Upper Limit on Outer Radiation Belt Electron Flux Based on Dynamical Equilibrium

Abstract

In the Earth’s radiation belts, an upper limit on the electron flux is expected to be imposed by the Kennel‐Petschek mechanism, through the generation of exponentially more intense whistler‐mode waves as the trapped flux increases above this upper limit, leading to fast electron pitch‐angle diffusion and precipitation into the atmosphere. Here, we examine a different upper limit, corresponding to a dynamical equilibrium in the presence of energetic electron injections and both pitch‐angle and energy diffusion by whistler‐mode chorus waves. We first show that during sustained injections, the electron flux energy spectrum tends toward a steady‐state attractor resulting from combined chorus wave‐driven energy and pitch‐angle diffusion. We derive simple analytical expressions for this steady‐state energy spectrum in a wide parameter range, in agreement with simulations. Approximate analytical expressions for the corresponding equilibrium upper limit on the electron flux are provided as a function of the strength of energetic electron injections from the plasma sheet. The analytical steady‐state energy spectrum is also compared with maximum electron fluxes measured in the outer radiation belt during several geomagnetic storms with strong injections, showing a good agreement at 100‐600 keV.This article is protected by copyright. All rights reserved.

Checking Key Assumptions of the Kennel‐Petschek Flux Limit With ELFIN CubeSats

“…To estimate the wave power gain G(t) and its relationship to the equatorial trapped electron flux J trap (α 0 = 90°) based on ELFIN CubeSat measurements, we need to calculate J prec1 /J trap and J prec2 /J trap at different successive times t i during each storm. Assuming a quasi-equilibrium pitch-angle electron distribution at 0.08 0.6 MeV (as justified by previous works, see Mourenas et al, 2021;Shane et al, 2023), quasi-linear theory (Kennel & Petschek, 1966; shows that the average precipitating to trapped flux ratio measured at ELFIN CubeSats can be approximately written as (Mourenas et al, 2023), equivalent to:…”

“…This energy range, ∼0.08 0.6 MeV, corresponds to the main population of cyclotron resonant electrons with equatorial pitch-angles α 0 > 60°efficiently generating parallel chorus waves of mean normalized frequency ω/Ω ce0 ≃ 0.2 0.25 (Agapitov et al, 2018;W. Li et al, 2016) for a typical plasma frequency to gyrofrequency ratio Ω pe0 /Ω ce0 ≃ 4.5 near L ∼ 5 during disturbed periods (Agapitov et al, 2019;Mourenas et al, 2023;.…”

“…The electron flux pitch-angle anisotropy s for a distribution of the form J trap (α 0 ) = sin 2s α 0 (Summers et al, 2009) for α 0 = 10°-90°(i.e., for most of the electron population providing the free energy for chorus wave growth, see Omura et al, 2008;Tao et al, 2017; can be similarly estimated based on quasilinear theory (Mourenas et al, 2023), giving:…”

“…As will appear below, this method requires energy-resolved measurements of trapped electron fluxes J trap at pitch-angles just above the loss cone (for ELFIN data, at α 0 ≃ 1.05 α 0,LC where α 0 and α 0,LC denote the equatorial pitch-angle and loss cone angle, respectively), and energy-resolved data of both precipitating electron fluxes J prec, meas measured above ELFIN within the local bounce loss cone and backscattered (by the atmosphere) electron fluxes J up measured below ELFIN within the local anti-loss-cone (see examples of measured trapped, precipitating and backscattered fluxes in Figure 1 from Mourenas et al, 2023), sufficiently well resolved in pitch-angle, to allow an accurate determination of the average (over local pitch-angles) effective net flux J prec ≃ J prec, meas J up of electrons precipitated by whistler-mode waves within the loss cone (Mourenas et al, 2023). The lowaltitude ELFIN CubeSats precisely provide the needed electron flux data with a good resolution in pitch-angle within the loss cone (Angelopoulos et al, 2020(Angelopoulos et al, , 2023).…”

“…It is worth noting that, in deriving Equation 6, we made use of normalized z 0 (t i ) and flux values. An advantage of this method, previously used for electromagnetic ion cyclotron (EMIC) waves (Angelopoulos et al, 2023), is that such a normalization leads to a cancellation of all parameters in z 0 that remain approximately constant during the examined time interval, such as the plasma frequency to gyrofrequency ratio or the wave frequency to gyrofrequency ratio (Mourenas et al, 2023). As a result, the wave power gain near the magnetic equator can be inferred from low-altitude measurements alone, without additional plasma or wave measurements in the equatorial and mid-latitude regions where wave generation and electron precipitation take place.…”