Earth’s radiation belt is a torus shaped region filled with energetic electrons and ions trapped by the geomagnetic field. The radiation belts occupy the space between ∼ 1–9 R E , where R E is the Earth radius (Bloch et al., 2021), and is comprised of an inner belt ( ∼ 1–2 R E ), an outer belt ( > 3 R E ), and a slot region in between. The shape and structure of the radiation belt depend largely on external drivers and particle energy (e.g., Ripoll et al. (2016); Mei et al. (2021)). The complex dynamics of this region is maintained by a competing balance between different acceleration, transport and loss processes (Reeves et al., 2003; Baker et al., 2004; Summers et al., 2004; Hudson et al., 2008; Li et al., 2019; Ripoll et al., 2020; Lejosne et al., 2022). The acceleration processes include local acceleration by whistler-mode chorus waves (Horne and Thorne, 1998; Summers et al., 1998; Horne et al., 2005; Thorne et al., 2010; Millan and Baker, 2012; Ukhorskiy and Sitnov, 2013; Artemyev et al., 2016; Allanson et al., 2021; Gao et al., 2022a; Gao et al., 2022b; Chakraborty et al., 2022b) and radial diffusion by ultra low frequency waves (Fälthammar, 1965; Elkington et al., 1999; Hudson et al., 2000; Elkington et al., 2003; Ozeke et al., 2012; Mann et al., 2013; Ozeke et al., 2014a; Ozeke et al., 2014b; Jaynes et al., 2018; Zhao et al., 2018; Ozeke et al., 2020). The loss of electrons from the radiation belts happens either through drift shell splitting and magnetopause shadowing (e.g., Tu et al. (2014), Turner et al. (2012)), or through wave-particle interactions with various plasma waves, such as the plasmaspheric hiss (e.g., Zhao et al. (2019)), chorus (e.g., Shprits et al. (2016), Chakraborty et al. (2022b), Drozdov et al. (2022)), electromagnetic ion cyclotron (EMIC) waves (e.g., Ross et al. (2021)), and very low frequency (VLF) transmitter waves (e.g., Hua et al. (2020)). During periods of strong geomagnetic activity, the trapped electron fluxes can increase by several orders of magnitude over a brief period that can damage instruments on board spacecraft orbiting in this region of space, and in some extreme cases, can even lead to complete failure. As our modern society has become increasingly reliant on space-based technologies, understanding the complex dynamics of the radiation belt is of utmost importance to mitigate space weather hazards.

One effective way to indirectly determine what physical processes are occurring in the radiation belts is to examine the pitch angle distributions (PADs) of the trapped electrons and ions. This is because different physical processes in the radiation belt can generate different types of PADs. Apart from radiation belts, electron PADs have also been studied in the outer magnetosphere to provide useful information about the underlying physical processes (e.g., Li et al. (2020); Liu et al. (2020). In the outer radiation belt, the three most prevalent types of electron PADs are pancake, butterfly, and flattop. Pancake distributions have a peak flux at 90 ° pitch angle with smooth decrease towards lower pitch angles (field aligned directions). Inward radial diffusion (causing betatron acceleration) and wave-particle interactions (causing loss of electrons at lower pitch angles) are thought to generate this type of PAD (e.g., Schulz and Lanzerotti (1974), Summers et al. (1998), Xiao et al. (2009a), Xiao et al. (2009b), Xiao et al. (2012); Xiao et al. (2014); Thorne et al. (2013b)). Pancake distributions are generated by drift resonance with ultra-low frequency (ULF) waves. Due to radial diffusion, the electrons transport radially inwards, generating pancake distributions through a process similar to betatron acceleration (e.g., Xiao et al. (2009b); Xiao et al. (2009a), Xiao et al. (2012); Xiao et al. (2014); Thorne et al. (2013a)). Cyclotron resonance with whistler mode chorus waves can also generate pancake distributions. Pitch angle scattering and consequent loss of electrons through the filling of the loss cone can result in narrow pancake distributions (e.g., Thorne et al. (2010), Chakraborty et al. (2022b)). Butterfly distributions have a flux minimum at 90 ° pitch angle with peak flux located at lower pitch angles, preferably around 45 ° or 65 ° . Drift shell splitting combined with magnetopause shadowing or wave-particle interactions heating off-equatorial electrons are believed to generate butterfly distributions (e.g., Sibeck et al. (1987), Selesnick and Blake (2002), Horne et al. (2005), Li et al. (2016), Ozeke et al. (2022)). Butterfly distributions are generated through wave-particle interactions with chorus waves, magnetosonic waves, and/or electromagnetic ion cyclotron (EMIC) waves (e.g., Xiao et al. (2014), Yu et al. (2016)). Resonance of electrons at lower pitch angles with these wave modes results in electron heating along field-aligned directions, thereby forming butterfly distributions. Flattop distributions have relatively similar flux values over a wide range of pitch angles around 90 ° . They are considered as an intermediate stage between the pancake and butterfly distributions, and wave-particle interactions are thought to be the primary driver (e.g., Horne and Thorne (2003), Zhao et al. (2017), Chakraborty et al. (2022a), Killey et al. (2023); Killey et al. (2024)). Flattop distributions are generated by pitch angle diffusion with chorus waves. Pitch angle diffusion results in flattening of the distributions, thereby, forming flattop PADs from initial pancake distributions (e.g., Horne and Thorne (2003), Zhao et al. (2017)).

Previous studies have tried to approximate observed PADs using several fitting functions. One of the most commonly used function is a sinusoidal function of the form s i n n ( α ) , where α is the electron pitch angle and n is a steepness parameter that provides an estimate of the pitch angle anisotropy (e.g., Vampola (1997), Gannon et al. (2007), Ni et al. (2015); Pandya et al. (2020), Greeley et al. (2021)). However, one major drawback of using a sinusoidal function is that it can not fit butterfly distributions. To overcome such limitations, some studies used a combination of two sinusoidal functions (e.g., Allison et al. (2018)). A more effective and widely applied method is using Legendre polynomials, where a combination of different orders of the Legendre coefficients are used to fit the observed PADs (e.g., Chen et al. (2014), Zhao et al. (2018), Zhao et al. (2020); Chakraborty et al. (2022a)). Recently, Smirnov et al. (2022a), Smirnov et al. (2022b) used a Fourier sine series expansion to fit the observed equatorial PADs, which was found to be effective in fitting all the different types of PADs prevalent in the outer radiation belt. Apart from these fitting methods, recent studies have also used machine learning techniques to identify different PADs in the outer radiation belt and study their storm time evolution (e.g. Killey et al. (2023), Killey et al. (2024)), to rectify some of the issues as mentioned above.

Previous studies have shown that electron PADs in the outer radiation belt are dependent on electron energy, L-shell, and MLT. For example, at tens of keV energies, the PADs are pancake at all L-shells and MLTs, while at higher energies, such as at hundreds of keV or several MeV, pancake PADs are observed on the dayside while butterfly PADs are observed on the nightside at larger L-shells (e.g., West et al. (1973), Gannon et al. (2007), Ni et al. (2015), Pandya et al. (2020), Chakraborty et al. (2022a), Killey et al. (2023); Killey et al. (2024)). The electron PADs also exhibit strong dependence on geomagnetic activity, with the anisotropy of the distributions increasing with enhanced activity level. Some studies have also reported the dependence of the evolution of electron PADs with different storm drivers (e.g., Pandya et al. (2020), Greeley et al. (2021), Chakraborty et al. (2022a)).

Although PADs in the Earth’s radiation belts have been extensively studied in the past, as mentioned before, most of them used fitting functions to examine the morphology of the electron PADs. One drawback of using fitting functions is that it requires examining the variation of a combination of multiple parameters. For example, while using Legendre polynomials, pancake PADs are categorized by large negative c2 (second order Legendre coefficient) and near-zero c4 (fourth order Legendre coefficient), butterfly PADs are categorized by large negative c4 and nearly negligible c2, and flattops are categorized by both negative c2 nd c4 (e.g., Zhao et al. (2018); Zhao et al. (2020); Zhao et al. (2021)). Similar is the case when using Fourier sine series expansion as used by Smirnov et al. (2022a), Smirnov et al. (2022b). This makes the interpretation of the results slightly complicated. To avoid such complications, in this study, instead of using the fitting methods described, the primary motivation was to use a simplified formula (Equation 1) to estimate an electron pitch angle anisotropy index purely from pitch angle resolved electron flux measurements, and then use that index to categorize the different electron PADs in the outer radiation belt to study their variation with geomagnetic activity. Towards that goal, we used pitch angle resolved electron flux measurements from Van Allen Probe-B spacecraft over its entire lifespan (2012–2019) to provide an extensive statistical survey of near-equatorial PADs of relativistic electrons, having energy ≥ 0.5 MeV, in the outer radiation belt (L ≥ 3).

In this study, we used ∼ 11 s resolution of pitch angle resolved (Level 3) electron flux measurements from both the Magnetic Electron Ion Spectrometer (MagEIS) and Relativistic Electron Proton Telescope (REPT) instruments, which are parts of the Energetic Particle, Composition and Thermal Plasma (ECT) Suite (Baker et al., 2013; Blake et al., 2013; Spence et al., 2013) onboard the Van Allen Probe-B spacecraft, during the entire period of operation (September 2012 - July 2019). First, we calculated 5-min average of the measured electron fluxes for each energy channel, specifically, 0.5 MeV, 0.6 MeV, 0.7 MeV, 0.9 MeV, 1.1 MeV and 1.5 MeV from MagEIS, and 1.8 MeV, 2.1 MeV, 2.6 MeV, 3.4 MeV and 4.2 MeV from REPT, as a function of pitch angle and time. Next, we used a selection criterion of considering measurements only when MLAT was less than ± 10 ° to limit our observations close to the geomagnetic equator, and used a lower threshold to remove bad data points: for MagEIS measurements, the lower threshold was set at 10 2 / c m 2 /s/sr/MeV (Ni et al., 2020), and for REPT measurements, the lower threshold was set at 10 − 2 / c m 2 /s/sr/MeV (Baker et al., 2013). Finally, we normalized the fluxes by the maximum flux value within the entire pitch angle range so that for each measurement (corresponding to the spatial location: L-shell, MLT and MLAT, and time), the flux values vary between 0 and 1.

We used the near-equatorial normalized electron fluxes to calculate the pitch angle anisotropy index. For the rest of this article, we will refer to the pitch angle anisotropy index as PAI. Considering a symmetric pitch angle distribution around 90 ° pitch angle (PA), PAI is calculated using the formula: P A I = f l u x 90 ° a v g f l u x P A a : P A b , 180 ° − P A b : 180 ° − P A a (1) where flux 90 ° is the normalized flux at 90 ° PA, and avg ( flux [ PA a : P A b ]) is the average of normalized fluxes within the PA range 40.91 ° ( PA a ) to 73.64 ° ( PA b ) for MagEIS, and 47.65 ° ( PA a ) to 79.41 ° ( PA b ) for REPT.

We used this PAI to categorize the different PADs of relativistic electrons (E ≥ 0.5 MeV) in the outer radiation belt (L ≥ 3). In this study, we choose L ≥ 3 as a fixed inner boundary to make sure that the electron flux measurements throughout the study are within the outer radiation belt, instead of being in the slot region or contaminated by inner radiation belt protons. Pancake distributions are categorized by PAI values ≥ 1.05, butterfly distributions are categorized by PAI values ≤ 0.95, and flat top distributions are categorized by PAI values within the range of 0.95–1.05. Here we choose an upper limit of PAI for butterfly distributions at 0.95 in agreement with Ni et al. (2016) who used a different methodology but the same threshold to distinguish between butterfly and non-butterfly distributions. The choice of a narrow range of PAI values for flattop distributions agrees with Yu et al. (2016), who also used a different methodology but assigned a similar narrow range of pitch angle indices for flattop PADs.

Figure 1 shows the average shape of the three PADs and the temporal evolution of PAD and PAI of 1.8 MeV electrons from 0320 UT to 0620 UT on 17 March 2015. Here we have used 1.8 MeV as a representative energy, as the average shapes of the three PADs at other energies are identical. In Figure 1 panels a, b, and c, the local pitch angle (in degrees) is along the x-axis and the normalized flux is along the y-axis. The filled circles are the median flux values, and the error bars denote the interquartile ranges (IQRs) at the measured pitch angles. In panel 1d, time in UT is along the x-axis, pitch angle in degrees is along the y-axis, and the colorbar at the right denotes the electron flux. The black dashed horizontal line shows 90 ° pitch angle. In panel 1e, time in UT is along the x-axis, and the PAI is along the y-axis. The two dotted horizontal lines denote the thresholds used to identify the different PAD types. Several important features can be noted from Figure 1: (1) for pancake PAD (panel 1a), the flux at 90 ° is always maximum, as evident from the disappearing IQR. (2) For butterfly PAD (panel 1b), the flux is less at 90 ° and peaks at ∼ 45 ° / 135 ° . However, the large IQRs at these PA values indicate that the flux is not always maximum at ∼ 45 ° / 135 ° . This supports the presence of the two types of butterfly PADs peaking at two different PAs as reported by Ozeke et al. (2022) and Killey et al. (2024). (3) For flattop PAD (panel 1c), the flux values remain almost similar over a broad range of PAs ( ∼ 60 ° - 120 ° ) as evident from the small IQRs within this PA range. Panels 1d and 1e depict the temporal evolution of the electron PADs and how the PAI is used to classify the electron PADs into three different types. From 0320 UT to 0400 UT, PAI values are greater than 1.05, thereby the electron PADs being classified as pancake distributions. Between 0400 UT and 0410 UT, PAI values are within the range of 0.95 and 1.05, leading to the electron PADs being classified as flattop distributions. After 0410 UT, PAI values are less than 0.95, the electron PADs are therefore classified as butterfly distributions.

The large IQRs in all three PADs indicate high variability in the electron flux, as well as in the PAI. This motivated us to study any existing correlation between the PAI, and solar wind drivers and geomagnetic indices. We used the 5-min resolution OMNI data of the z-component of the interplanetary magnetic field, solar wind dynamic pressure, SYM-H, and AL indices for this purpose (Papitashvili and King, 2020). Further, to study the variation of electron PAI with the solar wind parameters and geomagnetic indices, the OMNI data is interpolated to match the timestamp of the Van Allen Probe electron flux measurements. This processed combined dataset of the electron PAI, solar wind parameters, and geomagnetic indices is then used to examine the variation of electron PAI with geomagnetic activity.

In this Section, we present statistical results of the spatial (L, MLT), energy, and geomagnetic activity dependence of the different electron PADs, and their associated PAI values.

In this study, we used 7 years (2012–2019) of Van Allen Probe-B pitch angle resolved electron flux measurements to examine the spatial distribution and energy dependence of different types of relativistic ( ≥ 0.5 MeV) near-equatorial (MLAT ≤ ± 10 ° ) electron PADs in the outer radiation belt (L ≥ 3), and to investigate the variation of electron pitch angle ansiotropy with different levels of geomagnetic activity.

As the first step, we applied Equation 1 to calculate a pitch angle anisotropy index (PAI) which we used to categorize the electron PADs into three types, namely, pancake: PAI ≥ 1.05, butterfly: PAI ≤ 0.95, and flattop: 0.95 < PAI < 1.05. To obtain the spatial distribution of the electron PADs, we calculated MLT-averaged normalized occurrence of the PADs in 6 L-shell bins (3–6) and 11 energy bins (0.5–4.2 MeV). To investigate the variation of pitch angle anisotropy with different levels of geomagnetic activity, we used the 5 min OMNI database of solar wind parameters and geomagnetic indices to monitor P dyn , SYM-H, and AL, and split them into periods of low and high activity.

The major findings from this study can be summarized as follows:

1. In the dawn and dusk sectors, the L range of 5 - 6 is dominated by butterfly and pancake PADs at higher energies ( ≥ 1.5 MeV in the dusk sector, and ≥ 2.6 MeV in the dawn sector), and flattop and pancake PADs at the corresponding lower energies. The L range of 4–5 is dominated by pancake PADs at higher energies ( ≥ 1.5 MeV), and flattop PADs at lower energies. The L range of 3 - 4 is dominated by mostly pancake PADs.

The statistical results related to the spatial distribution of the different electron PADs (findings 1, 2, and 3) are consistent with previous findings, both using fitting methods or machine learning techniques (e.g., Chen et al. (2014), Zhao et al. (2018), Pandya et al. (2020), Greeley et al. (2021), Zhao et al. (2021), Chakraborty et al. (2022a); Killey et al. (2023); Killey et al. (2024), etc.). However, in most of the past studies, butterfly PADs are usually reported to be present on the nightside in outer L-shell ranges, ≥ 5 (e.g., Ni et al. (2016); Ozeke et al. (2022), etc.). Here, in addition, we found a second population of low-L (3 < L < 4) butterfly PADs across all MLTs and low energies ( < 1 MeV), with a lower occurrence rate compared to the outer-L butterflies (finding 4). This is consistent with the very recent findings of Killey et al. (2024). One possible mechanism might be wave-particle interactions heating off-equatorial electrons (e.g., Horne and Thorne (2003)), as magnetopause shadowing is unlikely to create butterfly PADs at such low L-shells. A detailed analysis is required to understand the generation mechanism of these low-L butterflies, which we plan to do in a future study.

The results related to the variation of outer radiation belt pitch angle anisotropy with solar wind forcing (finding 5) are consistent with Killey et al. (2024) who showed the storm time evolution of the relativistic electron PADs in the outer radiation belt using machine learning techniques (Figure 3 of Killey et al. (2024)). However, in Killey et al. (2024), the storm phases are defined based on the SYM-H index. In our study, we found that P dyn is the dominant factor in driving the radiation belt pitch angle anisotropy, compared to SYM-H and AL indices (finding 6). This agrees with Smirnov et al. (2022a) who studied the storm-time evolution of electron PADs during 129 geomagnetic storms in the Van Allen Probe era and found P dyn to be more effective than SYM-H and solar wind electric field in driving the observed enhancement in electron pitch angle anisotropy. Based on these results, in a companion paper, Smirnov et al. (2022b) developed an empirical model of the equatorial electron pitch angle distribution driven by P dyn . In another earlier paper, Yu et al. (2016) used 3 years of Van Allen Probe measurements (2012–2015) to study the effect of P dyn and IMF Bz on the outer radiation belt electrons. They found that during periods of enhanced P dyn , pancake PADs on the dayside become more 90 ° peaked, while the nightside butterfly PADs extend azimuthally and also radially inwards. Our results thus confirm such findings by providing evidence of the correlation between the calculated electron pitch angle anisotropy and the driving solar wind parameters or geomagnetic indices.

Physically, this implies that when a pressure impulse hits the Earth’s magnetosphere, it globally compresses the entire system. As a result of this global compression, some of the electrons are lost through magnetopause shadowing, while others get pushed radially inwards. As the electrons transport radially inwards, they move from a region of weaker to a stronger magnetic field. In the course of this motion, to preserve the first and second adiabatic invariants, they gain energy in the perpendicular direction more than that in the parallel direction through a process similar to betatron acceleration, leading to an enhanced anisotropy. This causes the pancake distributions to become more narrow. On the other hand, the 90 ° electrons that are lost through drift shell splitting and magnetopause shadowing into the interplanetary space lead to more deepening of the butterfly distributions. In addition, wave-particle interactions also play a significant role in the evolution of radiation belt electron pitch angle distributions. Magnetospheric plasma waves are generated either from particle injection leading to the enhancement of temperature anisotropy that provides free energy for the generation of waves, or from the global reconfiguration of the magnetosphere generating large-scale MHD waves. Resonant interaction with these waves violates the conservation of either one of the three adiabatic invariants of the resonating electrons, thereby leading to either loss or non-adiabatic heating, and thus generating different PAD types. This is depicted schematically in Figure 7.

To summarize, our results show that a simplified formula (Equation 1) could capture the overall spatial and energy dependence of the outer radiation belt relativistic electron PADs. The results also confirm that P dyn is the dominant parameter in governing the outer radiation belt pitch angle anisotropy, and thus can be used as a driver in radiation belt models, as used by Smirnov et al. (2022b). In future, we plan to extend our study to resolve the two butterfly distributions peaking at two different pitch angles, as reported by Ozeke et al. (2022) and Killey et al. (2024), and also to understand the generation mechanism of the low-L butterfly PADs.