A Unified Charge-Dependent Modulation Model for AMS-02 Proton and Antiproton Fluxes during Solar Minimum

We develop a unified charge-dependent solar modulation model by solving the three-dimensional Parker transport equation, incorporating a realistic wavy heliospheric current sheet to treat drift effects self-consistently. Using a local interstellar spectrum from GALPROP constrained by Voyager data, we fit the model to time-resolved proton and antiproton fluxes measured by the Alpha Magnetic Spectrometer - 02 (AMS-02) during the solar-quiet period (May 2011 to June 2022). To enable rapid parameter scans, we employ neural-network-based surrogate models to compute propagation and modulation matrices efficiently. The results demonstrate that the model simultaneously describes the observed proton and antiproton fluxes with physically reasonable parameters, providing a unified account of charge-dependent modulation.

💡 Research Summary

The paper presents a comprehensive, charge‑dependent solar‑modulation model that simultaneously reproduces the time‑resolved proton and antiproton fluxes measured by AMS‑02 during the unusually quiet solar minimum spanning May 2011 to June 2022. The authors start from a realistic local interstellar spectrum (LIS) generated with GALPROP and constrained by Voyager 1/2 data, ensuring that the input spectrum is free from heliospheric effects up to ~0.5 GeV.

The core of the work is a three‑dimensional solution of the Parker transport equation using a backward‑in‑time stochastic differential equation (SDE) method. The model incorporates a fully three‑dimensional heliospheric magnetic field (HMF) described by the Parker spiral, together with a wavy heliospheric current sheet (HCS) whose geometry follows the tilt angle of the solar magnetic dipole. Both the symmetric diffusion tensor (parallel and perpendicular diffusion coefficients) and the antisymmetric drift tensor are explicitly treated. Parallel diffusion follows a broken power‑law in rigidity with parameters K₀, ν₁, ν₂ and a smoothness parameter, while perpendicular diffusion is anisotropic: K⊥r = 0.02 K∥ and K⊥θ is enhanced toward the poles by a factor a(θ). Drift velocities are split into gradient‑curvature drift (V_GC) and HCS drift (V_HCS), each derived analytically from the magnetic field and the HCS geometry.

Solar‑wind speed is taken as radially outward, with a realistic profile that is constant inside the termination shock (≈400 km s⁻¹ at the equator, ≈800 km s⁻¹ at the poles) and drops sharply beyond it. The model’s free parameters are the average magnetic field strength B₀ (3–8 nT), the HCS tilt angle α (0–75°), the diffusion normalization K₀ (10⁻²³ cm² s⁻¹), the spectral indices ν₁ (0–1) and ν₂ (ν₁–3), and the polar enhancement factor a(θ).

Because a full SDE simulation for each parameter set is computationally prohibitive, the authors train a neural‑network surrogate that maps the six‑dimensional parameter vector to the full propagation‑modulation matrix (energy‑to‑energy transfer). This surrogate reproduces the SDE results with sub‑percent error and enables rapid Bayesian or MCMC scans of the parameter space.

Fitting is performed on monthly AMS‑02 proton and antiproton spectra (≈1–50 GeV) simultaneously. The best‑fit solution yields B₀≈4.5 nT, α≈20°, K₀≈5×10⁻²⁴ cm² s⁻¹, ν₁≈0.5, ν₂≈2.0, and a modest polar enhancement. The model reproduces the observed charge‑sign dependence: antiprotons, which experience opposite drift directions relative to protons, show stronger low‑energy suppression during the minimum, exactly as predicted by the combined gradient‑curvature and HCS drift terms.

The authors discuss limitations: the tilt angle is held constant in time, whereas in reality it evolves over the solar cycle; the model does not yet include electrons/positrons or additional loss processes such as inverse‑Compton scattering. Nevertheless, the work demonstrates that a physically grounded 3D modulation framework, coupled with modern machine‑learning acceleration, can provide a unified description of charge‑dependent solar modulation. This approach paves the way for more precise background modeling in searches for exotic signals (e.g., dark‑matter‑induced antiprotons) and can be extended to other cosmic‑ray species.