Ensemble Model Using the Fokker–Planck Equation and Alfvénic Drift for Estimating Proton Flux in Solar Energetic Particle Events

Abstract

Observations of solar energetic particles (SEPs) accelerated at collisionless shocks driven by coronal mass ejections (CMEs) highlight the importance of understanding proton transport in velocity space. In this study, we present an ensemble model based on the one-dimensional Fokker–Planck equation to estimate proton flux from CME-driven shocks propagating toward Earth. Using a CME analysis tool with coronagraph data, we derived initial conditions from key CME characteristics, including CME speed, angular width, Alfvén Mach number, and plasma beta. We then solved the one-dimensional Fokker–Planck equation under the test-particle regime, applicable to weak CME-driven shocks ($M_A\sim 1–4$) where the dynamical pressure of shock-accelerated particles is low compared to thermal pressure. Notably, our model incorporates the effects of Alfvén wave transmission and reflection at shocks, which significantly influence the efficiency of diffusive shock acceleration (DSA) and the transport of shock-accelerated particles in low-beta CME-driven shocks. To address systematic uncertainties from initial conditions obtained through the CME analysis tool and the nonlinear dynamics of the shock surface, including turbulence, we employed an ensemble approach for critical variables impacting DSA efficiency, such as the Alfvén Mach number, plasma beta, and upstream wave amplitude. By estimating proton flux during SEP events in 2024, our ensemble model produced predictions consistent with observations within a 1-sigma deviation, highlighting the importance of Alfvénic drift physics in SEP models of particle acceleration at shocks.