Betatron Acceleration of Suprathermal Electrons within a Small-scale Flux Rope in the Solar Wind

A growing body of evidence from observations, theories, and simulations indicates that particles can be effectively accelerated in solar wind regions filled with dynamic small-scale flux ropes (FRs). The main acceleration mechanisms identified in simulations include parallel electric field acceleration, first-order Fermi acceleration, and generalized betatron acceleration in contracting or merging small-scale FRs. However, direct identification of these acceleration mechanisms from in situ measurements remains a challenge. Here we present a distinct event of local betatron acceleration within a contracting small-scale FR in the solar wind, due to a local compression. In this event, the lower-energy halo electrons were effectively accelerated through the betatron mechanism, whereas the higher-energy suprathermal electrons predominated by the superhalo component were almost not energized. The halo electron energization processes via the betatron mechanism are reproduced using an analytical model. Further examination of small-scale FRs in the vicinity of the heliospheric current sheet over the period 1995–2020 indicates that in situ signatures of the betatron acceleration process are essentially elusive.


Introduction
Solar wind electrons typically consist of three (or four) components: a thermal core with energies below ∼50-100 eV, a suprathermal halo/strahl with energies from ∼100 eV to a few keV, and a suprathermal superhalo with energies above ∼2 keV (Feldman et al. 1975;Pilipp et al. 1987;Maksimovic et al. 2005;Lin 1998;Wang et al. 2012Wang et al. , 2015;;Tao et al. 2021).The three components can be distinguished from each other according to the electron velocity distribution function (VDF).Specifically, the core population VDF is characterized by a Maxwellian distribution (e.g., Štverák et al. 2009), the halo/ strahl population VDF can be described by a Kappa distribution (Maksimovic et al. 1997;Štverák et al. 2009), and the superhalo population VDF is approximated by a powerlaw distribution (Lin 1998;Wang et al. 2012).The core, halo, and superhalo are nearly isotropic, and thus show electron fluxes at all pitch angles (PAs), whereas the strahl is highly anisotropic and appears as a field-aligned beam (with relatively small PAs) streaming away from the Sun (Montgomery et al. 1968).The core population contains ∼90%-95% of the total electron density (Maksimovic et al. 2005), the halo/strahl component contains ∼5%-10% of the electron density, and the superhalo component takes up the remaining portion, which is many orders of magnitude lower than that for halo/strahl electrons (Luhmann et al. 2008).Despite a small fraction of the total electron density, suprathermal electrons (halo, strahl, and superhalo populations) are responsible for the majority of the solar wind heat flux transported away from the Sun because of their high energy, and therefore are important to the solar wind dynamics (Štverák et al. 2009).
Even with decades of in situ measurements of the solar wind electrons, we are still far from reaching a full understanding of the physical origin and formation of the suprathermal components, as well as their relative densities.It has long been regarded that suprathermal strahl and halo components originate in the solar corona (Pierrard et al. 1999;Crooker et al. 2004;ŠtveráK et al. 2008;Che & Goldstein 2014).Coronal electrons traveling outward into regions of decreasing magnetic field strength could experience strong adiabatic focusing, resulting in the formation of a highly field-aligned electron population, i.e., the strahl.This means that the strahl could result from the escape of thermal electrons from the corona (Feldman et al. 1975;Pilipp et al. 1987;Maksimovic et al. 2005;Tao et al. 2016).The halo is often explained by the scattering of suprathermal electrons via Coulomb collisions (Horaites et al. 2018) and wave-electron interactions, among which the more widely accepted is electron scattering by whistler waves (Vocks et al. 2005;Gary & Saito 2007;Saito & Gary 2007;Smith et al. 2012;Landi et al. 2012;Pavan et al. 2013;Kajdič et al. 2016;Verscharen et al. 2019;Vasko et al. 2019).There is substantial observational support for the scattering of strahl electrons in transit, which gives rise to the formation of part of the halo population.For example, multiple spacecraft observations between 0.3 and 4 au have shown that the relative number density of the strahl decreases with radial distance, whereas that of the halo increases (Maksimovic et al. 2005;Štverák et al. 2009;Tao et al. 2016).Moreover, the strahl Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.angular width is observed to increase gradually with radial distance (Hammond et al. 1996;Anderson et al. 2012;Graham et al. 2017).Despite these advances, the precise connection between halo and strahl populations is still debated (Graham et al. 2017).There might be more than one process contributing to the growth of the halo.A fraction of the core electrons could be accelerated to suprathermal energies by a resonant waveelectron interaction or other plasma processes, followed by scattering in the pitch angle to form the halo (Abraham et al. 2022).Concerning the formation of the superhalo, Wang et al. (2012Wang et al. ( , 2015) ) proposed that superhalo electrons might be produced by nonthermal electron acceleration processes in the solar wind source region, such as nanoflares and consequent isotropization processes due to electron scattering and reflection in the interplanetary medium.Another possibility is that they are produced by shock acceleration, resonant waveelectron interactions, and stochastic acceleration via the firstand second-order Fermi mechanisms in the interplanetary medium (Vocks et al. 2005;Fisk et al. 2010;Yoon et al. 2012;Yang et al. 2018Yang et al. , 2019)).
In recent years, it has become increasingly clear that the solar corona is not the only source of the strahl and halo components observed in the solar wind.Meanwhile, observational evidence is accumulating that suprathermal electrons can be created locally in solar wind regions filled with dynamic small-scale flux ropes (FRs), particularly at times when subject to strong plasma compression (i.e., FR compression acceleration; e.g., Khabarova et al. 2015Khabarova et al. , 2016;;Khabarova & Zank 2017).Smallscale FRs are commonly described as coherent structures advected with the solar wind flow, consisting of a twist magnetic field component in the 2D plane perpendicular to an out-of-plane axial magnetic field component (Cartwright & Moldwin 2010).It is known that these structures originate either in the solar corona or in the solar wind turbulent plasma through magnetic reconnection at large-scale (primary) current sheets (Borovsky 2008;Fu et al. 2017;Zank et al. 2017).The solar wind source viewpoint is supported by a statistical study of an unprecedented number of small-scale FRs identified using in situ measurements from the Wind spacecraft, showing that the number of small-scale FRs peak in the vicinity of primary current sheets (Hu et al. 2018;Zheng & Hu 2018).It is also supported by 3D compressible magnetohydrodynamic (MHD) turbulence simulations with a strong guide field, which suggest that turbulence in the solar wind is mainly quasi-2D with a dominant presence of coherent small-scale FRs (e.g., Dmitruk et al. 2004).Furthermore, evidence from simulations exists for the efficient acceleration of charged particles traversing regions filled with dynamic small-scale FRs, which results in the formation of power-law spectra (Matthaeus et al. 1984;Ambrosiano et al. 1988;Dmitruk et al. 2004;Drake et al. 2006Drake et al. , 2013)).The main small-scale FR acceleration mechanisms identified in particle simulations include (1) parallel guiding center motion acceleration by the parallel electric field generated when neighboring small-scale FRs form secondary reconnecting current sheets between them that merge (Oka et al. 2010), (2) curvature drift acceleration by the motional electric field generated when small-scale FRs contract or merge (in a first-order Fermi acceleration process; Drake et al. 2006Drake et al. , 2013;;Li et al. 2017Li et al. , 2018)), and (3) generalized or Lagrangian betatron acceleration (i.e., unified betatron and grad-B drift acceleration) by the motional electric field generated in contracting and merging small-scale FRs, which involves magnetic moment conservation when the magnetic field strength in the plasma drift flow frame slowly varies in time and space (Dahlin et al. 2016(Dahlin et al. , 2017;;Fu et al. 2019).Note that theoretical explanations of these acceleration mechanisms in the simulations often rely on kinetic transport theories (e.g., Le Roux et al. 2015Roux et al. , 2018Roux et al. , 2019;;Li et al. 2018).
Nevertheless, direct identification of these acceleration mechanisms from in situ measurements is often challenging, partly due to the limited spatial and/or temporal resolution of the measurement.However, there is an important exception, namely the adiabatic processes of betatron and Fermi (curvature drift) acceleration, which act, respectively, on the perpendicular and parallel components of suprathermal electron pitch angle distributions, allowing the role of these mechanisms to be distinctly identified (Dahlin 2020;Fu et al. 2020).In fact, the electron signatures of adiabatic acceleration mechanisms have been widely reported in planetary (mainly terrestrial) magnetotails (e.g., Fu et al. 2011Fu et al. , 2013;;Liu et al. 2018;Zhong et al. 2020;Guo et al. 2021).Here we present the direct observational evidence for the local betatron acceleration mechanism within a contracting small-scale FR in the solar wind, due to a local compression.In this event, the lowerenergy halo electrons were effectively accelerated through the betatron mechanism, whereas the higher-energy suprathermal electrons predominated by the superhalo component were almost not accelerated.The halo electron energization processes via the betatron mechanism are reproduced using an analytical model.

Data
In this work, we mainly use data from two instruments onboard the Wind spacecraft: the Three-Dimensional Plasma and Energetic Particle Investigation (3DP; Lin et al. 1995) and the Magnetic Field Investigation (MFI; Lepping et al. 1995).The Wind spacecraft was launched on 1994 November 1 into a highly elliptical Earth orbit with an apogee of about 80-250 Earth radii, and then was inserted into halo orbits around the Lagrange 1 point varying from 235 to 265 Earth radii.In the Wind 3DP instrument, electron electrostatic analyzers (EESA-L and EESA-H) provide three-dimensional measurements of electrons in the energy range of ∼3 eV-30 keV with an energy resolution of 0.2 (!E/E), and the Silicon Semiconductor Telescope measures ∼25 eV-400 keV electrons with an energy channel resolution of 0.3 (!E/E) and an angular resolution of 22°.5 × 36°.The 3D electron data are divided into eight PA bins with a resolution of 22°. 5 based on the magnetic field direction measured by the MFI instrument.The MFI instrument measures the direct current vector magnetic field with a maximum time resolution of up to 22 vectors per second.In addition, the proton data (proton density N p , proton velocity V p , and proton temperature T p ) calculated from the 3D proton distributions measured by the 3DP instrument are used in the identification of solar wind structures.We also use solar wind data measured from the ACE spacecraft located in a halo orbit around the L1 point (Stone et al. 1998).The plasma parameters (solar wind proton speed, density, temperature, and suprathermal electron PA) are obtained from the Solar Wind Electron, Proton, and Alpha Monitor (McComas et al. 1998), and the vector measurements of the interplanetary magnetic field are provided by the magnetometer (Smith et al. 1998).All data are utilized in geocentric solar ecliptic (GSE) coordinates, where the X-axis points from Earth to the Sun, the Y-axis is opposite to planetary motion in the ecliptic plane, and the Z-axis is parallel to the ecliptic pole.

Observation and Analysis
Figure 1 shows the solar wind plasma and magnetic field data from the Wind spacecraft near 1 au during the period from 2012 February 26 to March 1.At the time of the measurements, the Wind spacecraft was located upstream from the Earth at about (1.293, 0.503, − 0.078) × 10 6 km in GSE coordinates.A magnetic cloud (MC) arrived at ∼18:04 UT on February 27 and extended to ∼14:21 UT on February 28.Its signatures include the bidirectional streaming of strahl electrons, enhanced magnetic field strength |B|, smooth rotation of magnetic field azimuthal angle f, declining proton velocity V p , low proton temperature T p , and low proton β (Hirshberg & Colburn 1969;Gosling et al. 1973;Burlaga et al. 1981;Gosling et al. 1987).A forward shock driven by the MC arrived at ∼20:57 UT on February 26.The shock was followed by a sheath of shocked plasma characterized by enhanced fluctuating magnetic field strength, proton speed, density, and temperature.The MC was immediately followed by a heliospheric plasma sheet (HPS) that was straddling a heliospheric current sheet (HCS) where the IMF changed polarity.Notice that the HCS motion manifests itself as three crossings by the Wind spacecraft during the investigated interval, namely HCS1 at ∼15:28 UT on February 28, HCS2 at ∼05:00 UT on February 29, and HCS3 at ∼08:05 UT on March 1, possibly due to its irregular flapping and waving motion.The HCS roughly coincides with the sector boundary, where the PA of the strahl population changes from ∼0°to ∼180°or vice versa (Figure 1(b)).The first crossing of HCS occurred inside the HPS (i.e., HCS1 within HPS).Hereafter, when we speak of the HCS, we always mean the first crossing of HCS.The HPS, spanning ∼14:57 UT to 15:36 UT on February 28, is characterized by depressed magnetic field strength, enhanced proton density, and enhanced proton β (Winterhalter et al. 1994), which can be seen more clearly in the zoomed-in plot in Figure 2. We note that the leading edge of the HPS travels faster than the trailing edge of the MC (with a speed difference of about 20 km s −1 ), and the proton speed at the rear portion of the MC does not decline continuously but rather increases slightly (Figure 2(g)), implying a compression due to the overtaking HPS behind.Thus, it is reasonable to infer that the leading portion of the HPS was also undergoing a compression.An increase of the proton density in the leading portion of the HPS is clearly visible (Figure 2(f)), lending further confidence to our inference on the reverse compression.The compression could also lead to an intensification of the magnetic field strength (Figure 2(c)) in the leading portion, although not readily identifiable, owing to the combined effects of the depressed magnetic field inside the HPS (a defining characteristic of HPS mentioned above) and the compression of the HPS by the preceding MC.
It is interesting to note that a small-scale FR is embedded within the leading portion of the HPS.The FR is recognized in the data by the presence of smooth rotation of the magnetic field direction from 14:21 to 14:57 UT on February 28 (Figures 2(c) and (d)).Moreover, by applying the Grad-Shafranov-based algorithm to in situ measurements from the Wind spacecraft, Hu et al. (2018) found that the plasma structure in the leading portion of HPS can indeed qualify as a small-scale FR.The axis orientation of the FR in GSE spherical coordinates is represented by the polar angle of 80°and the azimuthal angle of 160°, meaning that its axis is mostly aligned with the Parker spiral magnetic field direction.The FR is probably produced and released through secondary magnetic reconnections in a reconnection exhaust, which corresponds to the observed high-density region defined as HPS and is driven by a primary reconnection near the tip of the helmet streamer, as proposed by Sanchez-Diaz et al. (2019) and Lavraud et al. (2020).The compression of the preceding MC could lead to the contraction of the FR.Thus, one could expect that the mean electric field induced by the contracting FR will accelerate suprathermal electrons inside the FR.
Figures 3(a) and (b) show the omnidirectional differential particle flux of suprathermal electrons in the energy range of 0.92-8.87keV and 103-1113 eV during the interval surrounding the FR.It is immediately evident that the omnidirectional electron fluxes of the 103-1113 eV electrons are enhanced within the FR, as marked by the upward arrow.Judging from the electron PA distributions displayed in Figures 3(c)-(h), the enhancement of the electron fluxes appears mainly at PAs near 90°.This implies that quasi-perpendicular electrons, i.e., halo electrons, were accelerated up to higher energy within the FR.This fact, in combination with the aforementioned magnetic field compression, indicates that the energization of halo electrons probably occurs through adiabatic betatron acceleration (conservation of the magnetic moment) within the FR, where the electron gyroperiod of ∼4 × 10 −3 s is much smaller than the timescale of the compression process or the characteristic time of FR contraction, and moreover, the characteristic spatial variance length, i.e., the scale size of the FR ∼3 × 10 5 km (Hu et al. 2018), is much larger than the electron Larmor radius (∼3.8 km for 100 eV electrons).The strahl electrons show a PA width 35°.For the beam-like electrons, the FR compression acceleration becomes inefficient, and thus their fluxes remain relatively constant.
In order to confirm and quantify the effect of betatron acceleration on the halo electrons within the FR, we attempt to reproduce the acceleration process with an analytical model based on Liouville's theorem, which states that the phase space density (PSD) is conserved along particle trajectories in phase space (see Pan et al. 2012).The analytical model is described by the equations where E is the electron energy, L is the electron bounce distance, and B is the magnetic field strength.The subscripts "0" and "1" correspond to the source population and consequence electron population, respectively.The symbols "⊥" and "∥" denote the perpendicular direction and field-aligned direction, respectively.F b and F f are acceleration factors connected with betatron and Fermi mechanisms, respectively.This model was initially advocated by Fu et al. (2011) and then was verified to be capable of tracking electron energy evolution in many studies (e.g., Fu et al. 2013;Zhong et al. 2020;Guo et al. 2021).
For this purpose, the source and resultant halo electrons (electrons before and after acceleration) need to be known first.We take the halo electrons measured at 15:00:09 UT (outside the FR but inside the HPS) as the source, and the halo electrons measured at 14:30:42 UT (inside the FR) as the consequence.
The assumption of the source is based on the viewpoint that FRs embedded within HPSs are likely released through sequential magnetic reconnection from the tip of helmet streamers (Sanchez-Diaz et al. 2019;Lavraud et al. 2020), which implies that the electrons inside the observed FR and those outside the FR but inside the HPS originated from the same source (i.e., helmet streamers).Note that no attempt has been made to separate beaming-like strahl electrons from isotropic halo electrons according to their different behaviors in the angular distribution.Instead, the halo electrons at 35°-180°P A are selected for the investigation of electron energy evolution.
Figures 4(a) and (b) show the PSDs of the source and consequence electrons in the energy range of ∼103-1113 eV, respectively.The modeled PSDs assuming adiabatic acceleration by betatron and Fermi mechanisms are also overlaid in Figure 4(b) for comparison.The best fit of the model to the electron observations at 35°-180°PA is obtained when F b is set to 1.16 and F f is set to 1.01.This result lends strong support to our earlier assertion that the observed electron energization within the FR is predominantly attributed to betatron acceleration by the electric field induced by the time variation in the magnetic field strength.The F b value of 1.16 means an increasing FR magnetic field strength by a factor of ∼16%.The magnetic field strength before compression is derived to be ∼7.8 nT.
Figure 4(c) shows the PSDs of quasi-perpendicular halo electrons measured at 15:00:09 UT (green solid curve) and 14:30:42 UT (blue solid curve) as a function of electron energy.The quasi-perpendicular direction corresponds to PAs ranging from 78°. 2 to 100°.7. It is immediately clear that the power-law index of the electron PSD remains essentially unchanged over time, in agreement with adiabatic acceleration theory.The two PSDs are closely approximated by a power law with a spectral index of −3.4.It is worth mentioning that the electron PSDs in this energy range can generally be fitted by either a Kappa distribution (mentioned earlier) or a power-law distribution (see Tao et al. 2016).Here, to permit a better comparison of the observations with the adiabatic theory, the electron PSDs are described by a power-law distribution.The calculated PSD of the source electron population after 1.16 times adiabatic acceleration in the quasi-perpendicular direction is overplotted as a red solid curve.It matches well with the consequence electron PSD, confirming that the betatron mechanism is the dominant process in the energization of the 103-1113 eV electrons.
We also inspect the in situ observations from the ACE spacecraft for this betatron acceleration event, during which ACE was located upstream from the Earth at about (1.533, −0.17, −0.087) × 106 km in GSE coordinates.The FR embedded within the leading portion of the HPS was encountered by ACE from 14:27 to 15:00 UT (not shown).We note that while there is a time shift between the FR encounters at the two spacecraft corresponding to the propagation of the FR from one point to another, the enhancements of halo electron fluxes were still maintained inside the FR.This provides evidence that the halo electrons were accelerated locally by betatron acceleration inside the FR, rather than being accelerated somewhere else and then entered into the FR.This scenario is similar to that of Zhong et al. (2020), who identified that enhanced energetic electrons in a reconnection-generated ion scale FR in the Earth's magnetotail were due to the local energization processes via betatron and Fermi (curvature drift) acceleration, and that most energetic electrons were produced by betatron acceleration.

Discussion
The Fermi factor obtained by fitting the analytical model to the observed halo electron PSD is very close to 1, meaning that the Fermi process did not operate on the local evolution of halo electron PSD.This is in agreement with the theoretical expectation for a region of open magnetic field lines, where the electrons could not be effectively trapped by magnetic mirroring.As mentioned above, the strahl electrons within the FR were observed field-aligned at 35°PA, indicating that the FR was magnetically connected to the Sun at one end, and the other end possibly extended to infinity in the heliosphere.It is possible, given that the magnetic field lines inside the FR have been stretched to infinity, that the contraction of distance between mirror points of electron bounce oscillations caused by the compression effect is relatively insignificant.More likely, the halo electrons inside the FR were not able to bounce back and forth many times, producing trapped electron distributions.Thus, the nonperpendicular electrons were not accelerated locally via the Fermi mechanism.Obviously, further observational studies of closed magnetic FRs are necessary to identify the in situ signatures of Fermi acceleration mechanism.
It is somewhat surprising to find that, unlike the lowerenergy (∼103-1113 eV) halo electrons, the higher-energy (>∼1113 eV) suprathermal electrons predominated by the superhalo component did not reveal any noticeable enhancement in their omnidirectional differential fluxes (Figure 3(a)) in response to the increasing magnetic field strength due to the preceding MC compression.This hints that the higher-energy electrons cannot gain much energy through adiabatic betatron acceleration.This is possible because the higher-energy electrons have shorter correlation times, which can be thought of as the time one particle needs to move from one magnetic FR (or island) to its neighboring region (see Pezzi et al. 2021), compared to the lower-energy electrons, and thus they can easily escape from the acceleration region and never turn back.More discoveries of such events are obviously required to develop a theoretical explanation of higher-energy electron energization.These advances, in turn, would contribute to our understanding of how the superhalo electrons are observed in the solar wind form.
Motivated by this, we have further examined small-scale FRs detected in the vicinity of HCS over the period 1995-2020 for the in situ signatures of betatron and Fermi acceleration processes.In addition to examining the Wind spacecraft data for the presence of FRs, we refer to available small-scale FR lists identified and compiled by Hu et al. (2018).The results indicate that almost all the small-scale FRs do not contain distinct electron distribution function signatures parallel and perpendicular to the local magnetic field.A possible explanation is that the suprathermal electrons experience significant PA scattering from the fluctuating magnetic mirroring forces inside dynamic FRs, as proposed by Le Roux et al. (2018Roux et al. ( , 2019)).It is worth mentioning here that, over the past decade, Parker-type kinetic transport theories for energetic particles experiencing PA scattering and energization in the solar wind region containing dynamic small-scale FRs have been developed and validated to reproduce the observed features of energetic particle acceleration and propagation (e.g., Zank et al. 2014Zank et al. , 2015;;Le Roux et al. 2015;Zhao et al. 2018;Adhikari et al. 2019).

Conclusions
In this Letter, we report the first in situ evidence for a local betatron acceleration mechanism within a contracting smallscale FR in the solar wind.The FR was embedded within the leading portion of an HPS, which was undergoing compression by a preceding magnetic cloud.The betatron acceleration is caused by an increasing FR magnetic field strength under the conservation of the first adiabatic invariant (magnetic moment) and results in the enhancement of halo electron fluxes in the perpendicular direction.Moreover, it is interesting to note that the lower-energy (∼103-1113 eV) halo electrons were effectively accelerated through the betatron mechanism, but the higher-energy (>∼1113 eV) suprathermal electrons predominated by the superhalo component were almost not accelerated, presumably because they could not be trapped within the acceleration region (i.e., the contracting FR) for a slightly longer time.The halo electron energization processes via the betatron mechanism are successfully reproduced using an analytical model.The examination of small-scale FRs in the vicinity of HCS over the period 1995-2020 indicates that the in situ signatures of betatron acceleration process are essentially elusive.This study improves our understanding of small-scale FR compression acceleration in the solar wind and also has profound implications for the origin and formation of the solar wind suprathermal electrons, as well as the physical processes contributing to the growth of the halo component.

Figure 1 .
Figure 1.Solar wind parameters observed by the Wind spacecraft from 2012 February 26 to March 1.From top to bottom: (a) omnidirectional differential flux of the 103-1113 eV electrons, (b) pitch angle (PA) distribution of the 427 eV electrons, (c) interplanetary magnetic field strength (|B|), (d) elevation angle (θ) and (e) azimuthal angle (f) of field direction in GSE coordinates, (f) proton density (N p ), (g) proton velocity (V p ), (h) proton temperature (T p ) overlaid with the expected temperature from the observed proton speed, and (i) proton β.The shaded regions indicate the magnetic cloud (MC) and heliospheric plasma sheet (HPS) intervals.The pink vertical dashed line marks the shock driven by the MC, and the blue dashed lines mark the multiple crossings of the heliospheric current sheet (HCS).

Figure 2 .
Figure 2. Wind observations of a flux rope (FR) embedded within the leading portion of an HPS (yellow shaded region) on 2012 February 28.The format is the same as Figure 1, except that magnetic field components in GSE coordinates are also given and overplotted in the magnetic field strength panel.Two purple vertical solid lines mark the boundaries of the FR.

Figure 3 .
Figure 3. Electron omnidirectional differential fluxes and electron pitch angle (PA) distribution during the interval surrounding the FR.From top to bottom: (a)-(b) omnidirectional differential fluxes of the 0.92-8.87keV and 103-1113 eV electrons, respectively, (c)-(h) PA distributions of electrons with central energies of 1113, 689, 427, 265, 165, and 103 eV, respectively, (i) magnetic field strength and three components in GSE coordinates.The yellow shaded region indicates the HPS interval.Two purple vertical solid lines mark the boundaries of the FR.The blue vertical dashed line marks the crossing of the HCS.

Figure 4 .
Figure 4. Electron phase space density (PSD) as a function of pitch angle (PA) and electron energy.(a) PSD outside the FR but inside the HPS (at 15:00:09 UT); it is treated as the "source."(b) PSD inside the FR (at 14:30:42 UT); it is treated as the "consequence" after acceleration from the assumptive source electrons.Solid curves represent the Wind observations, and dashed curves represent the modeling results.The betatron factor F b and Fermi factor F f are shown.(c) PSDs of quasiperpendicular electrons (103-1113 eV) at 14:30:42 UT (blue solid curve) and 15:00:09 UT (green solid curve).The red solid curve shows the predicted values assuming betatron acceleration.These PSDs follow approximately a power law with an index of ∼−3.4.