A reduced-order solar terrestrial recharge oscillator model for long-term modulation of inner Van Allen belt protons

Inner belt proton fluxes exhibit striking multi-year modulation, driven by solar EUV variability, geomagnetic disturbances, and thermospheric density variations. Here we introduce the Solar–Terrestrial Recharge Oscillator (STRO), a reduced-order dynamical model that captures this long-term variability with only two state variables and a minimal set of simple solar–geomagnetic inputs. The model couples proton population dynamics with an effective atmospheric state controlled by a smoothed F10.7 proxy, capturing nonlinear feedback between solar forcing and trapped-particle evolution. A global parameter optimization procedure is employed in order to derive physically consistent coefficients governing accumulation, loss, and solar-driven variability. The optimized STRO model mimics the observed multi-year proton flux record with remarkable fidelity, capturing both amplitude and phase structure, as well as the dominant solar cycle-scale spectral peak. Segmented comparisons over two separate observed intervals (2014–2020 and 2020–2025) illustrate stable cycle-to-cycle performance and minimal phase bias. STRO offers a computationally efficient, interpretable alternative to high-dimensional transport models and purely empirical predictors, holding promise for long-term reconstruction, sensitivity studies, and operational space environment applications.