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

arXiv:2601.07649 · astro-ph.HE, astro-ph.SR, physics.space-ph · Submitted 2026-01-12 · Read on arXiv

Listen

Radio episode about this paper

Transcript

Introduction to the show: ident: Astrophysics Radio. Generated commentary on the latest astrophysics papers.

Vera: I'm Vera, and with me are Jocelyn and Subrahmanyan, guest researcher.

Jocelyn: Today's paper: "A Unified Charge-Dependent Modulation Model for AMS-02 Proton and Antiproton Fluxes during Solar Minimum".

Vera: A unified charge-dependent solar modulation model is developed to explain monthly proton and antiproton fluxes measured by AMS-02 during solar quiet periods,

Jocelyn: First, who's behind it and why it matters.

Paper summary: Vera: Welcome back everyone, and I'm so excited to discuss this latest paper today. It’s a really interesting piece because it tackles how we model cosmic rays arriving at Earth, specifically focusing on charge dependence during solar minimum.

Jocelyn: I agree, Vera; this paper seems to be addressing a long-standing puzzle in cosmic ray physics by proposing a unified approach to understanding proton and antiproton fluxes measured by AMS-two <ref:2601.07649#pg0,proton and antiproton fluxes measured by>. It’s really important because it tries to provide a physically reasonable account of charge dependence that goes beyond simpler models.

Subrahmanyan: From a theoretical standpoint, the core idea presented in "A Unified Charge-Dependent Modulation Model for AMS-two Proton and Antiproton Fluxes during Solar Minimum" is to solve the three-dimensional Parker transport equation while including a realistic wavy heliospheric current sheet to handle drift effects self-consistently <ref:2601.07649#pg0,A Unified Charge-Dependent Modulation Model for AMS-02 Proton and Antiproton>. That level of detail in describing the magnetic field structure is crucial for getting accurate results, as suggested by the authors.

Vera: Exactly, and what makes it compelling is how they manage to combine these complex physical components—like the realistic HCS and the solar wind velocity distribution—into one cohesive model to explain both proton and antiproton fluxes. It claims this unified model outperforms simpler phenomenological models that only treat charge dependence in a superficial way.

Jocelyn: And it matters because it uses an input local interstellar spectrum derived from GALPROP, constrained by Voyager data, which allows them to describe the injection of cosmic rays into the Galaxy using a triple broken power-law spectrum defined by spectral indices −ν1, −ν2, and −ν3 <ref:2601.07649#pg0,from GALPROP, constrained by Voyager data>. That input is key to making their modulation results robust.

Subrahmanyan: The researchers also used neural network-based surrogate models, specifically calling them "PropMat," to compute the propagation and modulation matrices efficiently when solving the full three-dimensional Parker’s equation using a backward-in-time Stochastic Differential Equation method. This is a clever computational strategy to handle the complexity of the simulation.

Paper summary: Vera: I'm really interested in what they found regarding antiproton and proton fluxes, because that’s where many of these studies get tripped up when trying to link them together. The paper shows that HELPROP successfully describes the modulation of both protons and antiprotons within this unified framework, which is a significant finding.

Jocelyn: It seems the main result here is that their model incorporates tilt angle information and successfully reproduces the observed variations for both particle species simultaneously, proving that a realistic three-dimensional HCS drift treatment can quantitatively characterize those charge-dependent modulation effects.

Subrahmanyan: The correlation between antiproton and proton fluxes over the full observation period is naturally reproduced by HELPROP, which suggests that the physical mechanism they modeled—the wavy HCS drift—is indeed a valid way to characterize these effects in this context. Furthermore, the best-fit parameters they determined are stated as physically reasonable and their temporal evolution follows sensible trends.

Vera: That’s compelling; so essentially, this paper provides a framework where the complex physics of the heliosphere directly translates into observable differences between proton and antiproton fluxes at Earth during solar minimum. It moves the discussion from just fitting data to actually modeling the underlying transport process more thoroughly.

Jocelyn: And considering their findings on parameter constraints, they constrained injection parameters using both flux measurements and the B/C ratio, showing that for instance, a significant spectral break exists in their best-fit injection parameters with the index changing from two point zero two to two point two one at a rigidity of approximately two point four five GV.

Subrahmanyan: That spectral break they found is actually quite physically meaningful because it aligns with what’s expected from the rigidity spectrum of the Voyager LIS and the high-energy end of AMS-two data, which connects this solar modulation model to the broader cosmic ray propagation picture <ref:2601.07649#pg0>.

Vera: It seems like a lot of work went into setting up these constraints, especially linking those injection parameters back to real observational data points like those flux measurements from AMS-two and the B/C ratio <ref:2601.07649#pg0>. The authors did a very thorough job tying the theory back to the actual measurements they were trying to explain.

Paper summary: Jocelyn: Thinking about the broader impact, if this model is validated as robust, it suggests that we have a much better tool for predicting how secondary particles like antiprotons will behave when solar conditions change during periods of low solar activity. That has implications for understanding background noise in cosmic ray experiments.

Subrahmanyan: I see the potential here for refining our understanding of galactic cosmic-ray propagation because it forces us to consider the complex interplay between interstellar transport and heliospheric modulation more accurately, especially when dealing with charge effects. This work helps build a more complete picture of particle behavior from their source all the way to our detectors.

Vera: So, to wrap up this section, the paper "A Unified Charge-Dependent Modulation Model for AMS-two Proton and Antiproton Fluxes during Solar Minimum" provides a sophisticated way to model charge dependence by using a realistic wavy HCS structure within the Parker transport equation and fitting it against AMS-two data from May two thousand eleven to June two thousand twenty-two <ref:2601.07649#pg0,A Unified Charge-Dependent Modulation Model for AMS-02 Proton and Antiproton>.

Jocelyn: And what this means for us is that we can use these results to better understand how solar activity influences the fluxes of both protons and antiprotons when they pass through the heliosphere. It gives us a clearer picture of the modulation physics at play during quiet periods.

Subrahmanyan: The implication for future work I see is that we should look at how this model handles more extreme scenarios, perhaps periods of high solar activity, to see if those same physical components—like the HCS structure—still hold up under different conditions.

Vera: That sounds like a very logical next step for testing the limits of this specific modeling approach, pushing it beyond just fitting quiet period data.

Jocelyn: It’s exciting because it gives us a concrete mechanism to look for, instead of just seeing flux changes, which is what we've been doing before. This paper offers a new lens through which to view the interplay between solar conditions and cosmic ray arrival at Earth.

Conclusion: Vera: So, we’ve been diving deep into the technical details of this new model explaining proton and antiproton fluxes during solar minimum, and now it's time to wrap up with a look at what exactly this paper is about.

Jocelyn: It’s true, Vera; I think the title itself really sums up the main idea—a unified model that handles charge dependence for both particles. I wonder how they managed to tie everything together so cleanly across different species and charges?

Subrahmanyan: From a theoretical standpoint, the authors tackled that challenge by developing a single framework based on solving the Parker transport equation, incorporating drift effects in a very realistic way. This wasn't just throwing in some simple adjustment; they built the physics into the core equations.

Vera: That’s right; it sounds like they used a sophisticated mathematical tool to capture how those magnetic field structures affect particle movement differently based on their charge and sign. It really shows how essential that level of detail is for getting accurate results from the AMS-two data we observe.

Jocelyn: And looking at the authors, they’ve clearly put a lot of work into this, especially with the computational methods they used to test it; I hope their approach to fitting those complex parameters is sound. I'm curious about what those specific findings actually mean for our understanding of space weather impacts on cosmic rays.

Subrahmanyan: The implication is significant because if this unified model holds up, it gives us a much stronger tool for predicting secondary particle fluxes, like antiprotons, when the sun’s magnetic activity shifts during solar minimum periods. It helps us connect the local solar conditions directly to the particles we measure here on Earth.

Vera: Exactly; it means we can move beyond just seeing flux changes and start understanding the underlying physical reason why those changes happen in a predictable way. It connects our sky-view data directly to heliospheric physics.

Jocelyn: That connection is what gets me; understanding the solar wind’s effect on these particles helps us model how cosmic rays propagate through the entire system, from their injection point out into the heliosphere and finally reaching our instruments.

Subrahmanyan: Precisely; this work reinforces that galactic cosmic ray propagation isn't just about interstellar conditions; it’s intimately tied to the dynamic environment created by the solar activity we observe. It adds another layer of complexity that needs to be accounted for in our grand cosmic picture.

School of Physics and Astronomy, Sun Yat-sen University · School of Science, Shenzhen Campus of Sun Yat-sen University · Institute of Science and Technology for Deep Space Exploration, Nanjing University

astro-ph.HE, astro-ph.SR, physics.space-ph

Submitted: 2026-01-12

Updated: 2026-10-07

Comments: 24 pages, 10 figures

Code: https://github.com/linsujie/Helprop

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 79/100

The gist: A unified charge-dependent solar modulation model is developed to explain monthly proton and antiproton fluxes measured by AMS-02 during solar quiet periods, providing a physically reasonable account

Key concepts

Parker Transport Equation
This is the fundamental equation governing how cosmic ray particles move through the heliosphere. It accounts for diffusion (random scattering), adiabatic energy loss as particles move, and convection driven by the solar wind. The model solves this equation in three dimensions to track particle propagation.
Wavy Heliospheric Current Sheet (HCS)
The HCS is a realistic structure in the heliosphere that causes drift effects on charged particles. By incorporating a 'wavy' structure, the model self-consistently treats how these drifts influence particle transport, which is crucial for accurately modeling charge-dependent modulation.
GALPROP Package
This software package provides the local interstellar spectrum (LIS) used as the starting point for the cosmic ray model. It describes how cosmic rays are injected into the Galaxy via a triple broken power-law spectrum, allowing researchers to constrain and unify the description of different cosmic ray species.
PropMat Surrogate Models
These are artificial neural networks designed to speed up calculations. They learn to quickly map model parameters to transformation matrices for both GALPROP and HELPROP, significantly reducing the computational time needed to solve the complex three-dimensional propagation equations.

Terminology

Summary

A unified charge-dependent solar modulation model is developed to explain monthly proton and antiproton fluxes measured by AMS-02 during solar quiet periods, providing a physically reasonable account of charge-dependent modulation that outperforms simpler phenomenological models.

The gist: The model simultaneously describes the observed proton and antiproton fluxes with physically reasonable parameters, providing a unified account of charge-dependent modulation.

Model Development and Governing Equations

The research develops a unified charge-dependent solar modulation model by solving the three-dimensional Parker transport equation, incorporating a realistic wavy heliospheric current sheet (HCS) to treat drift effects self-consistently. The propagation of cosmic ray (CR) particles in the heliosphere is governed by Parker’s equation, which describes diffusion, adiabatic energy loss, and convection.

The model incorporates several key physical components:

  1. A realistic wavy HCS structure to account for drift effects.

  2. The solar wind velocity distribution (VSW), which varies with heliocentric distance and colatitude during solar minimum.

  3. A detailed diffusion model that includes both symmetric (diffusion) and antisymmetric (drift) components of the diffusion tensor, K = KS + KA, where KA is related to the drift velocity Vd ≡ ∇ × (KAeB).

Input Data and Local Interstellar Spectrum

The model utilizes a local interstellar spectrum (LIS) derived from the GALPROP package, constrained by Voyager data. The LIS serves as the fundamental input that determines the modulated spectra observed at TOA.

Key aspects of the LIS input include:

To achieve a physically consistent model for all these particles, we utilize the GALPROP package.

The injection of CRs into the Galaxy is described by a triple broken power-law spectrum, defined by spectral indices −ν1, −ν2, and −ν3...

The LIS input is crucial because direct measurements for secondary species like antiprotons are unavailable from Voyager data. The GALPROP framework allows for a unified description of modulation for all four elementary cosmic-ray species irrespective of charge or mass.

Computational Framework and Surrogate Models

To enable rapid parameter scans, the study employs neural-network-based surrogate models to compute propagation and modulation matrices efficiently. This addresses the computational cost associated with solving the full three-dimensional Parker’s equation using a backward-in-time Stochastic Differential Equation (SDE) method.

The architecture for these emulators is called PropMat, an artificial neural network designed to learn the mapping from model parameters to transformation matrices for both GALPROP and HELPROP.

  1. For GALPROP, the ANN learns to infer the implicit transformation matrix, M, from a set of LIS spectrum samples using a loss function defined by: LossG = ∑ j (log ∑ i M ij pred ψ i I ψ j O) squared + ∑ i j λ log(M ij dev) 2.

  2. For HELPROP, which tracks a large ensemble of particles, the ANN learns the transformation matrix directly using a loss function defined as: LossH = ∑ i j (log M ij pred M ij O) squared.

Fitting and Results

The study performs a global fit to determine the favored parameter spaces of both Galactic cosmic-ray propagation and solar modulation models by incorporating monthly proton and antiproton flux data from AMS-02, the B/C ratio, and the Voyager LIS proton spectrum. The analysis is confined to solar quiet periods (May 2015 to June 2022) to isolate charge-sign-dependent effects.

The results demonstrate that HELPROP successfully describes the modulation of both protons and antiprotons within a unified framework, outperforming the forcefield approximation which requires distinct modulation potentials for different charge signs.

Our model incorporates tilt angle information and successfully reproduces the observed variations for both both particle species.

The correlation between antiproton and proton fluxes over the full observation period is naturally reproduced by HELPROP, proving that a realistic three-dimensional HCS drift treatment can quantitatively characterize charge-dependent modulation effects. The best-fit parameters are physically reasonable, and their temporal evolution follows sensible trends.

Parameter Constraints and Conclusions

The fitting process constrains the injection and propagation parameters using the B/C ratio and flux measurements. The best-fit values for injection parameters show a significant spectral break, with the index changing from 2.02 to 2.21 at a rigidity of approximately 2.45 GV, which is effectively required by the spectral index of the Voyager LIS and the high-energy end of AMS-02 data. The reduced parameter κ2 for each Bartels rotation exhibits a significant correlation with the tilt angle α, indicating that periods with large α are associated with high solar activity and complex magnetic fields.

Improvements for AI systems

Based on my meticulous review of this scientific paper, here are specific improvements for an AI system designed to analyze or model cosmic ray data, and what the improved system can achieve:

  1. The core improvement lies in integrating the developed surrogate models (PropMat) into a unified machine learning pipeline capable of performing rapid global parameter searches.

  2. The improved AI system will be able to perform high-dimensional, computationally expensive global parameter scans—simultaneously fitting GALPROP (Galactic propagation) and HELPROP (Solar modulation) parameters against multi-wavelength data (AMS-02 fluxes, B/C ratios, Voyager LIS spectra).

Specific Capabilities of the Improved AI System:

  1. The system will execute a single-step inference process for complex astrophysical models. Instead of running GALPROP and HELPROP sequentially for every parameter set in an MCMC chain (which takes minutes per step), the AI will use the trained neural networks (PropMat) to compute the required transformation matrices in milliseconds, drastically reducing computational overhead.

  2. It can rapidly explore the high-dimensional parameter space defined by injection/propagation parameters and modulation parameters simultaneously. This allows for a comprehensive validation of whether a specific set of physical conditions (e.g., specific HCS drift characteristics) is consistent with all observed fluxes across different energy ranges and solar cycle phases (quiet vs. active).

  3. The system will be capable of isolating charge-dependent modulation effects with high fidelity, specifically distinguishing the influence of the realistic 3D HCS drift from simpler phenomenological models like the Force Field Approximation (FFA) by comparing its predictions against those derived from multiple, distinct physical mechanisms (as shown in Figure 6).

  4. It will provide quantitative, physically motivated parameter constraints for Galactic propagation parameters (e.g., spectral breaks in the injection spectrum) directly informed by the Voyager LIS data and AMS-02 B/C ratios, moving beyond simple phenomenological fitting to establish a self-consistent picture of cosmic ray transport from source to Earth.

  5. The AI can identify and quantify temporal trends in modulation parameters (like reduced variance, as shown in Figure 9) across entire solar cycles, enabling the prediction or classification of solar conditions based on the resulting modulation strength—for instance, correlating specific diffusion indices (like the low-energy index 'a') with periods of minimum solar activity.

Abstract

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) from May 2011 to June 2022, excluding the heliospheric magnetic field (HMF) polarity-reversal epoch (mid-2012 to early 2015). 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.

Sources

Related papers