Deduction of the rates of radial diffusion of protons from the structure of the Earth's radiation belts

Abstract. From the data on the fluxes and energy spectra of protons with an equatorial pitch angle of α₀ ≈ 90° during quiet and slightly disturbed (Kp ≤ 2) periods, I directly calculated the value D_LL, which is a measure of the rate of radial transport (diffusion) of trapped particles. This is done by successively solving the systems (chains) of integrodifferential equations which describe the balance of radial transport/acceleration and ionization losses of low-energy protons of the stationary belt. This was done for the first time. For these calculations, I used data of International Sun–Earth Explorer 1 (ISEE-1) for protons with an energy of 24 to 2081 keV at L = 2–10 and data of Explorer-45 for protons with an energy of 78.6 to 872 keV at L = 2–5. Ionization losses of protons (Coulomb losses and charge exchange) were calculated on the basis of modern models of the plasmasphere and the exosphere. It is shown that for protons with μ from  ∼ 0.7 to ∼ 7 keV nT⁻¹ at L ≈ 4.5–10, the functions of D_LL can be approximated by the following equivalent expressions: D_LL ≈ 4.9 × 10⁻¹⁴μ^−4.1L^8.2 or D_LL ≈ 1.3 × 10⁵(EL)^−4.1 or D_LL ≈ 1.2 × 10⁻⁹f_d^−4.1, where f_d is the drift frequency of the protons (in mHz), D_LL is measured in s⁻¹, E is measured in kiloelectronvolt and μ is measured in kiloelectronvolt per nanotesla. These results are consistent with the radial diffusion of particles under the action of the electric field fluctuations (pulsations) in the range of Pc6 and contradict the mechanism of the radial diffusion of particles under the action of sudden impulses (SIs) of the magnetic field and also under the action of substorm impulses of the electric field. During magnetic storms D_LL increases, and the expressions for D_LL obtained here can change completely.