Relativistic electron beams accelerated by an interplanetary shock

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 (more than 300 keV) by traveling shocks. 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. 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. 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.


Introduction 1
Shock waves are ubiquitous in space plasmas, and are the most 2 prolific particle accelerators in most systems.They can be di-3 rectly probed in the heliosphere, due to the presence of planetary 4 bow shocks (created due to the planetary obstacles in the solar 5 wind flow) and interplanetary shocks, which are driven by solar 6 activity such as coronal mass ejections (CMEs).CMEs are large-7 scale eruptions of plasma and magnetic fields that travel away 8 from the Sun; when their speed exceeds the information speed 9 of the medium (i.e., the fast magnetosonic speed), shock waves 10 are generated.While ion acceleration in such shocks has been 11 extensively studied (Lee et al. 2012, and references therein), electron acceleration is challenging due to the large separation 13 in scales 1 and physical limitations, such as their retention at the 14 shock and magnetization (for a review, see Lembege et al. 2004).
1 Ratio of gyro-radii (r L ) ∼ m i me = 42.8,where m i and m e are the mass of protons and electrons, respectively.
However, observational surveys have found a small number of interplanetary (IP) shocks that were associated with a significant increase in electron fluxes from subrelativistic 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 the acceleration of subrelativistic 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;Cairns et al. 2003;Jebaraj et al. 2020;Kouloumvakos et al. 2021;Jebaraj et al. 2021), the 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;Ken-nel 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 ( n), θ Bn .Quasi-perpendicular shocks (θ Bn > 45 • ) are often considered more efficient in accelerating electrons to high energies (Schatzman 1963;Bulanov et al. 1990;Mann et al. 2009).The typical widths 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 field-aligned beams (FABs) 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 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, which are known as sub-critical.This is characterized best by the ratio of the shock speed to the fast magnetosonic speed known as the Mach number M. Supercritical 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 traveling 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 on board Solar Orbiter (SolO; Müller et al. 2013).We utilized the Electron Proton Telescope (EPT) which covers electron energy in the range ≈ 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, and the results and its implications in Sect. 5.

99
On 25 July 2022, at 06:22:45 UT, the SolO spacecraft encoun-100 tered an IP fast-forward shock wave (see the last three panels of 101 Fig. 1).It was linked to a weak C1 solar flare and CME on 23 102 July 2022 at about 18:30 UT.A remarkable aspect related to the 103 in situ shock crossing (first two panels of Fig. 1,and Fig. A.1), is 104 the presence of upstream field-aligned beams of relativistic elec-105 trons (up to ∼ 1 MeV) indicated by the gray shaded region.The 106 shock was also associated with an energetic storm particle (ESP) 107 event for electrons, whose features are outside the scope of this 108 letter, but a short summary is provided in Appendix A.

109
The shock arrived at SolO approximately 36 hours after the 110 solar event, which results in an average transit speed of 1100 111 km s −1 .In order to obtain the in situ characteristics of the shock 112 wave, such as its speed and strength, a more robust physical ap-113 proach involving multiple steps was adopted.

114
First, the shock normal ( n) was estimated using a combina-115 tion of the mixed-mode method, magnetic and velocity copla-116 narity (Trotta et al. 2022, and references therein), and mini-117 mum variance analysis (MVA; Sonnerup & Scheible 1998) to 118 be n = [0.93 ± 0.05, 0.018 ± 0.12, −0.16 ± 0.25].Next, the 119 shock's obliquity was estimated to be θ Bn ∼ 64 • ± 6 • , indicat-120 ing a quasi-perpendicular geometry.The upstream and down-121 stream bulk flow speeds were then estimated in the shock rest 122 frame, V sh u ∼ 350 ± 50 km s −1 and V sh d ∼ 125 ± 50 km s −1 , 123 which results in a shock speed, V sh ∼ 775 ± 50 km s −1 in the 124 observers frame.The Alfvén Mach number (M A ) was then es-125 timated using the upstream Alfvén speed, c A = B up / √ µ 0 n i m i 126 (assuming Z = 1, m i and n i are proton mass and number den-127 sity), and proxies (for M A ) established in Gedalin et al. (2021)   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 at lower energies (subrelativistic, 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 (RPW; Maksimovic et al. 2020) instrument suite.Two wave packets (not shown), the first at 06:10:05 UT and then at 06:11:59 UT, is likely evidence of the FABs extending to subrelativistic energies.Other than small amplitude Langmuir waves, no strong wave activity was measured during this time, making it improbable that the FABs are generated locally.
The presence of Langmuir waves also indicates the remote origin of the beam, as it must have propagated at a distance that is, at the very least, on 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)

261
The primary aspect of super-critical shocks is that in order 262 to dissipate excess energy, they reflect a portion of upstream 263 ions forming an overshoot magnetic field (Krasnoselskikh et al. 264 2013).The steep magnetic gradient of the overshoot enables the 265 near-relativistic and relativistic electrons to conserve their mag-266 netic moment and as such, a relativistic approach to the fast-267 Fermi mechanism must be adopted (Appendix B).We note that 268 a sub-critical shock is also capable of reflecting electrons, but 269 to a far lesser degree as far fewer particles would undergo such 270 a process.The second important assumption is that in a near-271 perpendicular geometry, the shock speed along the magnetic 272 field line scales as a function of geometry.This limits acceler-273 ation to only those particles that exceed it, resulting in dilute 274 beams.

275
We performed semi-analytical modeling based on the rela-276 tivistic form of the reflection conditions provided in Leroy & 277 Mangeney (1984).When θ Bn > 89.5 • , the shock speed must also 278 undergo a Lorentz transformation as it is relativistic.The up-279 stream seed electron distribution function was assumed to be a 280 kappa-distribution (Appendix B) with κ ∼ 3 − 4 (considered re-281 alistic, Krauss-Varban & Burgess 1991).For the parameters of 282 the shock, an upstream magnetic field similar to that measured 283 in situ by SolO was used (i.e., ∼10 nT).The magnetic compres-284 sion including the overshoot magnetic field is considered to be 285 B overshoot /B = 7 (based on Mellott & Livesey 1987) yielding a 286 maximum B of 70 nT.Finally, the effect of the electrostatic po-287 tential cannot be ignored for electrons, and as such 100 eV was 288 used (obtained through proxies, Appendix B).

289
The modeling results are presented in the bottom two panels 290 of Fig. 2.An ensemble of runs was performed using kappa values 291 in the range κ ∼ 3 − 4, and shock obliquity in the range θ Bn ∼ 292 87 • − 89.5 • .The gray shaded region in the bottom left panel of 293 Fig. 2 shows the results and two different fits to the mean of all 294 runs, namely a single power law with a roll-over (blue dashed 295 line) and a broken power law (red dashed line).The first has a 296 power law with spectral index δ ∼ −2.4,which is followed by an 297 exponential roll-over, while the second has two power laws with 298 spectral indices δ 1 ∼ −2.5 and δ 2 ∼ −6.6.Both fits are used, but 299 the χ2 of the first fit is lower than that of the second.The mean 300 for each θ Bn .The effect of a changing κ index would then be responsible for the steepness of the spectrum since it determines the number of electrons in the tail.A κ = 2 can be characterized with a power-law index of δ ∼ −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 toward a Maxwellian distribution as κ → ∞.
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.

398
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)   The Electron Proton Telescope uses the so called magnet-foil 584 technique to separate and measure electrons and ions.This mea-585 suring principle has the inherent characteristic that energetic ions 586 will contaminate the electron channels.To determine reliable 587 electron fluxes, the ion contamination in the electron channels 588 must be taken into account.For this work, we used the ion re-589 sponse functions from EPT together with H and 4 He flux mea-590 surements from SIS-A in order to estimate and correct the ion 591 contamination.In particular, we calculated the count rate of con-592 taminating ions C ion observed in one of the electron channels of 593 EPT by 594 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 A.4 top left panel shows the PHA data of particles identified by the instrument as electrons penetrating the solid-state detectors A and B of HET, but stopping in scintillator C during a prolonged time interval on 25 July 2022.This representation (energy loss in A+B vs C) allows us to identify several different populations denoted as A) real electrons, which would extend farther to the right for higher energies; B) minimally ionizing particles, that is, protons with energies in the GeV range; and C) lower energy 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)).Although the data product was 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 A.4 bottom left panel shows the same representation, but 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.It is worth noting 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 cotemporally with the decay of the first ones.A VDA performed on the first peaks themselves also 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 statis-  If we were to work in the frame where V|| n (i.e., the NIF), 742 then there are two electric field components.One is associated 743 with the motion of the plasma E y , and the other is associated 744 with the shock transition E x .Transforming into the HT frame 745 then gives Using this equation, Goodrich & Scudder (1984) showed that 747 eE where the values of v x and v y are small compared to the upstream 748 electron thermal speed, and so the potential jump ∆Φ HT is equal 749 to the change in electron temperature: Here T e is in units of eV.The change in electron temperature across the shock usually ranges from tens of eV to about 100 eV (Lembege et al. 2004).Several more proxies can also be used to constrain the electrostatic potential, as done 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 more than a factor of five larger than 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.11 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∆Φ HT .Furthermore, the opening angle α of the socalled "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 that 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, α = 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 that can be defined as 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 → E κ , the distribution behaves as a Maxwellian.
The efficiency of acceleration without relativistic effects was originally estimated by Krauss-Varban et al. (1989).B.2, which shows that for a strong shock with a large magnetic 818 overshoot, its thickness is < d i (Hobara et al. 2010).The issue 819 of acceleration timescales becomes important when distinguish-820 ing between the various acceleration mechanisms such as fast-821 Fermi, and first-order Fermi acceleration.The latter, due to its 822 stochastic nature, happens over longer time periods.Meanwhile, 823 the former in the relativistic consideration is on the order of the 824 electron gyro period, and therefore at least two orders of magni-825 tude faster than in the non-relativistic case.

826
A matter of caution in the coordinate transformations con-827 sidered here is that the reflection process can be treated either 828 purely in the HT frame or the NIF.However, there are frame 829 specific effects such as the transmitted electrons gaining energy 830 in the HT frame due to the potential, while they are unaffected 831 in the NIF due to the convective electric field.If acceleration 832 is purely treated in the HT frame, then the description provided 833 in this section is satisfactory, that is, energy is gained via frame 834 transformation.However, if purely treated in the NIF, the mo-835 tional electric field ensures that the particles enter a gradient drift 836 motion.In this case, particles will be seen in the direction of the 837 non-coplanar component of the magnetic field.

Fig. 1 .
Fig. 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 indicated 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 180 • throughout the shown period).The upstream FABs are observed between 06:06:00 and 06:08:30 UT, and are indicated by the gray shaded region.Fig. A.1 shows the differential intensities across all viewing directions.The second panel presents the pitch-angle coverage 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 and 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.
, a velocity dispersion of the FABs allows the injection time and the propagated path length of the FABs to be estimated.The details of this analysis can be found in Appendix A.3.The results of the analysis shown in Fig.A.5  suggest that the electrons were accelerated in a region ∼ 13 R from the observer, and were injected at 06:05:46 UT.3.2.Assessing the characteristics of electron acceleration using peak-intensity energy spectraThe 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.2present 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 inDresing et al. (2020);Strauss et al. (2020) allows for different fit models.For the FABs, we used a broken power law defined by two different spectral indices δ 1 and δ 2 separated at a break-energy E b , a single power law (δ 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.2shows the fit applied to both peaks over all available EPT channels, and therefore also fitting 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 the HET points, it is hard to conclude what type of fit (either 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 the ion-contamination correction that was applied to the EPT data points was not perfect.The presence of a break or spectral rollover is without doubt, as it is already evident in the uncorrected EPT data (not shown).The cutoff energies E c for both the fits in the right panel are similar, ∼ 200 keV.However, the break energies E b for the fits in the left panel are both found to be above ∼ 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 ∼ 70 keV.The lower spectral index of the first peak in the right panel is δ = −1.19± 0.14, while that of the second peak is δ = −1.96± 0.54.For comparison, the spectral indices for the first peak in the left panel are δ 1 = −1.66±0.08 and δ 2 = −5.69±2.09,while that of the second peak are δ 1 = −0.94± 0.06 and δ 2 = −5.02± 0.79.

Fig. 2 .
Fig. 2. Comparison of the peak-intensity electron spectra: observation (top) vs model (bottom).In the top two panels the points in cyan 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 κ ∼ 3 − 4 and θ Bn ∼ 87 • − 89.5 • .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-dotted 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 κ ∼ 3.5 for different θ Bn (88 • − 89.5 • ).The corresponding break energies are indicated by the vertical dashed lines.
On 25 July 2022 a traveling IP shock wave accompanied by an 338 energetic storm particle event was recorded in situ by Solar Or-339 biter.Notably, field-aligned beams of relativistic electrons were 340 observed 14 minutes prior to the shock arrival.These are the first such observations of electron FABs measured at IP shocks 342 thanks to the high temporal resolution of the EPD instrument 343 suite on board SolO.This letter presents their observational char-344 acteristics and proposes a scenario for their acceleration at IP 345 shocks.evolving in a low-density inhomogeneous plasma might develop 357 decay instabilities that deform the laminar shock front.In higher 358 dimensions, large-scale deformations lead to curved fronts on 359 the scale of the upstream inhomogeneities, spanning from sev-360 eral R to tens of R (e.g., Wijsen et al. 2023), which play 361 a significant role in FAB acceleration (Bulanov & Sakharov 362 1986; Decker 1990; Bulanov & Krasnoselskikh 1999; Lario 363 et al. 2008).With any curved front, a finite region exists where 364 θ Bn ≥ 85 • and approaches 90 • , allowing efficient electron accel-365 eration along the tangent magnetic field line, as demonstrated in 366 this letter.When the observer is connected to the tangent field 367 line, the time-varying properties of the FABs are likely influ-368 enced by deformation-induced curvature and by variations in 369 the upstream field topology (e.g., Giacalone 2017; Trotta et al. observed and modeled peak-372 intensity spectra of the FABs serve as a demonstration of shock 373 acceleration at an IP shock under the right circumstances.The 374 model naturally forms a roll-over or break at higher energies (> 375 200 keV), which is also seen in observations.If a broken power 376 law were fit to the modeled results, the break-energy E b would 377 fall between 190 keV and 400 keV, comparable to the obser-378 vations.This is considerably different from the transport-related 379 breaks discussed in previous studies (Dresing et al. 2021), which 380 have suggested that the presence of two different spectral breaks 381 in peak-intensity spectra of solar energetic electron events are 382 caused by transport-related effects.The spectral fits and the 383 slopes obtained here are unique since it is the first such result 384 obtained close to the source, that is, the IP shock.In particular, 385 the spectral break observed at high energies (i.e., > 200 keV) 386 reported for the first time here is therefore likely a direct conse-387 quence of acceleration.388 The fast-Fermi and SDA mechanisms are both candidates for 389 electron acceleration.The characteristics of these mechanisms 390 depend on the conditions at the source, the seed distribution 391 function, and the shock properties (e.g., obliquity, criticality).392 The solar wind conditions in reality are constantly changing, al-393 tering the conditions for acceleration, favoring a rapid mecha-394 nism.Furthermore, it is possible that electrons accelerated in 395 such hot-spots can drift back onto the shock due to the E × B 396 drift at large distances and can undergo further acceleration via 397 a first-order Fermi process.

Fig. A. 1 .
Fig. A.1.Differential intensities of energetic electrons measured in 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.This figure is complementary to Fig. 1.

Figure A. 2
Figure A.2 presents the full electron event from the solar eruption until the shock arrival at PSP.The same energy channels as in Fig. 1 and Fig. A.1 are shown together with the pitch angles in the top panel for the three day period July 23 -26, 2022.As in Fig. A.1, a number of observations can be made about the presence of an energetic storm particle event for electrons, which are not the focus of this study.First, the ESP event is seen at all en- Fig. A.3.Ion contamination correction for EPD and EPT channels.The response matrices showing the mean response of all EPT foil channels to H (top left panel) and 4 He (bottom left panel) in the corresponding energy channels provided by SIS.(top right panel:) Corrected vs original data of both the electron flux and the background.(bottom right panel:) Ratio of original to corrected flux.

Figures
Figures A.4 top right and bottom right panels show PHA data of particles identified by the instrument as electrons penetrating
Fig. A.5. 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).The horizontal error bars represent the width of the energy channels, and the 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 in the fit have a white middle on the data points.
Fig. B.2. Acceleration timescale (τ) with respect to shock thickness (∆x) and Alfvèn Mach number (M A ). Based on Eq.B.16.Here the colorcoding follows a gradient from light to dark, which represents decreasing excursion time (faster acceleration) of the electron.

model for electron acceleration at IP shocks 246
We also fit an averaged spectrum of a time period of 2.5 min containing both 243 peaks, which results in a broken power law with δ 1 = −1.4±0.04,244δ 2 = −5.2±2.58, and E b = 261.8±47 keV (not shown).From the observations presented in this letter, it is understood 247 that the FABs were accelerated at a remote location on the shock 248 similar to how they are accelerated at the near-perpendicular ter-249 restrial bow shock.If so, then the curvature of the shock (or 250 large-scale deformations for the IP shock at 1 AU) and the time-251 varying properties of the upstream magnetic field become cru-252 cial.Acceleration happens under the exclusive assumption that 253 there exists a region on the shock wave separated by ∼ 13R 254 that is near-perpendicular (89.9 • > θ Bn > 85 • ) and super-255 critical 2 (Appendix B).When these conditions are met, accel-256 eration happens based on the conservation of magnetic moment 257 (the fast-Fermi mechanism, Sonnerup 1969; Wu 1984; Leroy & 258 Mangeney 1984) through which electrons can gain a significant 259 amount of energy in a single encounter with a near-perpendicular 260 shock.