# Capsule Electron Distributions Near the Diffusion Region of Magnetic Reconnection

Yong Ren, Lei Dai, Chi Wang, Benoit Lavraud, C. Philippe Escoubet, James

Burch

► To cite this version:

Yong Ren, Lei Dai, Chi Wang, Benoit Lavraud, C. Philippe Escoubet, et al. Capsule Electron Distributions Near the Diffusion Region of Magnetic Reconnection. Geophysical Research Letters, 2024, 51 (23), 10.1029/2024GL112074. http://dx.doi.org/10.1029/2024GL112074

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Distributed under a Creative Commons Attribution 4.0 International License
# Geophysical Research Letters®

## RESEARCH LETTER

10.1029/2024GL112074

**Key Points:**

* Non-Maxwellian capsule electron distributions near the electron diffusion region comprises an elongated component and a butterfly component

* Capsule-like electron distributions generate whistler emissions in the presence of unusual temperature anisotropy $T_{\parallel}/T_{\perp} < 1$

* Capsule electron distributions suggest electron trapping and preferential heating by parallel electric fields

**Supporting Information:**

Supporting Information may be found in the online version of this article.

**Correspondence to:**

L. Dai

ldai@spaceweather.ac.cn

**Citation:**

Ren, Y., Dai, L., Wang, C., Lavraud, B., Escoubet, C. P., & Burch, J. L. (2024). Capsule electron distributions near the diffusion region of magnetic reconnection. *Geophysical Research Letters*, *51*, e2024GL112074. https://doi.org/10.1029/2024GL112074

Received 27 AUG 2024

Accepted 20 NOV 2024

© 2024. The Author(s). This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

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## Capsule Electron Distributions Near the Diffusion Region of Magnetic Reconnection

Yong Ren<fnref n="1" />, Lei Dai<fnref n="1" /> Ⓒ, Chi Wang<fnref n="1" />, Benoit Lavraud<fnref n="2" /> Ⓒ, C. Philippe Escoubet<fnref n="3" /> Ⓒ, and James L. Burch<fnref n="4" /> Ⓒ

**Abstract** Understanding electron kinetic processes is crucial for elucidating the energy conversion mechanisms in magnetic reconnection. Non-Maxwellian electron distributions are strong indicators of kinetic-scale processes near the electron diffusion region, yet they remain incompletely understood. Using in-situ spacecraft data from 29 magnetopause reconnection events, we unambiguously identify a non-Maxwellian capsule electron distribution near the electron diffusion region. This distribution comprises an elongated component parallel with the magnetic field at lower energies and a butterfly component (with peaks at pitch angles near 45° and 135°) at higher energies. We provide evidence that these distributions are partly linked to electron trapping and preferential heating along the direction of magnetic fields. The parallel electric potentials needed for the parallel heating may be linked to kinetic Alfvén waves. These capsule-like electron distributions are also found to generate whistler emissions. Our results suggest that these kinetic processes are prevalent in magnetic reconnection.

**Plain Language Summary** Understanding how electrons behave in space plasmas, especially during magnetic reconnection, is crucial. Using data from spacecraft, we studied electron behavior near this reconnection region and identified a unique pattern called a capsule distribution. This distribution combines two shapes: an elongated component at lower energies and a butterfly like component at higher energies. These distributions are closely linked to whistler wave emissions. Our research provides evidence that these capsule-like distributions indicate electron trapping and heating by parallel electric fields. The parallel electric field and electric potentials may be related to kinetic Alfvén waves. These kinetic processes, captured by emitted whistler waves, are prevalent near the diffusion region, offering deeper insights into electron heating during magnetic reconnection.

## 1. Introduction

Magnetic reconnection is a fundamental process in solar-terrestrial coupling, transferring energy from the solar wind to Earth's magnetosphere and driving magnetospheric convection (Axford, 1969; Cowley & Lockwood, 1992; Dai et al., 2024; Dungey, 1961; Zhu et al., 2024). It emerges as a crucial mechanism facilitating energy transfer across multiple scales (Dai et al., 2020; Franci et al., 2017; Lee & Lee, 2020; Retinò et al., 2022; Zhang et al., 2020). On a large scale, free energy accumulates as magnetic energy is stored and intensifies the current layer (Dai et al., 2023; Han et al., 2024; Scholer et al., 2003; Sergeev et al., 1993). On a small scale, in the diffusion region within the current layer, kinetic-scale processes are required to violate the frozen-in condition, allowing the “breaking and reconnection” of magnetic field lines (Birn et al., 2001; Vasyliunas, 1975). During the change in magnetic configuration, stored magnetic energy is released and transferred into other forms of energy.

A strong indicator of kinetic processes arising from energy transfer during reconnection is non-Maxwellian velocity distributions, characterized by deviations from the familiar Maxwellian distribution. These distributions serve as diagnostic tools for kinetic processes and sources of free energy for plasma wave generation (Burch, Torbert, et al., 2016; Dai et al., 2021; Torbert et al., 2018; M. Fujimoto et al., 2011; Khotyaintsev et al., 2019; Ren et al., 2022). Through kinetic instabilities, non-Maxwellian velocity distributions can further feed energy back into various types of plasma waves, eventually becoming thermalized again (D. Graham et al., 2017; Khotyaintsev et al., 2020; Ren, Dai, Wang, & Guo, 2023; Ren, Dai, Wang, & Lavraud, 2023).

<footnote n="1">¹State Key Laboratory of Space Weather, National Space Science Center, Chinese Academy of Sciences, Beijing, China,</footnote>

<footnote n="2">²Institut de Recherche en Astrophysique et Planétologie, Université de Toulouse, CNRS, UPS, CNES, Toulouse, France,</footnote>

<footnote n="3">³European Space Research and Technology Centre, European Space Agency (ESA), Noordwijk, The Netherlands,</footnote>

<footnote n="4">⁴Southwest Research Institute, San Antonio, TX, USA</footnote>
Non-Maxwellian velocity distributions and their underlying processes have been studied in the context of magnetic reconnection. In the electron diffusion region (EDR), the Magnetospheric Multiscale (MMS) mission detected crescent-shaped distributions both parallel to and in the plane perpendicular to the magnetic field (Burch, Torbert, et al., 2016; Torbert et al., 2018; D. B. Graham, Khotyaintsev, Vaivads, et al., 2017; Lenouvel et al., 2021). The crescent-shaped distributions in the plane perpendicular to the magnetic field are described as “meandering particles” (Bessho et al., 2016; Lapenta et al., 2017) and have also been detected in non-reconnecting current layers (B. B. Tang et al., 2019). Additionally, electron crescents have been shown to result from diamagnetic effects associated with strong density asymmetry during reconnection at the dayside magnetopause (Egedal et al., 2016) and are not necessarily related to meandering electrons (Egedal et al., 2018). Multiple crescent distributions in the plane perpendicular to the magnetic field are seen around the X-line (Burch et al., 2019; Torbert et al., 2018). These crescent-shaped electron velocity distributions indicate electron acceleration by electric fields (Burch, Torbert, et al., 2016; Torbert et al., 2018), and can produce broad frequency spectra of upper hybrid waves (D. B. Graham, Khotyaintsev, Vaivads, et al., 2017; Burch et al., 2019). Elongated electron distributions along the magnetic field have been observed near diffusion regions (Egedal et al., 2008; Khotyaintsev et al., 2020; Ren et al., 2021; Wetherton et al., 2019, 2021; Zhong et al., 2022). The formation of such distributions represents parallel electron heating and is attributed to electron trapping by parallel electric fields (Egedal et al., 2008, 2013; Le et al., 2009; Wetherton et al., 2021). Electron parallel beams, pancake distributions, and loss-cone distributions are observed in dayside and nightside magnetic reconnection (Lavraud et al., 2016; D. B. Graham et al., 2016; Khotyaintsev et al., 2019; Wei et al., 2007; Zhou et al., 2011; Huang et al., 2016, 2017; Torbert et al., 2018; Ren et al., 2019). The field-aligned beam distributions arise from reconnection-induced acceleration or the redirection of perpendicular crescents, while pancake-like distributions result from betatron heating as reconnected field lines pile up (K. K. Fujimoto & Sydora, 2008; K. K. Fujimoto, 2014; Fu et al., 2014; Burch, Torbert, et al., 2016). Loss cone distributions accompanied by a cold electron beam are attributed to open magnetic field lines and electron acceleration (K. K. Fujimoto, 2014; D. B. Graham et al., 2016; Khotyaintsev et al., 2019). These electron parallel beams, pancake distributions, and loss-cone distributions provide free energy for whistler waves and electrostatic modes (X. Tang et al., 2013; Le Contel, Retinò, et al., 2016; Cao et al., 2017; Wilder et al., 2017; Huang et al., 2016; Torbert et al., 2018; Ren et al., 2019; Zhong et al., 2021; Wang et al., 2023; Choi et al., 2022).

Beyond the non-Maxwellian distributions mentioned above, this study identifies a capsule distribution near the diffusion region, characterized by an elongated component at lower energies and a butterfly like component (with peaks at pitch angles near 45 and 135° creating a shape of butterfly wings (Xiao et al., 2015; Ni et al., 2016; J. Li et al., 2016; Yang et al., 2017; Ozeke et al., 2022)) at higher energies. For the first time, we adopt the term “capsule” to effectively describe both energy components of these distributions. Capsule-like electron distributions have been previously observed in specific event studies (Khotyaintsev et al., 2020; Ren et al., 2021). However, the detailed pitch-angle and energy characteristics were not defined, nor was the term 'capsule' explicitly introduced. Moreover, the prevalence of these distributions during reconnection events remains uncertain. In this study, we quantitatively define capsule distributions near the electron diffusion region through a statistical analysis of their energy and pitch-angle characteristics. The pitch-angle distribution (PAD) of the capsule distribution gradually changes with increasing energy, evolving from an elongated shape with enhancements near 0° and 180° to a butterfly shape with enhancements near 45° and 135°. These capsule distributions are found to be prevalent near the electron diffusion region and are associated with whistler wave emissions.

## 2. Data and Method

### 2.1. Database

We utilized data from the Magnetospheric Multiscale (MMS) mission (Burch, Moore, et al., 2016). Three-dimensional magnetic field data were obtained from the fluxgate magnetometer (FGM) at 128 samples/s and the search-coil magnetometer (SCM) at 8,192 samples/s (Le Contel, Leroy, et al., 2016; Russell et al., 2016). For particle data, we used the Fast Plasma Investigation (FPI) (Pollock et al., 2016), which provided electron measurements at a temporal resolution of 30 ms and ion measurements at 150 ms. The database, including the EDR of 29 magnetopause reconnections, is detailed in (Ren et al., 2021; Webster et al., 2018).
## 2.2. Legendre Polynomials Fitting

We employed Legendre polynomials to fit the normalized pitch-angle distributions (PADs). The inherent symmetry of even-order Legendre polynomials is suitable for modeling symmetric distributions (Chen et al., 2014; Zhao et al., 2021). The fitting function for normalized PADs based on Legendre polynomials is given by:

$$f_{\text{norm}}(\theta_{\text{PA}}) = \sum_{n=0}^{\infty} C_n P_n(\cos(\theta_{\text{PA}})) \quad (1)$$

where $\theta_{PA}$ is the pitch angle, $f_{norm}(\theta_{PA})$ is the fitted distribution of normalized PADs at different energies, $C_n$ are the Legendre coefficients, and $P_n(\cos(\theta_{PA}))$ are the orthogonal Legendre polynomials. In our study, we applied Legendre polynomials ($P_2$, $P_4$, $P_6$, and $P_8$) with corresponding coefficients $C_2$, $C_4$, $C_6$, and $C_8$. The Root Mean Square Deviation (RMSD) was used to assess the fitting accuracy, with RMSD values below 0.1 indicating a good fit (Figure S1 in Supporting Information S1).

# 3. Results

## 3.1. Event Study of Capsule Distributions

An example of capsule distributions near the diffusion region of magnetic reconnection at Earth's magnetopause is analyzed in Figure 1. We converted the vector quantities into LMN coordinates using minimum variance analysis of the magnetic field. The EDR, as identified in (Webster et al., 2018), is marked by a vertical box. Near 07:41:15 UT and 07:42:10 UT, ion jets were accelerated to a velocity of ~400 km/s along the L direction as a result of magnetopause reconnection. Near the EDR, the parallel temperature anisotropy $T_\parallel/T_\perp > 1$ (Figure 1d) suggests preferential electron heating in the direction parallel to the magnetic fields. Figures 1e and 1f show the spectra of magnetic power and ellipticity. An ellipticity greater than 0.5 corresponds to strong right-handed polarization (Figure 1f). These emissions are identified as whistler waves based on their ellipticity and wave frequency, which falls between the lower hybrid frequency and the electron cyclotron frequency, as described by Ren et al. (2021). For whistler emissions near a few hundred Hz, their Poynting flux is bi-directional along the magnetic field (Figure 1g), suggesting local generation. This feature is consistent with the scenario where the spacecraft passes through a source region, locally generating wave emissions that propagate away from the source (Le Contel, Retinò, et al., 2016).

The electron distributions concurrent with the whistler waves are displayed in Figures 1h–1j. The 2D velocity distribution slice in $V_\parallel - V_\perp$ resembles a capsule (Figure 1h). Capsule distributions are observed in magnetosheath-like electrons over a broad region near the electron diffusion region, including the magnetosheath and the low-latitude boundary layer on the magnetosphere side of the current layer (Ren et al., 2021). Figures 1i and 1j show the energy and pitch-angle characteristics. To clearly demonstrate the pitch angle anisotropy of the phase space density (PSD), we present the normalized pitch angle distribution (PAD) as the electron PSD divided by the maximum PSD within the same energy range (Figure 1j). In the lower-energy range (<100 eV), the capsule distribution is characterized by an elongated shape with enhancements near pitch angles of 0° and 180°. As energy increases to 100–1,000 eV, these enhancements shift to pitch angles of 45° and 135°.

## 3.2. Statistical Characteristics of Capsule Distributions

We investigate the distribution of capsule electrons associated with whistler emissions within ±2 min of the EDR during the 29 magnetopause reconnection events during 2015–2016 (Webster et al., 2018). For all events of capsule electrons, whistler wave emissions are first identified following the methodology described by Ren et al. (2021).

To identify capsule distributions, we use Legendre polynomial fittings of PAD with the following criteria: 1. There are at least three low-energy bins (30–100 eV) $f_{norm}(\theta_{PA})$ displaying peak values at angles below 20° and above 160°. These energy bins correspond to elongated distributions. 2. There are at least three continuous medium-energy (100–1,000 eV) $f_{norm}(\theta_{PA})$ bins with peak values within the ranges of 20°–70° and 110°–160°. These energy bins correspond to butterfly distributions. 3. There is a dip in the range of 80°–110° in energy bins of butterfly distributions. The depth (DS) of butterfly distributions, defined as $DS = (f_{norm,peak} - f_{norm,peak,peak})$.
<figure><img src="image_3.png" /><figcaption>**Figure 1.** Capsule distributions co-occurring with whistler waves near diffusion region of magnetopause reconnection observed by MMS1 on 2015 September 19. (a) The magnetic field. (b) The ion bulk flow velocity in the LMN coordinate system. (c) The electron omni-directional differential energy flux. (d) The electron temperature anisotropy, $T_\|/T_\perp$. (e) The dynamic power spectra of the magnetic field. (f) The ellipticity spectra. (g) The spectra of Poynting flux in the direction parallel to magnetic fields. (h) An enlarged view of the ellipticity spectra near 07:43:48UT. (i) An enlarged view of the spectra of Poynting flux 07:43:48UT. (j) The capsule-like velocity distribution of electrons. (k) The pitch angle distributions (PADs) of electrons. (l) The normalized PADs.</figcaption></figure>
<figure><img src="image_4.png" /><figcaption>**Figure 2.** Statistical plasma properties associated with capsule electron distribution. The distribution of (a) $B_L/B_{L,max}$ and (b) $T_\parallel/T_\perp$ for capsule electron distributions.</figcaption></figure>

$f_{\text{norm},\theta_{\text{PA}}=90^{\circ}}/(f_{\text{norm,peak}} - f_{\text{norm,minimum}})$ (Ozeke et al., 2022), must be greater than 0.1. The criterion $DS > 0.1$ is applied to exclude flat-top type distributions, where the peaks at 45°/135° are relatively small.

A total of 509 capsule distribution samples, each averaged with a time window of 0.25 s, are identified in the database. Figure 2 shows the properties of magnetic fields and plasma associated with the capsule distribution. $B_L$ is the main component of the magnetic fields in the current layer. $B_L/B_{L,max} > 0$ and $B_L/B_{L,max} < 0$ correspond to the magnetosphere side and magnetosheath side of the current layer, respectively. Figure 2a demonstrates that capsule distributions are mostly located near the center ($-0.4 < B_L/B_{L,max} < 0.4$) of the magnetopause current layer. Figure 2b shows that capsule distributions are characterized by an electron temperature anisotropy ($T_\parallel/T_\perp > 1$) in over 80% of cases. For a small portion of capsule distributions, the elongated component may make less contributions to the overall temperature as compared with the butterfly component, reducing $T_\parallel/T_\perp$ to less than 1.

Figures 3a–3i present the superposed profiles of PADs for the identified capsule distributions. At lower energies (<100 eV), capsule distributions exhibit peaks near 0° and 180° (Figures 3d–3f), with a corresponding depletion in PSD near 90°. This depletion becomes more pronounced as energy increases. At higher energies (100–1,000 eV), PADs gradually develop into butterfly distributions with peaks near 45° and 135°, which shift toward 90° with increasing energy. Similar butterfly distributions have been observed in magnetosheath mirror structures (Breuillard et al., 2018; Kitamura et al., 2020; Kivelson & Southwood, 1996; Yao et al., 2018).

Figure 4a shows the averaged capsule distributions in 2D $V_\parallel - V_\perp$ slice. Figure 4b confirms that the depletion near 90° intensifies with energy in the lower energy range. Figure 4c shows that the peak of the butterfly distribution shifts toward 90° as energy increases.

For comparison, Figure 4 also displays PADs derived from kinetic modeling of elongated distributions, as detailed in studies (Egedal et al., 2008; Le et al., 2009; Wetherton et al., 2021). In the kinetic modeling, elongated distributions arise from trapping by large-scale parallel electric fields. The modeling PADs is (Wetherton et al., 2021):

$$f(\mathbf{x}, \mathcal{E}_\parallel, \mathcal{E}_\perp) = \begin{cases} f_\infty(0, \mu B_\infty), & \text{trapped} \\ f_\infty(\mathcal{E} - e\Phi_\parallel - \mu B_\infty - \Delta\mathcal{E}, \mu B_\infty), & \text{passing} \end{cases} \quad (2)$$

Here, $\mu$ denotes the first adiabatic invariant ($mv_\perp^2/(2B) = \mathcal{E}_\perp/B$), $B$ represents the magnetic field strength of the local region, $\Phi_\parallel$ stands for the acceleration potential ($\Phi_\parallel = \int_k^\infty \mathbf{E} \cdot d\mathbf{l}$), and $\mathcal{E}$ indicates kinetic energy. $B_\infty$ and $f_\infty$ represent the background magnetic field and the electron velocity distribution upstream of reconnection, respectively. $\Delta\mathcal{E}$ denotes the energy gained from magnetic compression ($\Delta B$) acting on the flux tube ($\Delta\mathcal{E} = \mu\Delta B = \mathcal{E}_\perp\Delta B/B$) (Wetherton et al., 2021). Trapped-passing boundaries are delineated by $\mathcal{E} - e\Phi_\parallel - \mu B_\infty = 0$ (indicated by white lines in Figure 4e). The model assumes a source $f_\infty(\mathcal{E}_\parallel, \mathcal{E}_\perp)$, an upstream distribution function with magnetic field strength $B_\infty$. In our study, this source $f_\infty(\mathcal{E}_\parallel, \mathcal{E}_\perp)$ is assumed to be isotropic, with $T_\infty = 40$ eV and $n_\infty = 25$ cm$^{-3}$. Trapped-passing boundaries and the predicted electron distributions are evaluated at a location specified by $B/B_\infty = 0.8$, with an average of $\Phi_\parallel = 100$ V, and $\Delta B/B = 0.1$ (Figures 4e–4g). The kinetic model used in this study is applicable to both inflow and exhaust regions (Wetherton et al., 2021).
<figure><img src="image_5.png" /><figcaption>**Figure 3.** Superpose profiles of normalized pitch-angle distributions for capsule electron distribution in observations (a–c) in the low energy 30–100 eV and (d–i) higher energy 100–1000 eV. The red, blue, and green line indicates the median (50th percentile), 25th and 75th percentiles, respectively.</figcaption></figure>

For both observations and models, the depletion depth in the PSDs below 100 eV increases with energy. At lower energies (<100 eV), the observed enhancement of PSD near 0° and 180° is also well captured by kinetic models. While there is a qualitative similarity between the low-energy component of the capsule distributions and the elongated distributions documented in previous studies (Egedal et al., 2008; Le et al., 2009; Wetherton et al., 2019, 2021), there are significant quantitative differences between the observed (Figure 4b) and modeling results (Figure 4f). Specifically, the theoretical model predicts a stronger decrease in phase space density at 90° and more elongated contours in the parallel and antiparallel directions than those observed. One potential reason for this discrepancy is that the kinetic model depends on several parameters, such as the position within the reconnection region ($B/B_\infty$), parallel electric potentials, source electron populations, and other factors. These parameters are not directly accessible through our in-situ observations in each event, which may probe a small region near the EDR. As a result, it is difficult to accurately implement all the required parameters based on the observational data. Our current approach is to use a set of parameters that are considered reasonable for magnetopause reconnection. We hope to address these parameters more comprehensively in future studies.

In the 100–1,000 eV range of butterfly distributions, a discrepancy exists between the kinetic models and observations in normalized PADs near 90° in particular. The kinetic modeling underestimates (<0.10) the normalized PAD compared to observations (0.65–0.75). As $Φ_‖$ increases, the distributions become more elongated. Adjusting parameters such as $B/B_\infty$ and $ΔB/B$ in the model does not effectively eliminate the underestimate in normalized PADs near 90°.
Observation

<figure><img src="image_6.png" /></figure>

The kinetic model

<figure><img src="image_7.png" /></figure>

**Figure 4.** Comparison of the capsule electron distribution in observations and kinetic modelings. Observation results of (a) the average profile of 2D capsule distribution in $V_\parallel - V_\perp$, (b) normalized PADs in 30–100 eV, and (c) normalized PADs in 100–1000 eV. Kinetic modelings of (d) normalized PADs in 30–100 eV, and (e) normalized PADs in 100–1000 eV. The modeling distributions are calculated for $T_\infty = 40$eV, $n_\infty = 25$cm$^{-3}$, $\Phi_\parallel = 100$V, $\Delta B/B = 0.1$ and $B/B_\infty = 0.8$. The white lines indicate the trapped-passing boundaries. See text for details.

3.3. Instability Analysis of Capsule Distributions

We analyze the instability of whistler waves for the average capsule distribution during magnetopause reconnection. Figure 5a presents a schematic of the capsule distributions in three-dimensional velocity space. As input to the WHAMP (Waves in Homogeneous, Anisotropic, Multicomponent Plasmas) (Roennmark, 1982), we model the capsule distribution as a superposition of three bi-Maxwellian populations with the following parameters: $-n_{e1} = 15$ cm$^{-3}$, $T_{e1,\perp} = 45$ eV, $T_{e1,\parallel}/T_{e1,\perp} = 1$, $V_{d1} = 0 - n_{e2} = 5$ cm$^{-3}$, $T_{e2,\perp} = 40$ eV, $T_{e2,\parallel}/T_{e2,\perp} = 2$, $V_{d2} = 3000$ km/s $- n_{e3} = 5$ cm$^{-3}$, $T_{e3,\perp} = 40$ eV, $T_{e3,\parallel}/T_{e3,\perp} = 2$, $V_{d3} = -3000$ km/s. Here, $V_d$ represents the drift velocities along the magnetic field, while $n_e$ and $T_e$ are the number density and temperature of electrons, respectively. The streaming of $n_{e2}$ and $n_{e3}$ populations is to approximate the elongated distributions in the parallel and anti-parallel direction in low energies, respectively. The temperature anisotropy of $n_{e2}$ and $n_{e3}$ is to mimic the butterfly effect in the high energies. For the magnetic field, we use an average value of $B = 20$ nT.

Figure 5c shows the normalized PADs of the modeled distribution. The modeled PAD closely resembles the observed average distribution shown in Figure 4a (see Figure S2 in Supporting Information S1). The black curve represents the first-order cyclotron resonant interaction (Thorne et al., 2005; W. Li et al., 2010). The resonance condition is expressed as $\omega - k_\parallel v_\parallel = n |\Omega_{ce}| \gamma^{-1}$, where $\Omega_{ce}$ is the electron gyrofrequency, $\gamma = (1 - v^2/c^2)^{1/2}$ is
<figure><img src="image_8.png" /><figcaption>**Figure 5.** The instability analysis of a typical capsule electron distribution. (a) Schematic illustration of the capsule distribution. (b) The 2D slices of 3D velocity distribution. (c) Energy-pitch distribution of the normalized phase space density for capsule electron distributions. The black curves indicate the first-order cyclotron resonance energies. (d) Dispersion analysis of two branch whistler waves propagating in parallel and anti-parallel direction. Frequency and growth rate as a function of the wave numbers.</figcaption></figure>

the relativistic factor, *c* is the speed of light, *v* is the electron velocity, *v*<sub>‖</sub> is the resonance velocity, and *ω*/*k*<sub>‖</sub> is the wave phase speed. The first-order cyclotron resonance (*n* = 1) primarily controls the growth of parallel propagating whistler waves (Kennel & Petschek, 1966).

In Figure 5c, the first-order cyclotron resonance occurs at the edge of the butterfly distribution (100 eV < *E*<sub>e</sub> < 1,000 eV), corresponding to a strong gradient over the pitch angle in the phase space density. This feature suggests that the capsule distribution drives whistler wave emissions through cyclotron resonant interaction. This result is consistent with numerical calculations from WHAMP, shown in Figure 5d, which indicate a positive growth rate of γ/Ω<sub>ce</sub> ~ 0.006 for whistler waves at frequencies near 0.13 Ω<sub>ce</sub>. The symmetric capsule distributions can produce whistler waves with a bi-directional Poynting flux (Figure 5d), and this instability mechanism is similar to the pancake-like electron velocity distributions described by Kitamura et al. (2020).

**4. Summary and Discussion**

Based on statistical in-situ data near the EDR of magnetic reconnection, we identify a distinctive capsule electron distribution associated with whistler waves. This non-Maxwellian distribution is detected is linked to electron temperature anisotropy (*T*<sub>‖</sub>/(*T*<sub>⊥</sub> > 1).

The capsule distribution exhibits energy-dependent variations in PAD. At energies below 100 eV, the PAD is elongated with enhancements near 0° and 180°. For energies between 100 and 1,000 eV, the PAD transitions to a butterfly pattern with enhancements near 45° and 135°. In the low-energy range, the depletion near 90° becomes more pronounced as energy increases. At higher energies, the peak of the PSD shifts toward 90° as energy increases. These features of PAD are compared in detail with kinetic models (Egedal et al., 2008; Le et al., 2009; Wetherton et al., 2019, 2021). The low-energy component of capsule electron distributions is qualitatively similar
to elongated distributions analyzed in previous studies (Egedal et al., 2008; Le et al., 2009; Wetherton et al., 2019, 2021), suggesting electron trapping and preferential electron heating along the magnetic field. However, the modeled distribution shows more pronounced elongation in the parallel and antiparallel directions than what is observed at lower energies. This discrepancy may result from the large-scale nature of several parameters used in the model, such as $B/B_\infty$ and parallel electric potentials. Accurately accounting for these parameters based on current in-situ observations, which may probe a small region near the EDR, is challenging. Future studies will need to address these complexities more comprehensively to improve model-data comparisons.

In the 100–1,000 eV range of butterfly distributions, the kinetic models significantly underestimate (<0.10) the normalized PADs at 90° pitch angle compared to observations (0.65–0.75), suggesting that other processes may contribute to the perpendicular heating of electrons. Interestingly, Lichko and Egedal (2020) observed that butterfly like electron distributions can arise from non-monotonic magnetic field evolution, where marginally trapped electrons encounter regions of increasing magnetic field strength. This suggests that additional factors, such as complex magnetic geometries, may contribute to the perpendicular heating of electrons. Furthermore, Egedal and Lichko (2021) proposed a more complete model that considers such complex magnetic geometries. Future work will explore this model further to address the discrepancies observed in the normalized PADs at 90° pitch angles.

Our statistical results suggest the prevalence of parallel electric potentials on the order of 100 V near the diffusion region of reconnection at the Earth's magnetopause. Producing this potential well requires parallel electric fields of large spatial scales. Such parallel electric fields, along with perpendicular Hall electric fields, are likely components of the kinetic Alfvén mode in the reconnection current layer (Dai & Wang, 2023; Lu et al., 2021). These parallel electric fields are balanced by the electron pressure gradient along the field lines (Dai, 2009; Dai et al., 2017; Duan et al., 2016; Wetherton et al., 2021).

We find that capsule distributions are unstable to whistler waves. Whistler waves are typically generated by a perpendicular temperature anisotropy ($T_\parallel/T_\perp < 1$). However, capsule distributions with $T_\parallel/T_\perp > 1$ have also been shown to be unstable to whistler waves. In this case, the butterfly like distributions, formed by the two anisotropic beam components in the parallel and antiparallel directions, lead to an increase in phase space density near 45° and 135° pitch angles. These butterfly like components supply the free energy necessary for whistler wave generation through cyclotron resonant interactions. Numerical calculations indicate that the generated whistler waves are near 0.1–0.2 $\Omega_{ce}$, consistent with observations in magnetopause reconnection (Ren et al., 2021) and lion roar emissions (Baumjohann et al., 1999; Giagkiozis et al., 2018).

## Data Availability Statement

We appreciate the MMS team and the MMS Science Data Center for providing the MMS data (https://lasp.colorado.edu/mms/sdc/public/datasets/). Data processing was done using SPEDAS available at https://themis.ssl.berkeley.edu/socware/bleeding_edge/.

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## Acknowledgments

The work at NSSC was supported by NNSFC grants (42425404, 42188101, 42204172, 42174207), the Specialized Research Fund for State Key Laboratories of China, and the Strategic Pioneer Program on Space Science II, Chinese Academy of Sciences, grants XDA15350201, XDA15052500.
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