Based on over 4 years of Van Allen Probes measurements, an empirical model of radiation belt electron equatorial pitch angle distribution (PAD) is constructed. The model, developed by fitting electron PADs with Legendre polynomials, provides the statistical PADs as a function of L-shell (L = 1–6), magnetic local time, electron energy (~30 keV to 5.2 MeV), and geomagnetic activity (represented by the Dst index) and is also the first empirical PAD model in the inner belt and slot region. For megaelectron volt electrons, model results show more significant day-night PAD asymmetry of electrons with higher energies and during disturbed times, which is caused by geomagnetic field configuration and flux radial gradient changes. Steeper PADs with higher fluxes around 90° pitch angle and lower fluxes at lower pitch angles for higher-energy electrons and during active times are also present, which could be due to electromagnetic ion cyclotron wave scattering. For hundreds of kiloelectron volt electrons, cap PADs are generally present in the slot region during quiet times and their energy-dependent features are consistent with hiss wave scattering, while during active times, cap PADs are less significant especially at outer part of slot region, which could be due to the complex energizing and transport processes. The 90°-minimum PADs are persistently present in the inner belt and appear in the slot region during active times, and minima at 90° pitch angle are more significant for electrons with higher energies, which could be a critical evidence in identifying the underlying physical processes responsible for the formation of 90°-minimum PADs.

Key Points

Megaelectron volt electrons have steeper pitch angle distributions (PAD) during active times and for higher-energy electrons, suggesting EMIC wave scattering
Cap PADs are generally present in the slot region during quiet times with energy-dependent features consistent with hiss wave scattering
The 90°-minimum PADs are persistently present in the inner belt, with more significant minima at 90° PA for electrons with higher energies

1 Introduction

The energetic electron pitch angle distribution (PAD) is an important characteristic of radiation belt electrons, as it provides valuable information on the source and loss processes in a specific region acting on the radiation belt electrons with specific energies. Typical electron equatorial PADs in the radiation belts include pancake PADs, butterfly PADs, and flattop PADs, as shown in Figures 1a–1c, respectively. It is worth noting that all PADs in Figure 1 are equatorial PADs, measured by Magnetic Electron Ion Spectrometer (MagEIS) instruments on the Van Allen Probes and propagated to the magnetic equator using T89D geomagnetic field model (Tsyganenko, 1989). The pancake PAD (Figure 1a) is the PAD that electron flux peaks at 90° pitch angle (PA) and smoothly decreases toward the field-aligned directions. It is one of the most prevalent PAD types in the outer radiation belt and also dominates on the dayside (e.g., Gannon et al., 2007; West et al., 1973). It is thought to form as a result of the PA diffusion caused by electromagnetic waves and subsequent loss of low PA electrons to the atmosphere. Inward radial diffusion can also cause the flux peak around 90° PA due to the conservation of the first two adiabatic invariants: when an electron transports inward, the perpendicular momentum increases more than the parallel component; thus, the equatorial PA of electron increases and a more 90°-peaked PAD forms (e.g., Schulz & Lanzerotti, 1974). The butterfly PAD (Figure 1b) has a minimum flux around 90° PA and a maximum flux at lower PAs. It is usually observed in the outer radiation belt at nightside, especially at high L-shells, and is thought to be caused by the drift-shell-splitting effect combined with magnetopause shadowing or strong negative radial flux gradient (e.g., Selesnick & Blake, 2002; Sibeck et al., 1987). Horne et al. (2005) have also suggested that chorus waves outside the plasmasphere could preferentially heat off-equator electrons and create butterfly PADs there. For the flattop PAD (Figure 1c) the electron flux remains almost the same for a relatively wide PA range. Gannon et al. (2007) showed that the flattop PADs exist in the outer belt at high L region as well as inside of the plasmasphere. It can be a transition between the pancake PAD and butterfly PAD or can be a result of strong wave-particle interactions (Horne et al., 2003).

Details are in the caption following the image — **Figure 1**
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Examples of (a) pancake, (b) butterfly, (c) flattop, (d) cigar, (e) cap, and (f) 90°-minimum pitch angle distributions of energetic electrons, measured by Magnetic Electron Ion Spectrometer instruments on the Van Allen Probes and propagated to the magnetic equator using T89D geomagnetic field model. Gray dots are propagated measurements over 1 min, black crosses show the data averaged in PA bins of 10° based on the propagated measurements, and red lines are the fits using Legendre polynomials. The coefficients of Legendre polynomials (c₁-c₁₀), root-mean-square deviation (RMSD), L-shell, and magnetic local time (MLT) are shown on the top of each panel. PA = pitch angle.

Besides these three main types of PADs, other PAD types, for example, cigar, cap, and 90°-minimum PADs, as shown in Figures 1d–1f, respectively, were also found for energetic electrons in specific regions as results of different physical processes. For tens to hundreds of kiloelectron volt electrons, cigar PADs (Figure 1d) with flux peaking along the local magnetic field direction were observed in the outer radiation belt at high L region and are thought to be related to the tail-like stretching of the nightside magnetic field prior to the substorms (Baker et al., 1978). The cap PAD (Figure 1e) has a bump around 90° PA on top of a pancake PAD (e.g., Lyons & Williams, 1975a; Sibeck et al., 1987). It is found to be present in the outer radiation belt as well as the slot region and mainly for tens to hundreds of kiloelectron volt electrons. As for the slot region, Lyons and Williams (1975a) showed comparisons between observations and modeling and the agreement between the two suggests that the cap PADs in the slot region form as a result of PA scattering caused by the plasmaspheric whistler mode waves. Similarly, Zhao et al. (2014b), through detailed event studies, showed that the cap PADs form as the electron fluxes decay after storm time injections and are likely caused by the hiss wave scattering. As for the outer radiation belt, Sibeck et al. (1987) investigated the cap PADs and suggested that those PADs in the outer radiation belt can be caused by a combination of the drift-shell-splitting effect and a substorm injection or a sudden magnetospheric compression. The 90°-minimum PAD (Figure 1f), as first reported by Zhao et al. (2014a) using data from MagEIS instruments, shows as a pancake PAD with a small bite out centered around 90° PA. It is generally present in the inner radiation belt and appears in the slot region during storm times. It is also only observed for tens to hundreds of kiloelectron volt electrons. The 90°-minimum PAD is distinct from the butterfly PAD by location—the butterfly PADs, mostly caused by the drift-shell-splitting effect, are generally present in the outer radiation belt, while the 90°-minimum PADs exist in the inner belt and slot region where the drift-shell-splitting effect is minimal due to dipole-like geomagnetic field—and by PA of the maximum flux—the maximum flux of 90°-minimum PAD is generally near or above 70° PA, while butterfly PADs usually have a maximum flux around 40°–45° PA. Though some physical mechanisms have been proposed to explain the formation of 90°-minimum PADs (e.g., Albert et al., 2016; L. Chen et al., 2015; J. Li et al., 2016; Zhao et al., 2014b), the cause of this type of PADs is still under considerable debate.

Many previous studies have focused on the characteristics and evolution of electron PADs in the outer radiation belt (e.g., Gannon et al., 2007; Ni et al., 2016; West et al., 1973). West et al. (1973), using ~80-keV to 2.8-MeV electron flux data from Ogo 5, performed a survey of the electron equatorial PADs at different local times with radial distances from ~3 to ~20 R_E on the nightside and to the magnetopause on the dayside. Their results suggest the normal PADs prevail in the inner magnetosphere and dayside of outer magnetosphere, and butterfly PADs dominate in the afternoon to midnight magnetosphere at extended distances, while inside ~9 R_E the PADs are quite energy dependent. Gannon et al. (2007), using ~150-keV to 1-MeV electron data from Combined Release and Radiation Effects Satellite (CRRES), conducted a statistical analysis on the radiation belt electron PADs. By categorizing electron PADs into 90°-peaked, flattop, and butterfly PADs, they showed that PADs at lower L values and on the dayside are mostly 90°-peaked PADs, and PADs tend to be more 90° peaked at lower L and for lower-energy electrons, while butterfly PADs are most commonly seen at higher L and exist over a wider L range during active times. Ni et al. (2016), using data from Relativistic Electron Proton Telescope (REPT) on the Van Allen Probes, investigate the occurrence pattern of butterfly PADs in the outer radiation belt. They found that the occurrence rate of butterfly PADs is highest (~80%) at ~20–04 magnetic local time (MLT) at L > ~5.5, while ~50% occurrence rate of butterfly PADs is also observed at 11–15 MLT at L ~ 4, and the outer belt butterfly PADs do not show significant correlation with the solar wind dynamic pressure. Some studies have also focused on electron PADs in the slot region and inner belt (e.g., Lyons & Williams, 1975a, 1975b; Shi et al., 2016; Zhao et al., 2014a, 2014b). Lyons and Williams (1975a, 1975b), using 35- to 560-keV electron data from Explorer 45, studied both quiet time and storm time structures of radiation belt electron PADs at 1.7 < L < 5.2. Their results show that, during geomagnetic quiet times, in the slot region and outer part of plasmasphere the PADs are consistent with predictions from plasmaspheric hiss wave scattering; during storm and prestorm times, electron PADs are greatly distorted due to injections, while after the storm the PADs return to their quiet time structures over a period of several days. Zhao et al. (2014a), using data from MagEIS instruments on the Van Allen Probes, reported a peculiar type of PADs in the inner belt and slot region, which has a minimum at 90° PA and cannot be well explained by any existing theory, suggesting the complicated dynamics of inner belt and slot region that is not well understood. Zhao et al. (2014b) also using data from MagEIS instruments, studied the statistics of radiation belt electron PADs at L < 4 and showed that most of PADs in this region can be categorized into normal, cap, or 90°-minimum PADs. They also showed that for ~460-keV electrons, 90°-minimum PADs are generally present in the inner belt and appear in the slot region during quiet times, normal PADs dominate at L ~ 3.5–4, and cap PADs generally appear in the slot region at the decay phase of storms and are likely caused by plasmaspheric hiss wave scattering. Shi et al. (2016), using data from MagEIS instruments, performed a statistical survey of radiation belt electron PADs and their correlation with electron energy, MLT, geomagnetic Kp index, and L-shell. They found that the electron PADs are more exceedingly peaked at 90° PA in the low L region, during active times, on the dayside, and for higher-energy electrons.

The PADs reflect the effects of physical processes on energetic electrons and the competition between different processes in a specific region. Understanding the distribution of energetic electron PADs can contribute to better identifying and understanding of those processes in the radiation belts. On the other hand, a number of instruments only make single-direction radiation belt electron flux measurements, and also due to the orbit, low Earth orbit satellite or medium Earth orbit satellite cannot measure electron fluxes near geomagnetic equator in the outer radiation belt. Thus, an empirical model of PADs of radiation belt electrons would be very useful as the electron fluxes at all locations can be derived based on single-direction measurements or low Earth orbit/medium Earth orbit satellite observations. Specifically, several studies have focused on the modeling of radiation belt electron PADs. Garcia (1996), using Geostationary Operational Environmental Satellite X-ray observations, deduced electron PADs and created a model of electron PADs with the form of sinⁿ(α) for the pancake PADs and sin^m(2α) for the butterfly PADs, where n and m are the anisotropy indices. With Geostationary Operational Environmental Satellite data, this model only focuses on the electron PADs at geosynchronous orbit. Vampola (1998), using CRRES observations, conducted a statistical survey of electron PADs and suggested that the electron PAD can be represented by sinⁿ(α) with n = (log(I_90/45) + 0.004105)/0.14303, where α is the electron PA and I_90/45 is the ratio of measured electron flux at 90° PA to that at 45° PA. However, this PAD model is not able to represent specific types of PADs, for example, the butterfly PADs. Y. Chen et al. (2014), using data from CRRES, Polar, and LANL-97A satellites and method of Legendre polynomial fitting, constructed an empirical model of relativistic electron PADs in the outer radiation belt as a function of energy (~150 keV to 1.5 MeV), L-shell (3–9), MLT, and geomagnetic activity represented by AE index. Though previous studies have focused on electron PAD models in the outer radiation belt, no empirical model has been constructed in the inner belt and slot region. Instruments in the inner belt are subject to influences from very energetic inner belt protons. Prior to the Van Allen Probes era, limited electron PAD information can be derived in the inner belt due to proton contamination. The MagEIS instruments on the Van Allen Probes give unprecedented measurements of inner belt electrons by removing inner belt proton contamination background and thus give us excellent opportunity to study the electron PADs in the inner belt.

In this study, using PA-resolved electron flux data for over 4 years from the Van Allen Probes, we show statistical representations of electron equatorial PADs as a function of L-shell, MLT, geomagnetic activity (represented by the Dst index), and electron energy. Following the method used by Y. Chen et al. (2014), we use Legendre polynomials to fit directional fluxes propagated to the magnetic equator, calculate the medians, means, and standard deviations of the coefficients of Legendre polynomials, and construct our empirical model using those results. This empirical radiation belt electron PAD model, comparing to previous models, covers wider energy range (~30 keV to 5.2 MeV) and L-shell range (L = 1–6) and, for the first time, shows statistical energetic electron PADs in the inner radiation belt and slot region. Based on the model results, underlying physical mechanisms affecting electrons with different energies in specific regions are examined. It is worth mentioning that all PADs shown in this paper and included in our model are equatorial PADs (propagated from the measured location to the magnetic equator using T89D model). In sections 2 and 3, we introduce the data and methodology used in constructing our model, respectively. In section 4, some intriguing model results are shown and underlying physical processes are discussed. Discussions and comparisons to previous models are shown in section 5, and the summary is in section 6.

2 Data

In this study, PA-resolved electron flux data from the MagEIS (Blake et al., 2013) and the REPT (Baker et al., 2012) instruments of Energetic Particle, Composition, and Thermal Plasma suite (Spence et al., 2013) on the Van Allen Probes from September 2012 to December 2016 are used. The Van Allen Probes, launched on 30 August 2012, operate in an elliptical orbit with an inclination of 10° and altitude of ~600 km × 5.8 RE (Kessel et al., 2013). With the spin axis approximately pointing to the Sun, the spacecraft is spinning with a period of ~12 s, which provides great PA coverage during most times and thus provides an ideal data set for PAD studies.

MagEIS instruments on the Van Allen Probes provide energetic electron flux measurements with energy range of ~30–4,000 keV. It contains four independent magnetic electron spectrometers on each spacecraft: one low-energy spectrometer (LOW), two medium-energy spectrometers (M75 and M35), and one high-energy spectrometer (HIGH). The LOW unit, HIGH unit, and one of the medium units (M75) are mounted with the field of view centered at 75° to the spin axis, while the field of view of the other medium unit (M35) is centered at 35° to provide more PA coverage. In this study, we use the PA-unbinned level 3 flux data from LOW and M75 units of MagEIS (with energy of ~30 keV to 1 MeV). Background-corrected data are available for MagEIS LOW and M75 units mostly, for which the contaminations from inner belt protons and bremsstrahlung radiation, have been removed (Claudepierre et al., 2015) and thus are preferentially used in our study, and the uncorrected data are only used for the time period when the background-corrected data are not available and only used at L > 3.

REPT instruments provide high-quality measurements of relativistic electrons with energy from 1.8 to ~20 MeV. However, since the counts for >6-MeV electrons are too low to show clear PAD patterns during Van Allen Probes era, only data for electrons with energy from 1.8 to 5.2 MeV are used. We also used the PA-unbinned level 3 electron data of REPT instruments in this study.

Figure 2 shows the daily-averaged fluxes of radiation belt electrons of different energies using MagEIS and REPT level 2 data from the Van Allen Probes-A from 1 September 2012 to 1 January 2017, the time period during which the electron PADs are used to construct our empirical model, along with the daily-averaged Dst index. Note that in Figure 2, top three panels show a combination of background-corrected and uncorrected data from MagEIS instruments: at L < 3, only the background-corrected data are used; at higher L, background-corrected data are preferentially used and uncorrected data are only used when corrected data are not available. It is clear from Figure 2 that energetic electron fluxes often exhibit great variations, and the variation of electron fluxes strongly depends on the geomagnetic activities (e.g., Baker et al., 1986, 1990, 1999; Blake et al., 1997; Borovsky & Denton, 2009; Boynton et al., 2013; X. Li et al., 2001, 2005, 2011; X. Li, 2004; Paulikas & Blake, 1979; Reeves, 1998; Reeves et al., 2011; Simms et al., 2016; Zhao, Baker, Jaynes, et al., 2017). Also, the behavior of energetic electrons is energy dependent. Generally, lower-energy electrons exhibit greater and faster variations than higher-energy electrons. Tens to hundreds of kiloelectron volt electrons penetrate into the slot region and inner belt quite often and abundant tens to hundreds of kiloelectron volt electrons exist in the inner belt (e.g., Claudepierre et al., 2017; Fennell et al., 2015; Zhao & Li, 2013b; Zhao et al., 2016; Zhao, Baker, Califf, et al., 2017). Megaelectron volt electrons can hardly penetrate into the low L region: during Van Allen Probes era, no deep penetration of electrons with energies of several megaelectron volts into L < 2.5 occurred, while previously the deep penetration of those electrons only occurred under very intense geomagnetic conditions (e.g., Baker et al., 2004, 2014; Blake et al., 1992; X. Li et al., 1993, 2017; Zhao & Li, 2013a). The differences between quiet and storm times and electrons with different energies indicate the existence of different physical processes in the radiation belts during different times working on electrons of different energies. Note that during the Van Allen Probes era the geomagnetic activity is relatively quiet and limited intense geomagnetic storms occurred; thus, the statistics of our model during disturbed times is not as good as that during quiet and moderately disturbed times.

3 Methodology

To construct a statistical model for energetic electron PADs, the PADs need to be quantified. One way to quantify PADs is to fit PADs to Legendre polynomials (also see Y. Chen et al., 2014). Using Legendre polynomials, the electron PAD can be expressed as

$urn:x-wiley:21699380:media:jgra54233:jgra54233-math-0001$

where j(α) is the flux of electrons as a function of PA α, P_n[cos(α)] is the nth-order Legendre polynomial, and C_n is the corresponding coefficient. The coefficient of each Legendre polynomial can be calculated using the orthogonal property:

$urn:x-wiley:21699380:media:jgra54233:jgra54233-math-0002$

The coefficients derived using this equation are then normalized as

$urn:x-wiley:21699380:media:jgra54233:jgra54233-math-0003$

Here $urn:x-wiley:21699380:media:jgra54233:jgra54233-math-0004$ is the directionally averaged flux. The normalization process aims to derive the PAD shape without taking the actual fluxes into consideration. Note that the main focus of this study is on the shape of electron PADs; thus, we did not take electron flux levels into consideration. However, the intensity of fluxes would also affect the shape of PAD and specific types of PADs can form as results of flux variations of electron of different PAs. For example, a more 90°-peaked PAD can form as a result of flux enhancements of nearly equatorially trapped electrons and can also be caused by flux decreases of electrons with low PAs.

It is worth to note that we chose to use Legendre polynomials in constructing our model because (1) theoretically every PAD can be fully represented by Legendre polynomials, (2) Y. Chen et al. (2014) study has validated that in the outer radiation belt most of electron PADs can be well represented by Legendre polynomials up to sixth order, and (3) by fitting to Legendre polynomials, derived coefficients are meaningful in identifying the PAD types.

The quality of fitting results can be examined by the root-mean-square deviation (RMSD) between data and fitting results, which can be calculated as

$urn:x-wiley:21699380:media:jgra54233:jgra54233-math-0005$

where j is the measured flux, $urn:x-wiley:21699380:media:jgra54233:jgra54233-math-0006$ is the fitting result, and m is the number of data points in the fitting.

Ideally, any PAD can be fully represented by a complete set of Legendre polynomials. However, in a statistical model we can only keep finite number of coefficients and the number of coefficients should be kept as small as possible. Based on the study of Y. Chen et al. (2014), c_n decreases quickly with increasing n, and including the Legendre polynomials up to order of 6 is enough to reproduce most observed PADs in the outer radiation belt. Thus, we fit measured outer belt electron PADs of greater than megaelectron volt electrons at L > 3 and PADs of tens to hundreds of kiloelectron volt electrons at L > 4 to a summation of zeroth- to sixth-order Legendre polynomials. The reason that we fit PADs of different energy electrons to zeroth- to sixth-order Legendre polynomials at different L-shell region is the dependence of slot region location on the electron energy: for electrons with higher energies, the location of slot region is closer to the Earth (as Figure 2 shows). Example fits of pancake, butterfly, and flattop PADs are shown in Figures 1a–1c, respectively, along with the coefficients of Legendre polynomials and RMSD of each fit. The RMSD overall is very small, showing good fitting results. As we mentioned before, the coefficients of Legendre polynomials are useful in identifying the PAD types. As shown in Figure 1, pancake PADs usually have large negative c₂ with near zero c₄, and butterfly PADs have large negative c₄ with negligible c₂, while flattop PADs are combinations of these two types, for which c₂ and c₄ are both negative.

However, the situation in the inner radiation belt and slot region is very different from that in the outer radiation belt. The electron PADs in the inner belt and slot region usually have some detailed features, for example, 90°-minimum PAD and cap PAD as shown in Figures 1e and 1f, which cannot be well represented by Legendre polynomials up to sixth order. Thus, for PADs in the low L region we use Legendre polynomials with higher orders to fit. Figure 3 shows the comparisons of fits of radiation belt electron PADs to Legendre polynomials of different orders in the inner belt and slot region. Black crosses show the 1-min averaged PADs (in 10° PA bins) using data from MagEIS instruments, blue lines show the fitting results using Legendre polynomials up to sixth order, and red lines show the results using Legendre polynomials up to tenth order for the inner radiation belt (left panel) and eighth order for the slot region (right panel). In the inner belt, as Figure 3 left panel shows, Legendre polynomials up to order 6 cannot represent the small dip around 90° PA shown in the data, while with higher-order Legendre polynomials the 90°-minimum PAD can be well represented and the RMSD between the data and fitting results can been significantly reduced. Similarly, in the slot region (Figure 3 right panel), Legendre polynomials up to eighth order can better represent the measured PAD than those up to only sixth order, especially at lower PAs. Thus, in our model, we fit PADs of tens to hundreds of kiloelectron volt electrons at L < 2 using Legendre polynomials up to tenth order and fit PADs at 2 ≤ L < 4 using Legendre polynomials up to eighth order. Very limited megaelectron volt electrons existed in the low L region during Van Allen Probes era (e.g., Claudepierre et al., 2017; Fennell et al., 2015; X. Li et al., 2015, 2017); thus, greater than megaelectron volt electron PADs at L < 3 are not included in our model due to poor statistics.

To derive the equatorial PADs with a wide PA coverage, in this study, we only use data when the Van Allen Probes were close to the magnetic equator with the absolute value of magnetic latitude less than 10° and propagate the measured local PADs to the magnetic equator using T89D magnetic field model. The electron PADs in the radiation belts are expected to be symmetric with respect to 90° PA, so to get better statistics, we average PADs in time bins of 1 min and in PA bins of 10° while forcing them to be symmetric with respect to 90° PA. Since only even order Legendre polynomials are symmetric, for these 1-min averaged PADs we expect c_nwith odd n to be 0, which can be validated from Figure 1. Also, since the electron fluxes inside the loss cone are expected to be very small, we ignore those data inside the drift loss cone (DLC), while sizes of DLC at different L-shells are calculated based on International Geomagnetic Reference Field (IGRF) model. Note that the sizes of DLC calculated using IGRF model only and using T89D model are very similar; thus, for simplicity we use DLC calculated using IGRF model in this study. The derived equatorial PADs are then fit to Legendre polynomials. However, those PADs with no data points within high PA range [80°, 100°] or low PA range [DLC, DLC + 20°]/[160° − DLC, 180° − DLC] are excluded from the statistics to make sure the fitting results capture real PAD shapes. Also only PADs with highest directional flux greater than 100 keV/s/sr/cm² for MagEIS and 0.1 keV/s/sr/cm² for REPT are used to ensure enough statistics. Finally, the accuracy of our model strongly depends on the accuracy of the fitting results, so we use the PADs that can be well represented by the Legendre polynomials to construct the statistical model: only good fits with RMSD < 0.1 are included in the statistics. According to our results, most fits are valid. For example, for REPT 2.1-MeV electrons, <10% fits are rejected due to poor fitting. This also validates our fitting method.

Based on this fitting method and the Van Allen Probes data from September 2012 to December 2016, a statistical radiation belt electron equatorial PAD model is constructed as a function of L-shell, MLT, geomagnetic activity, and electron energy. The model covers radiation belt electrons with energies of ~30 keV to 5.2 MeV with 19 energy channels, 13 of which (~30 keV to 1 MeV) are from MagEIS data and 6 (1.8–5.2 MeV) are from REPT. The model includes 26 L-shell bins from L = 1 to L = 6 with ΔL = 0.2 and 12 MLT bins with ΔMLT = 2. Note that for greater than megaelectron volt electrons our model only includes L-shell down to L = 3, since in the slot region and inner belt the fluxes of these electrons are usually too low to show clear PAD pattern. The geomagnetic activity, represented by the Dst index, is divided into three levels: Dst > −20 nT, −50 nT < Dst ≤ −20 nT, and Dst ≤ −50 nT. The medians, means, and standard deviations of coefficients of Legendre polynomials as well as the sample size are derived in each L and MLT bin for electrons with a specific energy under a specific geomagnetic activity condition. Using the equation $urn:x-wiley:21699380:media:jgra54233:jgra54233-math-0007$ , where f(α) is the model PAD as a function of PA α, P_n[cos(α)] is the nth-order Legendre polynomial, c_n is the corresponding normalized coefficient, and m is the highest order of Legendre polynomials included in the model (for tens to hundreds of kiloelectron volt electrons, m = 10 at L < 2, 8 at 2 ≤ L < 4, and 6 at L ≥ 4; for greater than megaelectron volt electrons, m = 6), the averaged PADs can be reconstructed using medians or means of coefficients at each L-shell, MLT, and geomagnetic activity level for electrons with a specific energy. More detailed information and example programs of model usage are included in the supporting information. Some results are shown and discussed in the next sections.

4 Model Results

Using the method described in section 3, we constructed an empirical model of radiation belt electron PADs as a function of L-shell, MLT, geomagnetic activity, and electron energy using data from the Van Allen Probes. In this section, some intriguing results from our model are shown and discussed. In section 4.1, we will focus on the PADs of megaelectron volt electrons in the outer radiation belt and show the dependence of PADs on the L-shell, MLT, geomagnetic activity, and electron energy. In section 4.2 we will focus on the hundreds of kiloelectron volt electrons in the inner radiation belt and slot region. In section 4.3 we will show some data-model comparison results.

4.1 Megaelectron Volt Electrons in the Outer Radiation Belt

4.1.1 Megaelectron Volt Electron PADs in the Outer Belt: Dependence on the L-Shell, MLT, and Geomagnetic Activity

Figure 4 shows model results for 2.1-MeV electrons during quiet times (Dst > −20 nT, top panels), moderately disturbed times (−20 nT ≥ Dst > −50 nT, middle panels), and disturbed times (Dst ≤ −50 nT, bottom panels). Medians of coefficients c₂, c₄, and c₆ are shown in left, middle, and right panels of Figure 4, respectively. In left panels, the sample size, which is the number of PADs include in each L and MLT bin, is shown on the top right corner.

Several things are worth noting in Figure 4. First, the day-night asymmetry can clearly be seen from each coefficient, especially at higher L-shells. Generally, the coefficient c₂ is more negative at dayside and c₄ is more negative at nightside, indicating that 2.1-MeV electron PADs peak at 90° PA at dayside while butterfly PADs are generally present at nightside at higher L-shells (as described in Figure 1). This is caused by the negative flux radial gradient and drift-shell-splitting effect as a result of asymmetric geomagnetic field (e.g., Selesnick & Blake, 2002; Sibeck et al., 1987). Second, comparing coefficients under different geomagnetic activity levels, it is clear that during moderately disturbed and disturbed times, c₂ tends to be more negative at dayside and c₄ tends to be more negative at nightside. This indicates that during storms, for 2.1-MeV electrons, PADs are more exceedingly peaked at 90° PA at dayside, and butterfly PADs are more significant at higher L-shells and even extend to lower L-shells, which could be due to the magnetic field configuration changes and/or electron flux radial gradient changes. It is worth mentioning that during the Van Allen Probes era, the geomagnetic activity was relatively quiet and not many intense geomagnetic storms were present. Thus, the statistics of our model is relatively low for geomagnetically disturbed times with Dst ≤ −50 nT. In the rest of this section, we will focus on model results during quiet times (Dst > −20 nT) and moderately disturbed times (−50 nT < Dst ≤ −20 nT).

Figure 5 shows the 2.1-MeV electron averaged PADs reconstructed from our model using medians of c₂, c₄, and c₆, at L = 4.0, 5.0, and 5.8 at different MLTs, while the results during quiet times are shown in black curves and those during moderately disturbed times are shown in red. It is worth mentioning that in the Figure 5 and following figures, the reconstructed PADs are plotted with x axis showing PAs and y axis showing log(f(α)) + 1, where f(α) is the model PAD. At quiet times, as shown in Figure 5, the averaged PADs in the outer radiation belt are mainly pancake PADs, except that the butterfly PAD is present at MLT = 0 at L = 5.8, while at MLT = 12 at L = 5.0 and 5.8 the PADs are more significantly peaked at 90° PA. This day-night asymmetry is expected from the drift-shell-splitting effect resulting from asymmetric magnetic field. During moderately disturbed times, such day-night asymmetry of electron PADs is more significant: the butterfly PADs are present at wider L range and wider MLT range. At L = 5.0 the transition between pancake PADs and butterfly PADs can be seen at MLT = 0 and 18, and at MLT = 6 the PAD tends to be more flattop than that during quiet times, while at L = 5.8 the butterfly PADs are not only present at midnight but also at MLT = 6 and 18. Such kind of differences between averaged PADs during quiet times and moderately disturbed times can be due to the enhanced drift-shell-splitting effect caused by more stretched geomagnetic field during active times, while magnetopause shadowing effect can also play a more significant role. Also, the PADs are mostly steeper during moderately disturbed times than quiet times. This phenomenon has also been reported by Y. Chen et al. (2014) using data from CRRES, Polar, and LANL-97A. It could be due to the stronger wave-particle interactions during active times, which could create excessive peaks around 90° PA or cause more losses at lower PAs. This could also be caused by the magnetic field configuration changes and/or the electron flux radial gradient changes during active times.

4.1.2 Megaelectron Volt Electron PADs in the Outer Belt: Dependence on Electron Energy

The dependence of relativistic electron PADs on electron energy is also investigated using model results. Figure 6 shows the comparison of averaged PADs of ~1- (top panels), 2.1- (middle panels), and 3.6-MeV (bottom panels) electrons reconstructed from our model at L = 4.0, 5.0, and 5.8 during quiet times (Dst > −20 nT). While the day-night asymmetry of PADs is still very clear for all three energy channels, it is more significant for electrons with higher energies, which could be due to the steeper radial gradient of fluxes for higher-energy electrons. Also, generally higher energy electrons have steeper PADs than lower-energy electrons, showing as lower normalized fluxes at lower PAs and higher normalized fluxes at higher PAs. This feature could be due to excessive losses of higher-energy electrons at lower PAs, and one potentially responsible physical process is electromagnetic ion cyclotron waves, which, under similar conditions, resonate with higher-energy electrons with larger PA range (e.g., Blum et al., 2015; Li et al., 2007; Usanova et al., 2014).

4.2 Hundreds of Kiloelectron volt Electrons in the Low L Region

The megaelectron volt electron PADs in the outer radiation belt have attracted a lot of attention in the past. However, hundreds of kiloelectron volt electron PADs in the low L region received little attention prior to the Van Allen Probes era mainly due to limited availability of high-quality data. As the launch of Van Allen Probes, MagEIS instruments provide unprecedented measurements of hundreds of kiloelectron volt electrons in the low L region, and many new features of hundreds of kiloelectron volt electrons have been unveiled. It has been found that, though limited megaelectron volt electrons are present in the low L region, abundant hundreds of kiloelectron volt electrons exist in the inner radiation belt almost all the time (Claudepierre et al., 2017; Fennell et al., 2015), and the peculiar PADs of hundreds of kiloelectron volt electrons with minimum flux at 90° PA were discovered for the first time in the low L region (Zhao et al., 2014a, 2014b). Based on measurements from MagEIS instruments, an empirical model of hundreds of kiloelectron volt electron PADs at L = 1–6 is constructed. This is the first empirical model of energetic electron PADs in the slot region and inner radiation belt. Intriguing results regarding hundreds of kiloelectron volt electron PADs in the low L region are discussed below based on the model results.

4.2.1 Hundreds of Kiloelectron Volt Electron PADs in the Low L Region: Dependence on the L-Shell, MLT, and Geomagnetic Activity

Figure 7 shows the averaged PADs of ~235-keV electrons at different L-shells and MLTs in the inner radiation belt and slot region measured by MagEIS instruments during quiet geomagnetic activities (Dst > −20 nT) and moderately disturbed times (−50 nT < Dst ≤ −20 nT). Note that in the low L region, as expected, the averaged PADs have almost no dependence on MLT, indicating an azimuthally symmetric geomagnetic field. As shown in Figure 7, the 90°-minimum PADs, as first reported by Zhao et al. (2014a), were clearly present in the inner radiation belt during both quiet and moderately disturbed times, and the averaged PADs in the inner belt have very week dependence on the geomagnetic activity. As the L-shell gets larger, the minima at 90° PA gradually disappear. At L = 2.4, during quiet times, the minima at 90° PA in PADs are not present; however, during moderately disturbed times, the minima at 90° PA appear again, showing the prevalence of 90°-minimum PADs in the slot region during moderately disturbed times. The wave-particle interaction between energetic electrons and fast magnetosonic waves is suggested as a possible mechanism responsible for the formation of this PAD type (L. Chen et al., 2015; J. Li et al., 2016; Zhao et al., 2014a, 2014b), and quasi-linear PA and energy diffusion has also been proposed as a potential mechanism causing the 90°-minimum PADs in the low L region (Albert et al., 2016), while the actual causes are still under considerable debate. Though the formation of such 90°-minimum PADs is not clear, the disappearance of this type of PADs at higher L-shells is likely to be caused by the plasmaspheric hiss wave scattering (Zhao et al., 2014a, 2014b). Our model results also support this hypothesis, which shows that as the L-shell gets larger, the cap PADs gradually appear (indicating the presence of hiss wave scattering) and the minima at 90° PA disappear. At L = 3.4, the averaged PADs are cap PADs during quiet times, suggesting the interaction between energetic electrons and plasmaspheric hiss waves, consistent with theoretical predictions (e.g., Lyons, Thorne, & Kennel, 1972). However, note that during moderately disturbed times, at L = 3.4, the averaged PADs of ~235-keV electrons do not show clear cap feature; instead, they are closer to pancake PADs. This is also consistent with some previous observations (e.g., Lyons & Williams, 1975b; Zhao et al., 2014b) and suggests that the complex energizing and transport processes, for example, inward radial diffusion and wave heating, play more important roles on energetic electron dynamics in the slot region than hiss wave scattering during active times.

4.2.2 Hundreds of Kiloelectron Volt Electron PADs in the Low L Region: Dependence on Electron Energy

Figure 8 shows the averaged PADs of ~105-, 235-, and 460-keV electrons at different MLTs in the low L region during quiet times, reconstructed using model results. As Figure 8a panels show, for electrons with higher energies, 90°-minimum PADs are more significant: for 105-keV electrons, almost no bite out can be seen around 90° PA; while for 460-keV electrons, the minima at 90° PA are very obvious. Such energy dependence, demonstrated here using statistical analysis for the first time, could be a critical evidence in identifying the underlying physical processes responsible for the formation of 90°-minimum PADs. At L = 2.4, the PADs are steeper for higher-energy electrons, which is consistent with theoretical predictions of wave-particle interaction caused by plasmaspheric hiss waves in the slot region. Figure 8c panels show the effect of plasmaspheric hiss waves on energetic electrons of different energies clearer: at L = 3.4, cap PADs are present for energetic electrons with all energies shown here, but the size (or PA range) of “cap” is shown to be smaller for higher-energy electrons and larger for lower-energy ones. Still, this is consistent with theoretically predicted PADs caused by hiss wave scattering in the slot region (e.g., Lyons et al., 1972).

4.3 Comparison of Model Results and Observations

Figure 9 shows comparison between data and our empirical model results for ~590-keV electrons at L < 6 during 1–2 March 2017, when a moderate geomagnetic storm occurred. Our model is constructed based on data from September 2012 to December 2016; thus, data shown in Figure 9 were not used to construct our model. The top panels show the model results reconstructed using the fluxes of locally mirroring electrons measured by MagEIS instruments on the Van Allen Probes-A, and the bottom panels show the MagEIS data after propagated to the geomagnetic equator using T89D model. The data and model results show good agreements. For example, in the inner belt both model and data show much higher fluxes for near-90° PA electrons; around 22:00 on 1 March and 2:00 on 2 March, both data and model show butterfly PADs.

5 Discussion

5.1 Comparison to Y. Chen et al. (2014) Model

As an important feature of radiation belt electrons, the relativistic electron PAD is a good indicator of specific physical processes in a specific region. Many previous studies have been focused on the radiation belt electron PADs and some empirical models have been constructed. Specifically, Y. Chen et al. (2014), using Legendre polynomials to fit radiation belt electron PADs from CRRES, Polar, and LANL-97A, constructed an empirical model of radiation belt electron PADs as a function of L-shell, MLT, geomagnetic activity represented by AE index, and electron energy. The model they constructed covers the whole outer radiation belt and electron energy ranges of ~150–1,600 keV. Our study, which uses similar techniques in conducting the statistical survey of radiation belt electron PADs, covers even wider energy range of electrons by using data from the Van Allen Probes, from ~30 keV all the way up to ~5.2 MeV. Moreover, our empirical model covers not only the outer radiation belt but also the inner radiation belt and slot region for tens to hundreds of kiloelectron volt electrons for the first time.

Figure 10 shows some comparison of ~1-MeV electron PADs reconstructed using the results of our model (solid curves) and the empirical model from Y. Chen et al. (2014; dashed curves) under quiet geomagnetic conditions, which is defined as Dst > −20 nT in our model while as AE < 100 nT in Y. Chen et al. (2014). It can be seen from Figure 10 that during quiet times, for ~1-MeV electrons, our model shows similar averaged PADs comparing to results from Y. Chen et al. (2014). As shown in Figure 10a panels, at L = 4, both models show pancake PADs for ~1-MeV electrons with MLT symmetry; at L = 5, as Figure 10b panels show, the averaged PADs of ~1-MeV electrons are still pancake, though at noon our model shows a more 90°-peaked PAD; at L = 5.8, both models show the butterfly PADs at midnight and more 90°-peaked PADs at noon. Good agreement between the two models validates both models and also confirms that the radiation belt electron PADs, especially during quiet times, can be well represented by empirical model results. The larger differences between our results and those from Y. Chen et al. (2014) at MLT = 0 and 12, especially at higher L-shells, are likely due to different definition of quiet times used when constructing the model. Y. Chen et al. (2014) used AE index less than 100 nT, which is extremely quiet times comparing to our definition of Dst index greater than −20 nT: from September 2012 to December 2016, at ~80% of time Dst index was greater than −20 nT, while only at ~50% of time AE index was less than 100 nT. Under quieter conditions, the magnetic field is less stretched and day-night asymmetry is less significant, thus the exceedingly 90°-peaked PADs at noon and butterfly PADs at midnight, which are formed as a result of drift-shell-splitting effect, are less significant. It is also worth mentioning that as shown in Figure 10, generally, our model results show relatively lower fluxes around lower PAs. This could be due to finer PA resolution and cleaner measurements of Van Allen Probes as near the loss cone low fluxes are expected.

Note that since the Van Allen Probes era is relatively quiet and the statistics of disturbed times in our electron PAD model is not as good as those of quiet times and moderately disturbed times, our model may not be favorably used during intense storms but rather during quiet times or small to moderate storm times.

5.2 Variabilities of Radiation Belt Electron PADs

Figures 11 and 12 show the variabilities of radiation belt electron PADs used in constructing our model during quiet times with Dst > −20 nT by showing the median, mean, 25th and 75th percentiles of all PADs used in one (L, MLT, Dst, and energy) bin in our model. Figure 11 shows results for megaelectron volt electrons at L = 5 at four different MLTs. It is clear from Figure 11 that though the variation of relativistic electron PADs can be quite significant at some time, statistically, they show good consistency under quiet geomagnetic conditions across a long period of time: the variations between 25th and 75th percentiles are generally small especially for lower-energy electrons. As the electron energy gets higher, the variation of PADs gets more significant. In Figure 11 only results at L = 5 during quiet times are shown, but the standard deviation of different coefficients can give us a hint on how this variability changes with L-shell and geomagnetic activity. Though not shown here, for megaelectron volt electrons, as L-shell gets higher, the variation of PADs gets more significant, while during storm times the variation of PADs is also more pronounced than that during quiet times. These results are expected since at higher L-shells, the geomagnetic field is more asymmetric and dynamic while subjects to more significant influence of solar wind/geomagnetic conditions, and during storm times the radiation belt also exhibits more variations as the effects of a lot of physical mechanisms are enhanced.

Figure 12 shows the variabilities of PADs of hundreds of kiloelectron volt electrons in the inner belt and slot region. It is evident that in the inner belt, hundreds of kiloelectron volt electron PADs show little variation, though some outliers are also present, while in the slot region, the variation of PADs gets more significant. This is also expected because the competition between different mechanisms, for example, inward radial diffusion and whistler mode wave scattering, is quite significant in the slot region.

As the variability of electron PADs varies under different conditions, our radiation belt electron PAD model, which gives the standard deviation of each parameter, can be very helpful when assessing the variability of PADs under a certain condition and also can provide an estimate of error when using this radiation belt electron PAD model in calculation.

6 Summary

In this study, we constructed an empirical model of radiation belt electron PADs based on the Van Allen Probes measurements from September 2012 to December 2016. The model, using the method of Legendre polynomial fitting, provides the medians, means, and standard deviations of coefficients of Legendre polynomials as a function of L-shell, MLT, geomagnetic activity, and electron energy, with which the statistical PADs can be reconstructed and variability of PADs can be assessed. Comparing to previous empirical models of radiation belt electron PADs, our model covers wider L-shell range (L = 1–6) and wider energy range (~30 keV to 5.2 MeV) with higher energy resolution (a total of 19 energy channels). Our model is also the first empirical model on the radiation belt electron PADs in the inner radiation belt and slot region.

Model results show important features of PADs of electrons with different energies at different regions. Day-night asymmetry of megaelectron volt electron PADs in the outer radiation belt are shown to be consistent with theoretical prediction from drift-shell-splitting effect combining with negative flux radial gradient. The asymmetry is found to be more significant during active times than quiet times, which can be due to geomagnetic field configuration changes and/or the flux radial gradient changes potentially caused by enhanced magnetopause shadowing effect; the asymmetry is also more significant for higher-energy electrons due to more negative flux radial gradient. Megaelectron volt electron PADs are also shown to be steeper during active times and for higher-energy electrons, which could be due to electromagnetic ion cyclotron wave scattering. For hundreds of kiloelectron volt electrons in the low L region, 90°-minimum PADs are shown to be persistently present in the inner belt and appear in the slot region during active times, while during quiet times the cap PADs caused by the hiss wave scattering are prevalent in the slot region. During quiet times, the energy-dependent features of cap PADs are consistent with theoretical predictions of hiss wave scattering, indicating the critical role of hiss waves in creating cap PADs in the slot region; during active times, cap PADs are less significant especially at outer part of slot region, which could be due to that the complex energizing and transport processes, for example, inward radial diffusion and wave heating, play more important roles on energetic electron dynamics in the slot region than hiss wave scattering during active times. Specifically, for the 90°-minimum PADs in the inner belt, it is revealed for the first time on a statistical basis that the minima at 90° PA are more significant for higher-energy electrons, which could be a critical evidence in identifying the underlying physical processes responsible for the formation of 90°-minimum PADs and is worth to be further explored in the future work. Overall, our model provides valuable information for identifying and examining the physical processes acting on radiation belt electrons.

Acknowledgments

The research presented here was supported by RBSP-ECT funding through JHU/APL contract 967399 (under prime NASA contract NAS5-01072). The parameters of radiation belt electron pitch angle distribution model constructed in this study are included in the supporting information. Van Allen Probes REPT data and MagEIS level 2 data used in this paper are available from the ECT Science Operations and Data Center (http://www.rbsp-ect.lanl.gov). MagEIS level 3 unbinned pitch angle data can be made available by contacting the MagEIS team (S. G. Claudepierre, J. B. Blake, and J. F. Fennell). The Dst index is provided by OMNIWeb (http://omniweb.gsfc.nasa.gov/).

Supporting Information

References

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Volume123, Issue5

May 2018

Pages 3493-3511

Filename	Description
jgra54233-sup-0001-2018JA025277-ds01.zipapplication/x-zip-compressed, 423.9 KB	Supporting Information S1
jgra54233-sup-0002-2018JA025277-SI.pdfPDF document, 59.5 KB	Data Set S1

An Empirical Model of Radiation Belt Electron Pitch Angle Distributions Based On Van Allen Probes Measurements

Abstract

Key Points

1 Introduction

2 Data

3 Methodology

4 Model Results

4.1 Megaelectron Volt Electrons in the Outer Radiation Belt

4.1.1 Megaelectron Volt Electron PADs in the Outer Belt: Dependence on the L-Shell, MLT, and Geomagnetic Activity

4.1.2 Megaelectron Volt Electron PADs in the Outer Belt: Dependence on Electron Energy

4.2 Hundreds of Kiloelectron volt Electrons in the Low L Region

4.2.1 Hundreds of Kiloelectron Volt Electron PADs in the Low L Region: Dependence on the L-Shell, MLT, and Geomagnetic Activity

4.2.2 Hundreds of Kiloelectron Volt Electron PADs in the Low L Region: Dependence on Electron Energy

4.3 Comparison of Model Results and Observations

5 Discussion

5.1 Comparison to Y. Chen et al. (2014) Model

5.2 Variabilities of Radiation Belt Electron PADs

6 Summary

Acknowledgments

Supporting Information

References

Citing Literature

Figures

References

Information

RESOURCES

PUBLICATION INFO

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An Empirical Model of Radiation Belt Electron Pitch Angle Distributions Based On Van Allen Probes Measurements

Abstract

Key Points

1 Introduction

2 Data

3 Methodology

4 Model Results

4.1 Megaelectron Volt Electrons in the Outer Radiation Belt

4.1.1 Megaelectron Volt Electron PADs in the Outer Belt: Dependence on the L-Shell, MLT, and Geomagnetic Activity

4.1.2 Megaelectron Volt Electron PADs in the Outer Belt: Dependence on Electron Energy

4.2 Hundreds of Kiloelectron volt Electrons in the Low L Region

4.2.1 Hundreds of Kiloelectron Volt Electron PADs in the Low L Region: Dependence on the L-Shell, MLT, and Geomagnetic Activity

4.2.2 Hundreds of Kiloelectron Volt Electron PADs in the Low L Region: Dependence on Electron Energy

4.3 Comparison of Model Results and Observations

5 Discussion

5.1 Comparison to Y. Chen et al. (2014) Model

5.2 Variabilities of Radiation Belt Electron PADs

6 Summary

Acknowledgments

Supporting Information

References

Citing Literature

Figures

References

Related

Information

RESOURCES

PUBLICATION INFO