namely, a 1D shock profile, and that the electron’s magnetic
moment is conserved. In reality, super-critical shocks are complex
(Krasnosel’skikh, 1985; Lembege & Savoini, 1992; Balikhin et al., 1997), and
acceleration by multiple reflections and subsequent gradient drift (SDA) may
also take place (see, Appendix B for more details). In such cases the
resultant distribution will deviate from what is presented here and may
resemble a broken power-law.

##  5 Discussion of the results

On 25 July 2022 a travelling IP shock wave accompanied by an energetic storm
particle event was recorded in-situ by Solar Orbiter. Notably, field-aligned
beams of relativistic electrons were observed 14 minutes prior to shock
arrival. These are the first such observations of electron FABs measured at IP
shocks thanks to the high time-resolution of the EPD instrument suite onboard
SolO. This letter presents their observational characteristics and proposes a
scenario for their acceleration at IP shocks.

A combination of a lack of upstream magnetic structures, and the velocity
dispersion of the FABs lead to the conclusion that they originated at a remote
location of the shock. Velocity dispersion analysis suggested that the beams
originated $\sim$13$R_{\odot}$ away from the spacecraft. Further corroboration
was provided by the presence of plasma waves (Langmuir waves, Filbert &
Kellogg, 1979; Kasaba et al., 2000), which are a consequence of beam
propagation and relaxation (quasi-linear relaxation, Vedenov et al., 1962).

Galeev & Karpman (1963) initially postulated that shocks evolving in a low-
density, inhomogeneous plasma might develop decay instabilities that deform
the laminar shock front. In higher dimensions, large-scale deformations lead
to curved fronts on the scale of the upstream inhomogeneities, spanning
several $R_{\odot}$ to tens of $R_{\odot}$ (e.g., Wijsen et al., 2023) which
play a significant role in FAB acceleration (Bulanov & Sakharov, 1986; Decker,
1990; Bulanov & Krasnoselskikh, 1999; Lario et al., 2008). With any curved
front, a finite region exists where $\theta_{Bn}\geq 85^{\circ}$ and
approaches $90^{\circ}$, allowing for efficient electron acceleration along
the tangent magnetic field line as demonstrated in this letter. When the
observer is connected to the tangent field line, the time-varying properties
of the FABs are likely influenced by both deformation induced curvature and
variations in the upstream field topology (e.g. Giacalone, 2017; Trotta et
al., 2023).

Similarities between the observed and modeled peak-intensity spectra of the
FABs serve as a demonstration of shock acceleration at an IP shock under the
right circumstances. The model naturally forms a roll-over or break at higher
energies ($>200$ keV), which is also seen in observations. If a broken power-
law were fit to the modeled results, the break-energy, $E_{b}$ would fall
between 190 keV and 400 keV comparable to the observations. This is
considerably different from the transport-related breaks discussed in previous
studies (Dresing et al., 2021), which have suggested that the presence of two
different spectral breaks in peak-intensity spectra of solar energetic
electron events are caused by transport-related effects. The spectral fits and
the slopes obtained here are unique since it is the first such result obtained
close to the source, that is, the IP shock. Particularly, the spectral break
observed at high energies (i.e. $>200$ keV) reported for the first time here
is therefore likely a direct consequence of acceleration.

Both fast-Fermi and SDA mechanisms are candidates for electron acceleration.
The characteristics of these mechanisms depend on the conditions at the
source, the seed distribution function, and the shock properties (e.g.,
obliquity, criticality, etc.). The solar wind conditions in reality are
constantly changing, altering the conditions for acceleration, favoring a
rapid mechanism. Furthermore, it is possible that electrons accelerated in
such hot-spots can drift back onto the shock due to the
$\textbf{E}\times\textbf{B}$ drift at large distances and undergo further
acceleration via a first-order Fermi process.

This letter presents novel observations of relativistic electron beams and
demonstrates the efficiency of IP shock acceleration of electrons at 1 AU. The
results highlight the importance of shock geometry, large-scale deformations,
and the resultant curvature in electron acceleration. This research paves the
way for investigating similar phenomena closer to the sun, where shock waves
are inherently curved, predominantly near-perpendicular, and are in proximity
to dense populations of energetic particles (Jebaraj et al., 2023a).

###### Acknowledgements.

I.C.J and N.D. are grateful for support by the Academy of Finland (SHOCKSEE,
grant No. 346902). We also acknowledge funding by the European Union’s Horizon
2020 research and innovation program under grant agreement No. 101004159
(SERPENTINE). I.C.J acknowledges support from ISSI as part of the visiting
scientist program. I.C.J, V.K, and O.V.A. acknowledge support from the
International Space Science Institute (ISSI) in Bern, through ISSI
International Team project number 23-575, “Collisionless Shock as a Self-
Regulatory System”. V.K., acknowledges financial support from CNES through
grants, “Parker Solar Probe”, and “Solar Orbiter”. V.K., and O.V.A. were
supported by NASA grants 80NSSC20K0697 and 80NSSC21K1770. O.V.A was partially
supported by NSF grant number 1914670, NASA’s Living with a Star (LWS) program
(contract 80NSSC20K0218), and NASA grants contracts 80NNSC19K0848,
80NSSC22K0433, 80NSSC22K0522. A.L. is supported by STFC Consolidated Grant
ST/T00018X/1. A.K. acknowledges financial support from NASA NNN06AA01C (SO-SIS
Phase-E, PSP EPI-Lo) contract. N.W. acknowledges support from the Research
Foundation - Flanders (FWO-Vlaanderen, fellowship no. 1184319N). D.L.
acknowledges support from NASA Living With a Star (LWS) program
NNH19ZDA001N-LWS, and the Goddard Space Flight Center Heliophysics Innovation
Fund (HIF) program. I.C.J is grateful for the fruitful discussion with Dr.
Jens Pomoell regarding IP shocks. Solar Orbiter is a space mission of
international collaboration between ESA and NASA, operated by ESA. The open-
source Python packages SEPpy (Palmroos et al., 2022), solo-epd-loader
(Gieseler & Palmroos, 2023), and sunpy (Barnes et al., 2023) have been used in
obtaining and processing Solar Orbiter data.

## References

  * Agapitov et al. (2023) Agapitov, O. V., Krasnoselskikh, V., Balikhin, M., et al. 2023, ApJ, 952, 154
  * Armstrong & Krimigis (1976) Armstrong, T. P. & Krimigis, S. M. 1976, J. Geophys. Res., 81, 677
  * Bale et al. (1999) Bale, S. D., Reiner, M. J., Bougeret, J. L., et al. 1999, Geochim. Res. Lett., 26, 1573
  * Balikhin et al. (1989) Balikhin, M. A., Krasnosel’Skikh, V. V., & Woolliscroft, L. J. C. 1989, Advances in Space Research, 9, 203
  * Balikhin et al. (1997) Balikhin, M. A., Walker, S. N., de Wit, T. D., et al. 1997, Advances in Space Research, 20, 729
  * Ball & Melrose (2001) Ball, L. & Melrose, D. B. 2001, PASA, 18, 361
  * Barnes et al. (2023) Barnes, W. T., Christe, S., Freij, N., et al. 2023, Frontiers in Astronomy and Space Sciences, 10, 1076726
  * Bulanov et al. (1990) Bulanov, S., Kovrizhnykh, L., & Sakharov, A. 1990, Physics Reports, 186, 1
  * Bulanov & Sakharov (1986) Bulanov, S. & Sakharov, A. 1986, JETP Letters, 44, 543
  * Bulanov & Krasnoselskikh (1999) Bulanov, S. V. & Krasnoselskikh, V. 1999, in ESA Special Publication, Vol. 448, Magnetic Fields and Solar Processes, ed. A. Wilson & et al., 1047
  * Decker (1990) Decker, R. B. 1990, J. Geophys. Res., 95, 11993
  * Dimmock et al. (2019) Dimmock, A. P., Russell, C. T., Sagdeev, R. Z., et al. 2019, Science Advances, 5, eaau9926
  * Dresing et al. (2020) Dresing, N., Effenberger, F., Gómez-Herrero, R., et al. 2020, ApJ, 889, 143
  * Dresing et al. (2022) Dresing, N., Kouloumvakos, A., Vainio, R., & Rouillard, A. 2022, ApJ, 925, L21
  * Dresing et al. (2016) Dresing, N., Theesen, S., Klassen, A., & Heber, B. 2016, A&A, 588, A17
  * Dresing et al. (2021) Dresing, N., Warmuth, A., Effenberger, F., et al. 2021, A&A, 654, A92
  * Filbert & Kellogg (1979) Filbert, P. C. & Kellogg, P. J. 1979, J. Geophys. Res., 84, 1369
  * Fleth et al. (2023) Fleth, S., Kuehl, P., Kollhoff, A., et al. 2023, A&A, 676, A58
  * Galeev & Karpman (1963) Galeev, A. & Karpman, V. 1963, Sov. Phys.–JETP, 17, 403
  * Gedalin & Balikhin (2004) Gedalin, M. & Balikhin, M. 2004, Journal of Geophysical Research (Space Physics), 109, A03106
  * Gedalin et al. (2021) Gedalin, M., Russell, C. T., & Dimmock, A. P. 2021, Journal of Geophysical Research (Space Physics), 126, e29519
  * Giacalone (2017) Giacalone, J. 2017, ApJ, 848, 123
  * Gieseler & Palmroos (2023) Gieseler, J. & Palmroos, C. 2023, solo-epd-loader: v0.3.6
  * Goodrich & Scudder (1984) Goodrich, C. C. & Scudder, J. D. 1984, J. Geophys. Res., 89, 6654
  * Hanson et al. (2019) Hanson, E. L. M., Agapitov, O. V., Mozer, F. S., et al. 2019, Geochim. Res. Lett., 46, 2381
  * Hanson et al. (2020) Hanson, E. L. M., Agapitov, O. V., Vasko, I. Y., et al. 2020, ApJ, 891, L26
  * Hobara et al. (2010) Hobara, Y., Balikhin, M., Krasnoselskikh, V., Gedalin, M., & Yamagishi, H. 2010, Journal of Geophysical Research (Space Physics), 115, A11106
  * Hudson (1965) Hudson, P. D. 1965, MNRAS, 131, 23
  * Huttunen-Heikinmaa et al. (2005) Huttunen-Heikinmaa, K., Valtonen, E., & Laitinen, T. 2005, A&A, 442, 673
  * Jebaraj et al. (2023a) Jebaraj, I. C., Kouloumvakos, A., Dresing, N., et al. 2023a, A&A, 675, A27
  * Jebaraj et al. (2021) Jebaraj, I. C., Kouloumvakos, A., Magdalenic, J., et al. 2021, A&A, 654, A64
  * Jebaraj et al. (2023b) Jebaraj, I. C., Magdalenic, J., Krasnoselskikh, V., Krupar, V., & Poedts, S. 2023b, A&A, 670, A20
  * Jebaraj et al. (2020) Jebaraj, I. C., Magdalenić, J., Podladchikova, T., et al. 2020, A&A, 639, A56
  * Karpman (1964) Karpman, V. 1964, Soviet Physics-Technical Physics, 8, 715
  * Kasaba et al. (2000) Kasaba, Y., Matsumoto, H., Omura, Y., et al. 2000, J. Geophys. Res., 105, 79
  * Kennel (1981) Kennel, C. F. 1981, J. Geophys. Res., 86, 4325
  * Kennel et al. (1985) Kennel, C. F., Edmiston, J. P., & Hada, T. 1985, Washington DC American Geophysical Union Geophysical Monograph Series, 34, 1
  * Kis et al. (2013) Kis, A., Agapitov, O., Krasnoselskikh, V., et al. 2013, ApJ, 771, 4
  * Kouloumvakos et al. (2021) Kouloumvakos, A., Rouillard, A., Warmuth, A., et al. 2021, ApJ, 913, 99
  * Kouloumvakos et al. (2019) Kouloumvakos, A., Rouillard, A. P., Wu, Y., et al. 2019, ApJ, 876, 80
  * Krasnoselskikh et al. (2013) Krasnoselskikh, V., Balikhin, M., Walker, S. N., et al. 2013, Space Sci. Rev., 178, 535
  * Krasnoselskikh et al. (1985) Krasnoselskikh, V. V., Kruchina, E. N., Volokitin, A. S., & Thejappa, G. 1985, A&A, 149, 323
  * Krasnoselskikh et al. (2002) Krasnoselskikh, V. V., Lembège, B., Savoini, P., & Lobzin, V. V. 2002, Physics of Plasmas, 9, 1192
  * Krasnosel’skikh (1985) Krasnosel’skikh, V. 1985, Sov. Phys. JETP, 62, 282
  * Krauss-Varban & Burgess (1991) Krauss-Varban, D. & Burgess, D. 1991, J. Geophys. Res., 96, 143
  * Krauss-Varban et al. (1989) Krauss-Varban, D., Burgess, D., & Wu, C. S. 1989, J. Geophys. Res., 94, 15089
  * Lario et al. (2008) Lario, D., Decker, R. B., Malandraki, O. E., & Lanzerotti, L. J. 2008, Journal of Geophysical Research (Space Physics), 113, A03105
  * Lee et al. (2012) Lee, M. A., Mewaldt, R. A., & Giacalone, J. 2012, Space Sci. Rev., 173, 247
  * Lembege et al. (2004) Lembege, B., Giacalone, J., Scholer, M., et al. 2004, Space Sci. Rev., 110, 161
  * Lembege & Savoini (1992) Lembege, B. & Savoini, P. 1992, Physics of Fluids B: Plasma Physics, 4, 3533
  * Leroy & Mangeney (1984) Leroy, M. M. & Mangeney, A. 1984, Annales Geophysicae, 2, 449
  * Lin (1974) Lin, R. P. 1974, Space Sci. Rev., 16, 189
  * Lopate (1989) Lopate, C. 1989, J. Geophys. Res., 94, 9995
  * Magdalenić et al. (2014) Magdalenić, J., Marqué, C., Krupar, V., et al. 2014, ApJ, 791, 115
  * Maksimovic et al. (2020) Maksimovic, M., Bale, S. D., Chust, T., et al. 2020, A&A, 642, A12
  * Maksimovic et al. (1997) Maksimovic, M., Pierrard, V., & Lemaire, J. F. 1997, A&A, 324, 725
  * Mann et al. (2009) Mann, G., Warmuth, A., & Aurass, H. 2009, A&A, 494, 669
  * Masters et al. (2017) Masters, A., Sulaiman, A. H., Stawarz, Ł., et al. 2017, ApJ, 843, 147
  * Mellott & Livesey (1987) Mellott, M. M. & Livesey, W. A. 1987, J. Geophys. Res., 92, 13661
  * Mitchell et al. (2021) Mitchell, J. G., De Nolfo, G. A., Hill, M. E., et al. 2021, ApJ, 919, 119
  * Müller et al. (2013) Müller, D., Marsden, R. G., St. Cyr, O. C., & Gilbert, H. R. 2013, Sol. Phys., 285, 25
  * Oka et al. (2018) Oka, M., Birn, J., Battaglia, M., et al. 2018, Space Sci. Rev., 214, 82
  * Palmroos et al. (2022) Palmroos, C., Gieseler, J., Dresing, N., et al. 2022, Frontiers in Astronomy and Space Sciences, 9
  * Posner (2007) Posner, A. 2007, Space Weather, 5, 05001
  * Pulupa & Bale (2008) Pulupa, M. & Bale, S. D. 2008, ApJ, 676, 1330
  * Rodríguez-Pacheco et al. (2020) Rodríguez-Pacheco, J., Wimmer-Schweingruber, R. F., Mason, G. M., et al. 2020, A&A, 642, A7
  * Ryutov (1969) Ryutov, D. D. 1969, Soviet Journal of Experimental and Theoretical Physics, 30, 131
  * Sagdeev (1966) Sagdeev, R. Z. 1966, Reviews of Plasma Physics, 4, 23
  * Sarris & Krimigis (1985) Sarris, E. T. & Krimigis, S. M. 1985, ApJ, 298, 676
  * Schatzman (1963) Schatzman, E. 1963, Annales d’Astrophysique, 26, 234
  * Shimada et al. (1999) Shimada, N., Terasawa, T., Hoshino, M., et al. 1999, Ap&SS, 264, 481
  * Sonnerup (1969) Sonnerup, B. U. Ö. 1969, J. Geophys. Res., 74, 1301
  * Sonnerup & Scheible (1998) Sonnerup, B. U. Ö. & Scheible, M. 1998, ISSI Scientific Reports Series, 1, 185
  * Strauss et al. (2020) Strauss, R. D., Dresing, N., Kollhoff, A., & Brüdern, M. 2020, ApJ, 897, 24
  * Talebpour Sheshvan et al. (2023) Talebpour Sheshvan, N., Dresing, N., Vainio, R., Afanasiev, A., & Morosan, D. E. 2023, A&A, 674, A133
  * Trotta & Burgess (2019) Trotta, D. & Burgess, D. 2019, MNRAS, 482, 1154
  * Trotta et al. (2023) Trotta, D., Pezzi, O., Burgess, D., et al. 2023, MNRAS, 525, 1856
  * Trotta et al. (2022) Trotta, D., Vuorinen, L., Hietala, H., et al. 2022, Frontiers in Astronomy and Space Sciences, 9, 1005672
  * Vainio et al. (2013) Vainio, R., Valtonen, E., Heber, B., et al. 2013, Journal of Space Weather and Space Climate, 3, A12
  * Vasko et al. (2018) Vasko, I. Y., Mozer, F. S., Krasnoselskikh, V. V., et al. 2018, Geochim. Res. Lett., 45, 5809
  * Vedenov et al. (1962) Vedenov, A. A., Velikhov, E. P., & Sagdeev, R. Z. 1962, Soviet Journal of Experimental and Theoretical Physics
  * Voshchepynets et al. (2015) Voshchepynets, A., Krasnoselskikh, V., Artemyev, A., & Volokitin, A. 2015, ApJ, 807, 38
  * Walker et al. (1999) Walker, S. N., Balikhin, M. A., & Nozdrachev, M. N. 1999, Geochim. Res. Lett., 26, 1357
  * Wijsen et al. (2023) Wijsen, N., Lario, D., Sánchez-Cano, B., et al. 2023, ApJ, 950, 172
  * Wilson et al. (2016) Wilson, L. B., Sibeck, D. G., Turner, D. L., et al. 2016, Phys. Rev. Lett., 117, 215101
  * Wu (1984) Wu, C. S. 1984, J. Geophys. Res., 89, 8857

##  Appendix A Experimental details of the energetic electron measurements

Figure 3: Differential intensities of energetic electrons measured by all four
viewing directions of EPT and HET at the time of shock arrival (gray vertical
dashed line). EPT intensities were corrected for ion contamination. To be used
complementary to Fig. 1.  Figure 4: Differential intensities of energetic
electrons measured by all four viewing directions of EPT and HET for the whole
event time (shock arrival indicated by gray vertical dashed line). EPT
intensities were corrected for ion contamination. To be used complementary to
Fig. 1.

During shock arrival, a steady increase in the differential intensities of the
energetic electrons was observed by all four viewing directions of EPT and
HET, namely, sun, anti-sun, north, and south, respectively. This letter
focuses on the upstream beam observed uniquely by the south telescope. This
section provides supportive information for the electrons observed during the
event. Complementary to Fig. 1 presented in the main text, Fig. 3 shows the
pitch angle coverage and the same energy channels as observed by all four
telescopes. The energy channels are the same as the ones presented in the
bottom panel of Fig. 1.

In Fig. 3 a number of features related to electron anisotropy at the vicinity
of the shock aside from the FABs can also be identified. Firstly, an increase
in the intensity of relativistic electrons along PA $180^{\circ}$ (second
panel from bottom) after the arrival of the shock at 06:22:50 UT. Another one
is the bi-drectional beams observed at $\sim$06:18 UT. Identification of such
features is possible thanks to the enhanced resolution of the EPD
instrumentation.

Figure 4 presents the full electron event from the solar eruption till the
shock arrival at PSP. The same energy channels as in Fig. 1 and Fig. 3 are
shown together with the pitch angles in the top panel for the three day period
between July 23 – 26, 2022. As with Fig. 3, a number of observations can be
made about the presence of an energetic storm particle event for electrons
which are not the focus of the study. Firstly, the ESP event is seen at all
energy ranges presented here which range from near-relativistic to
relativistic. The second panel from bottom shows minor increases in the
$\sim$0.5 – 1 MeV electron intensities above the background level. Such
relativistic electron observations are extremely rare in IP shocks (Talebpour
Sheshvan et al., 2023). Secondly, the relativistic electrons (panel above the
bottom panel) show a noticeable anisotropy for almost the entire event.

###  A.1 Ion contamination of Solar Orbiter EPD/EPT measurements

Figure 5: Ion contamination correction for EPD/EPT channels. Response matrices
showing the mean response of all EPT foil channels to H (left top panel) and
4He (left bottom panel) in the corresponding energy channels provided by SIS.
(right top panel) Corrected vs original data of both the electron flux and the
background. (right bottom panel) Ratio of original vs corrected flux.

EPT uses the so called magnet-foil technique to separate and measure electrons
and ions. This measuring principle has the inherent characteristic that
energetic ions will contaminate the electron channels. To determine reliable
electron fluxes, the ion contamination in the electron channels must be taken
into account. For this work, we use the ion response functions from EPT
together with H and 4He flux measurements from SIS-A in order to estimate and
correct the ion contamination. In particular, we calculate the count rate of
contaminating ions $C_{\mathrm{ion}}$ observed in one of the electron channels
of EPT by

$C_{\mathrm{ion}}=\sum_{x}R^{\mathrm{H}}_{x}I^{\mathrm{H}}_{x}\Delta
E_{x}+\sum_{y}R^{\mathrm{He}}_{y}I^{\mathrm{He}}_{y}\Delta E_{y},$ (1)

where $I^{\mathrm{H}}_{x}$ and $I^{\mathrm{He}}_{y}$ are the H and 4He fluxes
observed by SIS in the energy channels x and y, $\Delta E_{x}$ and $\Delta
E_{y}$ are the energy widths of the channels x and y and $R^{\mathrm{H}}_{x}$
and $R^{\mathrm{He}}_{y}$ are the mean responses of EPT for H and 4He in the
energy ranges of the channels x and y. Figure 5 left panels shows the mean
response factors $R^{\mathrm{H}}_{x}$ and $R^{\mathrm{He}}_{y}$ for each EPT
foil channel in a response matrix.

The ion count rate $C_{\mathrm{ion}}$ in each electron channel can be
subtracted from the observed count rate in order to obtain an almost clean
electron count rate. This electron count rate can be calibrated into electron
fluxes using the electron calibration factors from EPT.

Although this method should in principle produce clean electron fluxes, there
are some limitations that need to be considered. SIS and EPT have different
fields of view, which can lead to over- or under-correction of EPT electron
measurements in the case of anisotropic ion fluxes. Furthermore, the energy
coverage and energy resolution of SIS are limited, which also limits the
accuracy of the correction. Lastly, all ions other than H and 4He are
neglected here.

For the time periods investigated here, the anisotropy of the ions was so
small, that the assumption of isotropic ion fluxes seemed reasonable for the
correction of the ion contamination. Ion species other than H and 4He were not
considered here because their flux is generally small compared to H and 4He
fluxes and no significant contribution to the observed count rates in EPT is
expected.

The correction made based on this method during the investigated period is
shown in the right panels of Fig. 5. The correction is minimal ($\sim$ 1% or
lower) up to 100 keV. This ratio increases to $>$5% for the energy channels
covering $>$ 200 keV energies.

###  A.2 Ion contamination of Solar Orbiter EPD/HET measurements

Figure 6: Pulse height analysis (PHA) of HET electrons. (top left) PHA data of
electrons stopping in detector C, 25 July 2022, 6:00-14:00. (bottom left) PHA
data of electrons stopping in detector C, 25 July 2022, 6:00-6:10. (top right)
PHA data of electrons stopping in detector B, 25 July 2022, 6:00-14:00.
(bottom right) PHA data of electrons stopping in detector B, 25 July 2022,
6:00-6:10.

The higher energy electron channels of the HET instrument have been found to
suffer from proton contamination during strong shock passages. While this
issue will be solved in the near future by a software upload from the HET
team, careful investigation of the already taken Pulse Height Analysis (PHA)
data allows to identify time intervals featuring this contamination.
Figure 6 top left panel shows PHA data of particles identified by the
instrument as electrons penetrating the solid state detectors A and B of HET
while stopping in the scintillator C during a prolonged time interval on July,
25th, 2022. This representation (energy loss in A+B vs C) allows to identify
several different populations, annotated by A) real electrons which would
extend further to the right for higher energies, B) minimal ionizing
particles, that is, protons with energies in the GeV range, and C) a track of
lower energetic protons. All populations have been reproduced by Monte Carlo
simulations. The dashed red lines indicate the energy ranges of the HET
electron channel, that is, particles between the right and center red line
will be counted in the E5700 channel (the naming scheme refers to Fleth et al.
(2023)). While the data product has been crafted such that population B) does
not spoil the channel, a strong proton contamination due to track C) can be
seen during this day. Figure 6 bottom left panel shows the same
representation, however limited to 6:00-6:10, that is, the time of interest
for this study. While the PHA statistic is limited here, it is obvious that
the higher electron channels are also heavily contaminated by protons during
this time series.
Figures 6 top right and bottom right panels show PHA data of particles
identified by the instrument as electrons penetrating detector A but stopping
in detector B (without reaching the scintillator C), corresponding to HET’s
lowest electron energy channel employed in this study. In contrast to the ABC
coincidence discussed above, no clear track as expected from a possible proton
contamination is visible in either of the time series. Furthermore, Monte
Carlo simulations do not indicate any possibility of proton contamination in
this channel.

###  A.3 Velocity dispersion analysis

Figure 7: VDA based on onset times of Solar Orbiter EPT and HET electron
channels. The horizontal axis shows the inverse of the average unit-less speed
of electrons as observed in each channel. The vertical axis presents the
determined onset time for each energy channel. Onset times observed by EPT
(HET) are marked in green (blue). Horizontal error bars represent the width of
the energy channels, and vertical error bars represent the time resolution
used to determine the onsets (1 second). The orange line represents the linear
fit to the first 16 energy channels of EPT and the first channel of HET,
respectively. The onset times that are not considered to the fit have a white
middle on the data points.

Based on the assumption that electrons of all energies were injected at the
same time (Vainio et al., 2013), the result of the velocity dispersion
analysis (VDA) presents an estimation of the injection time of the energetic
electrons and the associated path length from the source to the spacecraft.
Figure 7 shows a well-defined velocity dispersion of the beam in the EPT
channels $<300$ keV where ion-contamination is low. A good agreement between
the EPT channels and the first HET channel is also observed. In this case, the
estimated path length is $\sim 13\leavevmode\nobreak\ R_{\odot}$, which is 6%
of the Sun-Earth distance, making the beam a local phenomenon.

It is worth stating that the VDA could only be performed for the onset times
of the first peaks, as it was not possible to obtain reliable starting times
for the second peaks that occur on top of the decay of the first ones. Also, a
VDA performed upon the first peaks themselves exhibited no velocity
dispersion. In this regard, the estimation is approximated to the entirety of
the FAB. The onset times were identified using the SEPpy software package
(Palmroos et al., 2022) which employs the statistical Poisson-CUSUM method
(see e.g., Huttunen-Heikinmaa et al., 2005) to automatically identify the time
of the onset.

##  Appendix B The relativistic fast-Fermi model for electron acceleration

The dynamics of particle acceleration in the fast-Fermi model are most
effectively described in the de Hoffmann-Teller (HT) frame. In this frame, the
upstream plasma bulk velocity is parallel to the magnetic field, leading to no
upstream motional ($\mathbf{-V}\times\mathbf{B}$) electric field. The velocity
transformation to the HT frame from the normal incidence frame (NIF), a shock
rest frame where the upstream plasma flow is directed along the shock normal,
is given by:

$\mathbf{u}_{HT}=\frac{\hat{\mathbf{n}}\times(\mathbf{V}\times\mathbf{B})}{\mathbf{B}\cdot\hat{\mathbf{n}}}$
(2)

where $\hat{\mathbf{n}}$ is the shock normal, and $\mathbf{V}$ and
$\mathbf{B}$ denote the upstream bulk velocity and the magnetic field,
respectively. $\mathbf{u}_{HT}$ lies in the shock plane, parallel to the
projection of the upstream magnetic field into this plane. The energization of
electrons in the upstream solar wind frame arises from transformations to the
HT frame, reflection, and then transformation back to the solar wind frame.
Denoting the upstream plasma speed in the HT frame by $V_{HT}$, it was first
demonstrated by Sonnerup (1969) that a larger $V_{HT}$ leads to greater energy
gain. The relationship between $V_{HT}$ and $V$ can be expressed as:

$V_{HT}=\frac{V}{\cos\theta_{Bn}}$ (3)

An electron entering with a parallel velocity $v_{\parallel}$in the upstream
plasma frame will reflect upstream with a parallel velocity of
$-v_{\parallel}+2V_{HT}$. In relativistic scenarios, this can be described in
terms of momentum $p$ as:

$\Delta p_{\parallel,HT}=-2\gamma_{HT}mv_{\parallel,HT}=-2p_{\parallel,HT},$
(4)

where, $\gamma$ denotes the Lorentz factor. The associated energy change in
the upstream plasma frame is:

$\Delta E=-\Gamma V_{HT}\Delta p_{\parallel,HT}=2\Gamma
V_{HT}p_{\parallel,HT}$ (5)

Here, $\Gamma$ represents the Lorentz transformation of $V_{HT}$, defined as
$\left(1-V_{HT}^{2}/c^{2}\right)^{-1/2}$. As indicated in Figure 8, for
$\theta_{Bn}$ approaching $90^{\circ}$, the shock becomes superluminal, making
$\Gamma$ significant. The transformation becomes invalid at
$\theta_{Bn}=90^{\circ}$.

The change in particle momentum due to reflection is given by Eq. 4. To
comment on the transformation between the HT and the upstream flow frame, one
can use:

$\displaystyle\gamma=\Gamma\left(\gamma_{HT}+p_{\parallel,HT}V_{HT}\right)$
(6) $\displaystyle
p_{\parallel}=\Gamma\left(p_{\parallel,HT}+V_{HT}\gamma_{HT}\right).$ (7)

The process of reflection itself happens in the HT frame where the
conservation of both the energy and the magnetic moment apply,

$p^{2}_{\parallel,HT}+p^{2}_{\perp,HT}-e\Phi_{HT}=\text{constant}$ (8)

Here, $p$ is the momentum and is divided into $p_{\parallel}$ and $p_{\perp}$
corresponding to parallel and perpendicular momentum. $e\Phi_{HT}$ is the
electrostatic potential in the HT and is well known to impact the reflection
process (Gedalin & Balikhin, 2004).

For relativistic particles, we must consider the following form for energy:

$E=\sqrt{m^{2}c^{4}+p_{HT}^{2}c^{2}}+e\Phi_{HT}$ (9)

Next, the magnetic moment of the electron may be given as,

$I_{B}=\frac{p^{2}_{\perp,HT}}{B}$ (10)

This implies that the particle continues to gain perpendicular momentum till a
characteristic location at which $p_{\parallel,HT}=0$. This is chosen in such
a way that it is implicitly assumed that $e\Phi_{HT}$ and $B$ are maximum in
the same region, i.e. the reflection point. An important point to consider
here is the value of $e\Phi_{HT}$ in comparison with the $p_{\parallel}$ of
the electron. A smaller $e\Phi_{HT}$ would be insignificant to the incident
electron population. This would immediately ensure that the core electrons in
any distribution will not be reflected.

It is now possible to make use of the Eq. 10 and the adiabatic invariance of a
relativistic electron in the upstream plasma ($-\infty$) to that of one at the
ramp under the influence of the magnetic mirror ($B_{overshoot}$),

$\displaystyle\frac{p^{2}_{\perp,HT}(-\infty)}{B}=\frac{p^{2}_{\perp,HT}(overshoot)}{B_{overshoot}}$
$\displaystyle
p^{2}_{\perp,HT}(overshoot)=p^{2}_{\perp,HT}(-\infty)\frac{B_{overshoot}}{B}$
(11)

Substituting Eq. 9 in Eq. B one gets,

$\displaystyle\sqrt{m^{2}c^{4}+p^{2}_{\parallel,HT}(-\infty)c^{2}+c^{2}p^{2}_{\perp,HT}(-\infty)}=$
$\displaystyle\sqrt{m^{2}c^{4}+c^{2}p^{2}_{\perp,HT}(-\infty)\frac{B_{overshoot}}{B}}+e\Delta\Phi_{HT}$
(12)

If we were to work in the frame where $\textbf{V}||\hat{\textbf{n}}$, i.e. the
NIF, then there are two electric field components. One associated with the
motion of the plasma $E_{y}$, and another associated with the shock
transition, $E_{x}$. Transforming into the HT frame then gives,

$E_{x,HT}=E_{x}+\left(\frac{\textbf{V}_{HT}}{c}\times\textbf{B}\right)$ (13)

Using this, Goodrich & Scudder (1984) showed that,

$\displaystyle eE_{x,HT}=$ $\displaystyle-\frac{\partial}{\partial
x}\left[\frac{m}{2}\left(v^{2}_{y}+v^{2}_{z}\right)+mV_{HT}v_{z}\right]-\frac{1}{n}\frac{\partial
p_{e}}{\partial x}=$ $\displaystyle-e\frac{\partial\Phi_{HT}}{\partial x}$
(14)

$v_{x}$ and $v_{y}$ are small compared to the upstream electron thermal speed
and so the potential jump $\Delta\Phi_{HT}$ is equal to the change in electron
temperature,

$e\Delta\Phi_{HT}=\Delta T_{e}$ (15)

Here, $T_{e}$ is in units of eV. The change in electron temperature across the
shock is usually in the order of 10s of eV to about 100 eV (Lembege et al.,
2004). Several more proxies can also be used to constrain the electrostatic
potential as done so in Hanson et al. (2019) based on the methods established
in Gedalin & Balikhin (2004). They are based on ion velocity, ion density,
pressure balance, and the electron equation of state. In the case of the shock
analyzed here, the potential was found to be between 100 – 200 eV. This
estimate is large by more than a factor of 5 compared to the IP shock
evaluated by Hanson et al. (2019). The effect of the electrostatic potential
is that the acceleration is completely restricted to only the tail of the seed
distribution (Leroy & Mangeney, 1984).

It is quite clear from this entire ordeal that the most important variable in
Eq. B is the mirror ratio $B_{overshoot}/B$. In the case of a super-critical
quasi-perpendicular shock, the overshoot magnetic field can exceed the simple
discontinuity limit (MHD) of 4 (for a gas with adiabatic index $5/3$). The
reflected electrons therefore form a loss cone distribution which is dependent
upon $V_{HT}$ and $e\Delta\Phi_{HT}$. Furthermore, the opening angle, $\alpha$
of the so-called “loss-cone” itself is set by the ratio of upstream to
downstream magnetic field. The loss-cone angle is such that the resultant
distribution is purely made of particles which exceed a certain pitch angle
and perpendicular velocity.

Ideally, this may never exceed a certain limit under MHD conservation laws,
however, for the considerations made here where the presence of
$B_{overshoot}$ is acknowledged, $\alpha=1/\sin(B/B_{overshoot})$. As such, it
is evident that the upstream distribution of electrons plays an important role
in determining whether the acceleration is significant or not. The “seed”
distribution is based upon the in-situ spacecraft measurements by Lin (1974)
who showed that the electrons have an enhanced supra-thermal tail in the solar
wind as opposed to a purely thermal Maxwellian distribution. The supra-thermal
tail can be represented by a single kappa distribution can be defined as,

$f=A\cdot\left[1+\frac{E}{\kappa E_{\kappa}}\right]^{-\kappa-1}$ (16)

where $A$ is the normalization constant, and $E$ is the kinetic energy
(Maksimovic et al., 1997) in the seed distribution function. For energies
where $E\rightarrow E_{\kappa}$, the distribution behaves as a Maxwellian

Figure 8: de Hoffmann-Teller transformation velocity (in units of light speed,
$c$) for different upstream speeds, $V$ (NIF) as a function of $\theta_{Bn}$.
The relationship between these parameters is shown in Eq. 3. The vertical
dashed lines indicate the $\theta_{Bn}$ beyond which the $V_{HT}$ is
relativistic. The colors correspond to the different upstream speeds, $V$.

The efficiency of acceleration without relativistic effects was originally
estimated by Krauss-Varban et al. (1989). Figure 8 plots Eq. 3 which shows
that as $\theta_{Bn}$ increases above $86^{\circ}$, the HT transformation
velocity alone may result in an exponential energy gain of the particle.
Therefore, the role of $\theta_{Bn}$ is crucial, as only the electrons which
exceed $V_{HT}$ may interact with the shock, resulting in a loss-cone
distribution. At a certain limit ($\theta_{Bn}=90^{\circ}$), none of the
electrons are reflected, and they are transmitted downstream (Ball & Melrose,
2001). This aspect of the acceleration mechanism results in the formation of
“dilute” beams, that is, beams with very small $n_{b}/n$ (ratio of beam
density to the background).

Two important issues not discussed here are the so-called “injection” problem
and the acceleration time scales. The first issue is fundamental and implies
that shock acceleration requires a pre-existing population of supra-thermal
particles which can participate in the process. Here, it is overcome by the
fact that the acceleration mechanism ignores the thermal electrons and only
acts on the enhanced tail population of the kappa distribution. The choice of
a kappa distribution further simplifies the issue where the supra-thermal tail
is enhanced. The reflection time scale for fast-Fermi is dependent upon
several factors. The most important of which are the shock strength $M_{A}$,
the thickness of the shock $\Delta x$, and $B_{overshoot}$. Under the non-
relativistic limit for a weak shock with $\Delta x$ in the order of the ion-
inertial length ($d_{i}=c/\omega_{pi}$) and for a quasi-perpendicular shock,
this may be approximated to depend directly on the ion gyro period
($\Omega_{ci}$, Leroy & Mangeney, 1984),

$\tau\sim 4\Omega_{ci}^{-1}\frac{\Delta x}{M_{A}}\frac{B_{overshoot}}{\Delta
B}\left[1+\frac{2}{3}\frac{B_{overshoot}}{B}\right]$ (17)

here, $\Delta B=B_{overhsoot}-B$. The relationship is visualized in Fig. 9
which shows that for a strong shock with a large magnetic overshoot, its
thickness is $<d_{i}$ (Hobara et al., 2010). The issue of acceleration time
scales becomes important when distinguishing between the various acceleration
mechanisms such as fast-Fermi, and first-order Fermi acceleration. The latter
due to its stochastic nature, happens over longer time periods. Meanwhile, the
former in the relativistic consideration is in the order of the electron gyro
period and therefore at least two orders of magnitude faster than in the non-
relativistic case.

Figure 9: Acceleration time scale ($\tau$) with respect to shock thickness
($\Delta x$) and Alfvèn Mach number ($M_{A}$). Based on Eq. 17. Here, the
color coding follows a gradient from light to dark which represents decreasing
excursion time (faster acceleration) of the electron.

A matter of caution in the coordinate transformations considered here is that
the reflection process can be treated either purely in the HT frame or the
NIF. However, there are frame specific effects such as the transmitted
electrons gaining energy in the HT frame due to the potential, while being
unaffected in the NIF due to the convective electric field. If acceleration is
purely treated in the HT frame, then the description provided in this section
is satisfactory, that is, energy is gained via frame transformation. However,
if purely treated in the NIF, the motional electric field ensures that the
particles enter a gradient drift motion. In such a case, particles will be
seen in the direction of the non-coplanar component of the magnetic field.