## Introduction

Discovered at the start of the space age, the Van Allen radiation belts consist of relativistic (500 keV) and ultra-relativistic (3MeV) electrons1,2. Energization mechanisms for these particles remain a topic of intense debate. Historically, inward radial diffusion was considered as the main acceleration process, whereby a source population at large radial distances diffuses toward the Earth due to electric and magnetic field fluctuations3,4. By moving into regions of stronger magnetic field, the kinetic energies of the particles are increased. However, this mechanism alone was shown to be insufficient to fully explain observed enhancements in the relativistic electron flux5. Later work demonstrated that electrons could also be accelerated locally via resonant interactions with naturally occurring whistler-mode electromagnetic waves6, primarily very low frequency chorus waves7, which can energize electrons to greater than a few MeV.

Distinguishing the relative importance of local acceleration and inward radial diffusion in energizing radiation belt electrons is possible by considering the radial profile of the electron differential flux divided by the particle momentum squared8 (given in the magnetic coordinates that constrain electron motion: μ, K, and L*—see Supplementary Note 1), a quantity known as phase space density. The L* coordinate is the radius (in Earth radii) of a particle’s drift around the Earth if the magnetic field adiabatically relaxed to a dipole configuration. Radial diffusion moves electrons across L* while conserving μ and K. The resulting radial profile is sloped, lacking growing peaks. Conversely, local acceleration occurs at constant L* whilst altering μ and K, producing growing peaks. By considering the evolution of phase space density across L*, studies have mostly shown local acceleration to be the primary mechanism for generating 1–3 MeV electrons8,9,10. However, above energies of   ~3 MeV, current theory indicates that acceleration via resonant interactions with chorus waves tends to decline11,12 and how electrons come to be heated to higher energies is unclear.

Variable factors, such as the electron number density and the duration of chorus activity, influence the maximum energy achieved by local acceleration5,12 and it is therefore possible that chorus interactions may be responsible for  >3 MeV enhancements. Thorne et al.6 used a 2-D diffusion model at fixed L* = 5 and found that, under low plasma density conditions, chorus acceleration reproduced observed enhancements up to energies of  ~7 MeV. However, to date, there have been no observations of local acceleration acting for these ultra-relativistic energies and analysis of phase space density profiles across a range of μ values is required to determine whether local acceleration can generate these populations.

A multistep acceleration process has recently been suggested13,14,15, with particles first accelerated to relativistic energies locally by chorus waves and the newly formed population then further energized by inward radial diffusion16. However, radial profiles of phase space density relating to >3 MeV electrons have not been extensively studied and it remains unclear if relativistic electrons are accelerated to >1 MeV and then transported inwards, conserving the first and second invariants, or whether they can be locally accelerated directly to ultra-relativistic energies. If the multistep process is indeed the generation mechanism for ultra-relativistic electrons, then growing phase space density peaks would first be seen at large radial distances and subsequently broaden due to radial diffusion.

Here, we present observations from the Van Allen Probe satellites which show local acceleration acting to energize electrons to up to ultra-relativistic energies. For a broad range of the first adiabatic invariant, μ, the evolution of electron phase space density across the L* parameter shows signatures of local acceleration. Radiation belt electrons with energies up to 7 MeV can therefore be enhanced in situ.

## Results

### Overview of the storm event

An intense geomagnetic storm occurred on the 9 October 2012, during which the Relativistic Electron–Proton Telescope (REPT)17 on the NASA Van Allen probes recorded flux enhancements across a range of energies, including at 7.7 MeV (Fig. 1). The storm was triggered by two periods of largely southward directed interplanetary magnetic field (Fig. 1h) and is characterized by two decreases in Dst, a measure of the geomagnetic storm intensity (Fig. 1i). During the second decrease in Dst, the electron flux showed prompt energization. As shown in Fig. 1b, early on the 9 October the 2.6 MeV flux at L* = 4 rose by three orders of magnitude in less than 12 h, signifying an intense acceleration event. Several studies have previously focused on this storm, demonstrating that local acceleration mechanisms were both active during the period6,18 and provided the primary energization process for μ = 3433 MeV G−1, K = 0.11 REG1/2 electrons9. However, the electron energy corresponding to a particular μ changes with L* (Fig. 2) and analysis over a range of μ is required to determine how electrons at different energies come to be heated. In Fig. 1 enhancements in the 6.3 and 7.7 MeV flux are seen around L* = 4 and, from Fig. 2, therefore correspond to μ values of 8000–12,000 MeV G−1 for K = 0.11 REG1/2. Here, we follow the work of previous authors and use K = 0.11 REG1/2 which is sampled over a broad range of L* and generally corresponds to equatorial pitch angles greater than 45°9,10,13 (see also Supplementary Fig. 13).

A second geomagnetic storm occurred on the 13 October, and crucially, in the 2 days following the storm, the flux at 6.3 and 7.7 MeV exhibited a second enhancement. The combination of the two successive geomagnetic storms resulted in increased levels of ultra-relativistic radiation belt electrons, which persisted for more than a week, presenting an interesting period for study.

### Analysis of phase space density

In previous work, phase space density at fixed μ and K, measured through a number of outbound and inbound legs of the Van Allen Probes orbits, were analyzed for temporally growing peaks9,10. Generally, only a single μK pair is studied in this manner, owing both to the challenges involved in calculating phase space density, and because measurements of  >6 MeV ultra-relativistic electrons are seldom distinguishable from the background level of the instrument. Focus has primarily been on lower values of μ  (<6000 MeV G−1), leaving the dynamics of high energy particles largely unexplored. Here, we study the October 2012 period, where 7.7 MeV electron flux values, greater than the background threshold [16], are observed during multiple days, and calculate the phase space density values relating to several different μ, contrasting how electrons are enhanced over a broad range of energies.

Figure 3 is a schematic illustration showing the evolving L* coverage of phase space density values within a factor of five of the maximum phase space density, for various acceleration mechanisms. In this format, inward radial diffusion appears as a contour which gradually expands to smaller L*, whilst also extending out to the last L* sampled (Fig. 3a). Local acceleration, on the other hand, presents as an enhancement appearing away from the maximum L*, remaining at approximately the same location (Fig. 3b). However, peaks formed by local acceleration are likely to be redistributed by radial diffusion. A localized phase space density peak, formed in the heart of the outer radiation belt and gradually diffusing to higher L* is shown in Fig. 3c, while a peak at larger L*, radially diffusing inwards, is shown in Fig. 3d.

The temporal evolution of the L* extent of phase space density maxima during the October 2012 period is considered for eight values of μ (Fig. 4), in the format of Fig. 3, highlighting enhancements. The multistep formation method for >3 MeV electrons, suggested by several authors16,19, would appear as the profile shown in Fig. 3d. However, over a time-span of several days, the profiles for both storms shown in Fig. 4a more closely resemble local acceleration followed by outward radial diffusion (Fig. 3c) during the recovery period. For the enhancement during the first storm, contours are concentric with increasing μ and do not extend to the last L* sampled (black region) indicating that a peak arose for a broad range of energies at the same location. A temporal shift is observed, with the peak forming at high energies (up to 7 MeV) later than at lower energies, consistent with energy diffusion at this point in space. Note that for μ = 10,000 MeV G−1, the L* extent of the phase space density enhancement rapidly shifts inwards across  ~0.3L*. This L* variation may be the result of inward radial diffusion or may indicate local acceleration from a shifting source region. From Fig. 2, an inward radial motion from L* ~ 4.5–4.2 constitutes an energy increase of  ~600 keV at μ = 10,000 MeV G−1, and so a source population of ~6.4 MeV would be needed to produce the 7 MeV enhancement at L* = 4.2. Van Allen Probes passes during this acceleration period (first region shaded gray) are analyzed more closely for μ = 8000 and 10,000 MeV G−1 electrons in Fig. 5a, b.

## Discussion

The phase space density along the inbound pass of Van Allen Probe A and outbound pass of probe B are shown prior to the enhancement (black lines, Fig. 5a, labeled 20:45 to 23:10 8 October and 22:35 8 October to 01:30 9 October). These phase space density values are close to the background thresholds of the instrument but are included here to illustrate the upper bound of the pre-existing phase space density level. Over the next few passes of both probes, a peak emerges which grows with time. The broadening of the phase space density peak is likely a result of both inward and outward radial diffusion, or a dynamically evolving acceleration region, however, the presence of a growing peak is a signature of local acceleration. The primary L* coverage of the peak is L* = 3.8–5.0, corresponding to electrons up to  ~7.5 MeV.

As was also shown in Fig. 4a, at μ = 10,000 MeV G−1, enhancements were observed at later times than at μ = 8000 MeV G−1. Reeves et al.9 reported that the μ = 3433 MeV G−1 phase space density showed an enhancement between the 8 and 9 October (we also see similar behavior at this μ. See Supplementary Fig. 3a), while in Fig. 5, increases at μ = 8000 and 10,000 MeV G−1 were primarily seen at later times. The enhancement therefore occurred first at lower μ values. During local acceleration, electrons are accelerated across energy space, and an observed delay for enhancements to reach higher values of μ is consistent with this heating mechanism.

During the 07:20 to 10:35 9 October outbound pass of Van Allen Probe B, the phase space density peak had increased at μ = 8000 MeV G−1 and an enhancement was also seen for μ = 10,000 MeV G−1. Negative phase space density gradients were observed around L* ~ 4.7 for both values of μ, however, there is a very sharp increase toward the end of the orbit, outside L* ≈  4.8 (see Supplementary Fig. 12). Supplementary Note 7 examines this pass in more detail and it was concluded that the enhanced phase space density at L*  4.8 originates from an observed increase in the Bz component of the magnetic field, that is not fully captured by the field model, together with an inversion of the local pitch angle distribution. Note that the increase is preserved at a similar L* on the following inbound pass of probe B and is not observed at all by probe A. As it would appear that the enhancement at L* ≈ 4.8 is an artifact of field model, we do not show this in Fig. 5a, b. In addition, Supplementary Fig. 9 shows that the last closed drift shell was at L* = 5.7 for K = 0.11 REG1/2. The lowest energies at μ = 10,000 MeV G−1 and μ = 8000 MeV G−1 are therefore  ~4.2 and ~3.8 MeV, respectively (from Fig. 2). If the rapid change in phase space density does not originate from magnetic field discrepancies, then this raises an interesting question on the origin of the population, but is beyond the scope of this study.

After the phase space density enhancement, the peak broadens over time and becomes more of a plateau (see Supplementary Fig. 4). The radiation belt electrons therefore experienced radial diffusion, transporting them outwards. While the maximum phase space density contours shown in Fig. 4 expand toward higher L* for all values of μ shown, after 11 October, for μ ≤ 5000 MeV G−1, broader L* profiles are observed than for higher μ. The different behavior exhibited by μ ≤ 5000 MeV G−1 and μ > 5000 MeV G−1 electrons may be indicative of energy dependent loss mechanisms, such as electromagnetic ion cyclotron waves20,21.

Following the second storm in October 2012, the phase space density again shows peaked structures centered on similar L* for a broad range of μ (Fig. 4a). As for the first storm, we further analyze this period by considering individual passes of the Van Allen probes (Fig. 5c and d). Prior to the acceleration, magnetopause shadowing22 and outward radial diffusion23 greatly reduced the pre-existing levels of phase space density and the initial profile is shown in black. At later times, a peak forms for both μ = 8000 MeV G−1 and μ = 10,000 MeV G−1, centered at L* ~ 4. Over the subsequent passes, this initial peak broadens in L* before growing further, concentrated around L* ≈ 4.3. While the broadening suggests a contribution from radial diffusion, the growing peaked structure indicates that local acceleration was an active process during the second storm, forming the enhancements seen at μ = 8000 MeV G−1 and μ = 10,000 MeV G−1. After the localized increase in phase space density, a radial shift toward larger L* can be seen (Fig. 4) starting on 15 October. The rate of outward radial motion appears relatively invariant with μ, suggesting energy independent diffusion, consistent with radial diffusion driven by magnetic pulses3 rather than electrostatic pulses, or an azimuthal electron field power spectral density that is independent of frequency24. After 17 October, the dynamics of the peaks exhibit more variation with μ.

Phase space density peaks may, in some cases, also arise due to a rapid enhancement at the outer boundary followed by fast loss25. The twin Van Allen probes sample only two locations at any one time and, owing to the spatial and temporal limitations, it is possible that a rapid enhancement and loss could occur that is missed by the spacecraft sampling. As discussed in Supplementary Note 5, the orbits of the two Van Allen Probes impose times restrictions on this process, making such a scenario unlikely. Furthermore, the last closed drift shell location is at L*  ~ 5.5–6.5 for the first storm and L* ~ 6–6.5 for the second (Supplementary Note 6). The trends shown in Fig. 2 then suggest that the minimum energy of the source population required for the observed 7 MeV enhancements is likely to be several MeV. Observations from geostationary orbit of the   >2 MeV flux measured by the Geostationary Observational Earth Satellites (GOES) 13, 14, and 15 are shown in Supplementary Figs. 10 and 11. When considered alongside the changes in the L* measured, these additional observations do not suggest a rapidly appearing source population for radial diffusion during either event. Furthermore, the GOES flux is higher in the days following the acceleration occurring at L* ~ 4 than it was during the enhancement period, consistent with a locally accelerated source that diffuses outwards. However, we note that these are again in situ measurements and therefore also subject to certain spatial limitations. The GOES spacecraft may not have been appropriately situated to measure a source population during the period. In addition, because radiation belt electrons often show steep falling energy spectra, the integral energy channels may not necessarily show the source populations of >7 MeV electrons in the outer belt.

To determine the phase space density at the target μ and K values, we first interpolate the logarithm of the phase space density, and the corresponding μ and L* values, to the target value of K. The logarithm of the phase space density is then interpolated again to the chosen value of μ. We never use phase space density data that do not directly surround the chosen values of μ and K. If one or both of the immediate neighbors to the target invariant values has no corresponding phase space density data, then the interpolated phase space density was returned as missing data. A model pitch angle distribution was not used when interpolating to the target K value. However, as shown in Supplementary Fig. 1, using a $$f(\alpha )=A\sin (\alpha )+C$$ function between the two pitch angle channels immediately surrounding the target K has a very limited effect on the results.