Unified Charge-Dependent Solar Modulation Model

• Unified charge-dependent solar modulation models describe cosmic-ray propagation through deterministic and stochastic treatments of heliospheric electromagnetic fields.
• They combine regular-field approaches and Parker transport equation methods to predict species-specific energy losses and drift effects over the solar cycle.
• The models are crucial for accurately deriving local interstellar spectra and improving dark matter searches by explaining observed charge-sign spectral differences.

Unified Charge-Dependent Solar Modulation Model

A unified charge-dependent solar modulation model describes cosmic-ray propagation and energy loss in the heliosphere via physically-motivated electromagnetic interactions, incorporating explicit charge-sign, time, and rigidity dependence. This approach includes deterministic and stochastic treatments—either via the regular heliospheric fields or the full Parker transport equation with charge-dependent drift—yielding time-resolved predictions for positive and negative cosmic-ray species that match observed charge-dependent modulation phenomena across the solar cycle. Such models are critical for interpreting fluxes measured at Earth, validating local interstellar spectra (LIS), and informing dark matter searches.

1. Physical Basis: Heliospheric Electromagnetic Field and Charge-Dependent Modulation

The unified charge-dependent framework builds on the properties of the large-scale Parker spiral magnetic field, embedded in the radially expanding solar wind. In the regular-field (deterministic) model (Lipari, 2014), the heliosphere is assumed to have negligible field turbulence, and particles spiral through a magnetized plasma at velocity $V_{sw}$. The solar wind frame is ideal-conducting, so transforming to the Sun’s frame yields a stationary, curl-free motional electric field: $$\mathbf{E}(\mathbf{r}) = -\frac{\mathbf{V}_{sw}}{c} \times \mathbf{B}(\mathbf{r})$$ The propagation of a particle with charge $q$ through this field incurs energy losses purely via interaction with this large-scale electric field, with the loss: $$\Delta E = -q \int_{T} d\mathbf{\ell} \cdot \mathbf{E}$$ Particle trajectories depend on the sign of $q$ relative to the solar magnetic polarity $A$, causing charge- and polarity-dependent energy loss: for $qA>0$ losses are constant; for $qA<0$ losses are reduced and depend on the heliospheric current sheet (HCS) tilt angle $\alpha$.

2. The Parker Transport Equation and Charge-Dependent Drifts

The conventional stochastic approach describes cosmic-ray transport via the Parker equation (Potgieter, 2013): $$\frac{\partial f}{\partial t} = -(\mathbf{V}_{sw} + \langle \mathbf{v}_d \rangle) \cdot \nabla f + \nabla \cdot (\mathbf{K}_s \cdot \nabla f) + \frac{1}{3} (\nabla \cdot \mathbf{V}_{sw}) \frac{\partial f}{\partial \ln p}$$ where $f$ is the phase-space density, $\mathbf{K}_s$ the diffusion tensor, and $⟨\mathbf{v}_d⟩$ the mean drift velocity (gradient, curvature, HCS drifts). Charge-sign dependence enters via the antisymmetric part of the diffusion tensor, producing $$⟨ \mathbf{v}_d \rangle = \nabla \times (K_d \, \mathbf{e}_B)$$, where $K_d$ scales as $\beta P / (3 B)$ and flips sign with $qA$. The HCS tilt angle $\alpha$ globally modifies drift patterns, leading to time- and charge-dependent access to Earth.

3. Analytic and Semi-Analytic Unified Modulation Potentials

Unified modulation models such as the time-, charge-, and rigidity-dependent force-field approximation (FFA) (Cholis et al., 2015, Cholis et al., 2022, Cholis et al., 2020, Duan et al., 16 Jun 2025, Zhao et al., 14 Apr 2025) generalize the standard parameterization by expressing the modulation potential $\Phi$ as: $$\Phi(R, t) = \phi_0 \frac{|B_{tot}(t)|}{4\,\mathrm{nT}} + \phi_1\, H[-qA(t)] \frac{|B_{tot}(t)|}{4\,\mathrm{nT}} \frac{1 + (R/R_0)^2}{\beta (R/R_0)^3} \left( \frac{\alpha(t)}{\pi/2} \right)^4$$ Here $H[-qA]$ is a Heaviside function encoding charge-polartity selection, $R$ the rigidity, $\beta$ the velocity, and $\phi_0, \phi_1, R_0$ model parameters fit to data. This framework reproduces the observed alternation of modulation for $qA>0$ (pole access, weak modulation) and $qA<0$ (sheet access, strong modulation), unifying protons, antiprotons, electrons, and positrons in a single analytic prescription.

4. Numerical Solution Strategies and Surrogate Modeling Techniques

Cosmic-ray modulation is solved numerically by discretizing the full Parker transport equation or by backward-in-time stochastic differential equations (SDEs) (Kappl, 2015, Adriani et al., 2023, Aslam et al., 2020). The SDE approach tracks pseudo-particle trajectories, integrating drift and diffusion with charge-dependent boundary and initial conditions. Modern methodologies leverage precomputed modulation matrices, with machine-learning surrogate models trained to reproduce the outputs of GALPROP and HELPROP (modulation codes) for global parameter scans (Zhang et al., 12 Jan 2026). These surrogate models can capture dependencies on $B$, $\alpha$, diffusion normalization $K_0$, rigidity indices $a$, $b$, etc., enabling sub-percent-accurate, rapid evaluations for large data sets and global fits.

Model Type	Charge Dependence Mechanism	Free Parameters Example
Deterministic (EM field only)	Path-dependent energy loss $\Delta E(q,A,\alpha)$	$B_0$, $\Omega$, $r_0$, $A$, $\alpha$
Parker/SDE (drift-diffusion)	Drift tensor: sign of coefficient, polarity via HCS	$K_0$, $a$, $b$, $A$, $\alpha$, $K_{A0}$
Analytic Unified FFA	Analytic $\Phi(R, q, t, A)$, Heaviside drift switch	$\phi_0$, $\phi_1$, $R_0$

This table summarizes the three classes of unified models and their parameters.

5. Empirical Validation and Observed Charge-Sign Effects

Solar modulation models are validated by fitting time-resolved fluxes from AMS-02, PAMELA, CALET, BESS, Voyager, and neutron monitor data (Adriani et al., 2023, Felice et al., 2016, Zhao et al., 14 Apr 2025, Duan et al., 16 Jun 2025, Zhang et al., 12 Jan 2026). Observed effects include:

• Systematic spectral differences between protons and antiprotons, electrons and positrons, alternating with solar cycle polarity.
• For $qA>0$ epochs (e.g., $A>0$ for protons), modulation is minimal at low rigidities; for $qA<0$ (e.g., $A>0$ for antiprotons/electrons), tilt-dependent modulation produces decreased fluxes and larger time variation.
• Simultaneous fits to multi-species data (e.g., proton, antiproton, He, C, O, B/C) constrain modulation parameters, with best fits yielding modulation potentials $0.2$–$0.8$ GV for solar minimum periods, matching observed flux shifts.
• In the deterministic EM-field model, the energy shift $\Delta E$ matches the FFA parameter $\phi$ for fixed $qA$; explicit drift models reproduce the observed anticorrelation between count rates and $\alpha$ (Adriani et al., 2023), with electrons showing nearly fourfold larger amplitude than protons.

6. Limitations, Extensions, and Applications

Unified models, whether analytic, deterministic, or numerical, have several constraints:

• Rigid single-parameter FFA fails during high solar-activity epochs when rigidity- and charge-dependent effects are pronounced (Zhao et al., 14 Apr 2025). Energy-dependent diffusion and time-varying drift suppression are required for accurate prediction at high activity.
• Full Parker–SDE codes capture stochastic and spatially dependent effects, but are computationally intensive; surrogate modeling via neural nets enables practical global fits (Zhang et al., 12 Jan 2026).
• Drift suppression at very low rigidity and during field reversals (polarity flip epochs) must be explicitly handled; otherwise systematic discrepancies arise, especially for electrons (Cholis et al., 2022).
• A unified, charge-dependent modulation scheme, calibrated against multi-species and multi-epoch data, is essential for robust LIS determination and for discriminating possible dark-matter signals in antiproton or positron channels (Duan et al., 16 Jun 2025).

Extensions include non-parametric LIS determination with high-resolution AMS-02 data (Zhu et al., 2020), direct propagation of uncertainty via nuisance parameters (Duan et al., 16 Jun 2025), and application to future missions (Solar Orbiter, next-generation AMS) for predicting the evolution of charge-dependent modulation throughout the solar cycle.

7. Physical Interpretation and Broader Impact

Unified charge-dependent solar modulation models clarify the physical origin of the force-field approximation, linking the observed effective modulation potential to large-scale heliospheric EM fields and charge-dependent guiding-center drifts. They reconcile the previously ad hoc drift-convection terms with underlying magnetohydrodynamic structure, naturally explain the observed charge- and cycle-dependent spectral differences, and provide rigorous foundations for the derivation of the LIS for all charge species needed for galactic propagation studies. By enabling consistent demodulation of time-resolved cosmic-ray spectra, these models are essential for quantifying secondary production, constraining pulsar and dark-matter contributions, and improving the accuracy of indirect astrophysical searches (Lipari, 2014, Cholis et al., 2015, Zhang et al., 12 Jan 2026, Zhao et al., 14 Apr 2025).