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

This paper presents a unified charge-dependent solar modulation model that successfully describes time-resolved AMS-02 proton and antiproton fluxes during solar minimum by solving the 3D Parker transport equation with realistic drift effects and utilizing neural-network surrogates for efficient parameter fitting.

The Gist

The Big Picture: A Cosmic Traffic Jam

Imagine the universe is a giant, bustling city. Galactic Cosmic Rays are like millions of tiny, high-speed delivery trucks (particles) zooming through the galaxy. They come from distant explosions (supernovas) and carry energy.

When these trucks try to enter our solar system, they hit a massive, invisible traffic circle controlled by the Sun. This is the Heliosphere. The Sun isn't just a ball of fire; it's also a giant magnet blowing a constant wind (the Solar Wind). This wind creates a magnetic "fence" that pushes back against the incoming cosmic trucks.

The problem? The Sun's magnetic fence is messy. It twists, turns, and has a wavy "equator" called the Heliospheric Current Sheet (HCS). Because of this messiness, the traffic behaves differently depending on the "charge" of the truck:

• Protons (positive charge) get pushed one way.
• Antiprotons (negative charge) get pushed the opposite way.

For a long time, scientists used a simplified map (called the "Force Field Approximation") to predict how many trucks would get through. But that map was like using a flat, 2D drawing to navigate a 3D rollercoaster. It worked okay for some trucks, but it failed to explain why positive and negative trucks arrived in different numbers and at different times.

The Solution: A 3D GPS with a Smart Assistant

This paper introduces a new, highly sophisticated model called HELPROP. Think of it as a high-tech, 3D GPS that simulates the exact path of every single cosmic truck as it navigates the Sun's wavy magnetic maze.

The "Wavy Skirt" Analogy:
Imagine the Sun's magnetic equator (the HCS) is a giant, wavy skirt worn by a ballerina (the Sun). As the Sun spins, this skirt flaps and waves.

• Positive trucks (Protons) tend to stay away from the skirt and take a shortcut over the "poles" (the top and bottom of the solar system).
• Negative trucks (Antiprotons) get stuck in the folds of the skirt, drifting along the waves.

Because the skirt's shape changes over time (getting flatter or wavier), the number of trucks that make it to Earth changes. The old models couldn't explain why the negative trucks were more sensitive to the skirt's shape than the positive ones. The new model does.

The "Magic Trick": Neural Networks as Speedsters

Here is the tricky part: Running this 3D simulation for every single truck takes a long time. If you tried to calculate the path for millions of trucks for every month of data, it would take a supercomputer years to finish.

To solve this, the authors used Artificial Intelligence (Neural Networks).

• The Analogy: Imagine you are trying to learn how to bake a cake. Instead of baking 10,000 cakes from scratch to learn the perfect recipe, you bake 20,000 cakes, take photos of the results, and train a robot to look at the ingredients and instantly guess the outcome.
• The Result: The authors trained two "robots" (called PropMat). One robot learned how the galaxy shapes the particles, and the other learned how the Sun's magnetic field slows them down.
• The Payoff: Instead of taking hours to calculate a result, these robots do it in milliseconds. This allowed the scientists to test millions of scenarios quickly to find the one that perfectly matches reality.

What They Found

They fed their new model data from two sources:

1. Voyager Probes: The only human-made objects that have left the solar system, giving us a view of the "traffic" before it hits the Sun's fence.
2. AMS-02: A super-precise detector on the International Space Station that counts the trucks arriving at Earth.

The Discovery:
The new model successfully explained the data for both protons and antiprotons at the same time using a single set of rules.

• It showed that the "wavy skirt" (the tilt of the Sun's magnetic field) is the main reason why antiproton numbers fluctuate more than proton numbers.
• It proved that you don't need two different rules for positive and negative particles; you just need a realistic 3D map of the magnetic field.

Why Does This Matter?

This is a big deal for two reasons:

1. Cleaner Data: By understanding exactly how the Sun messes with the cosmic rays, scientists can "subtract" the Sun's effect. This leaves a cleaner picture of what's happening in deep space.
2. Hunting for Dark Matter: Scientists are looking for strange signals in cosmic rays that might be caused by Dark Matter (the invisible stuff that holds galaxies together). If we don't perfectly understand the Sun's "traffic control," we might mistake a solar glitch for a Dark Matter signal. This new model clears the fog, making it much easier to spot the real anomalies.

In a Nutshell

The authors built a super-fast, AI-powered simulator that treats the Sun's magnetic field like a complex, wavy 3D maze. They proved that this maze treats positive and negative particles differently, and by accounting for this, they can finally explain the cosmic ray traffic we see on Earth without needing to invent fake rules.

Technical Summary

1. Problem Statement

Galactic Cosmic Rays (GCRs) observed at the top of Earth's atmosphere are modulated by the heliospheric magnetic field (HMF) and solar wind. This modulation process is charge-dependent due to particle drifts caused by the gradient and curvature of the HMF and the wavy Heliospheric Current Sheet (HCS).

• The Challenge: Traditional phenomenological models, such as the Force-Field Approximation (FFA), assign distinct modulation potentials to different charge signs to fit data. However, the FFA does not elucidate the underlying physical drift mechanisms and fails to naturally predict the differences in modulation between protons and antiprotons.
• The Gap: While 3D stochastic models exist, they are computationally expensive, making global parameter scans (fitting both Galactic propagation and solar modulation simultaneously against large datasets like AMS-02 and Voyager) prohibitively slow.
• Goal: Develop a unified, physically consistent model that simultaneously describes proton and antiproton fluxes using realistic 3D drift physics, overcoming computational bottlenecks to perform rigorous global fits.

2. Methodology

The authors propose a multi-stage framework combining physical modeling, machine learning surrogates, and statistical inference.

A. Physical Modeling (HELPROP & GALPROP)

• Solar Modulation (HELPROP): The authors solve the 3D Parker transport equation using a backward-in-time Stochastic Differential Equation (SDE) method.
  • HMF Structure: Incorporates a realistic "wavy" HCS (modeled as a ballerina's skirt) and a Parker spiral magnetic field.
  • Drift Mechanisms: Explicitly calculates gradient-curvature drifts and HCS drifts. The HCS drift is parameterized based on the distance to the current sheet, allowing for charge-dependent transport (e.g., antiprotons drift inward along the HCS during $A>0$ polarity, while protons drift outward).
  • Solar Wind: Uses a velocity profile varying with latitude and radial distance, consistent with solar minimum observations.
• Galactic Propagation (GALPROP): Uses the GALPROP package to model the Local Interstellar Spectrum (LIS).
  • Injection: Triple broken power-law spectrum.
  • Propagation: Includes diffusion, reacceleration (Alfvén speed), and convection.
  • Constraints: The LIS is constrained by Voyager 1/2 data (for low energies) and AMS-02 Boron-to-Carbon (B/C) ratios.

B. Machine Learning Surrogate Models (PropMat)

To address the computational cost of running GALPROP and HELPROP millions of times for MCMC sampling, the authors developed PropMat, a neural network-based surrogate framework.

• Architecture: An encoder-decoder structure using Multi-Layer Perceptrons (MLP) and controlled multi-scale block operations (PixelShuffle) to generate transformation matrices.
• Function:
  • GALPROP Surrogate: Learns the mapping from propagation parameters to the LIS transformation matrix. It predicts the deviation from a baseline matrix to ensure stability.
  • HELPROP Surrogate: Learns the mapping from modulation parameters (magnetic field strength $B_0$, tilt angle $\alpha$, diffusion coefficients) to the heliospheric modulation matrix.
• Performance: Trained on ~25,000 (GALPROP) and ~35,000 (HELPROP) samples. The models achieve prediction errors of <1% (GALPROP) and <2% (HELPROP) relative to the true spectra, with inference times reduced from minutes to milliseconds ($O(ms)$).

C. Global Fitting Strategy

• Data: Monthly proton and antiproton fluxes from AMS-02 (May 2011 – June 2022), Voyager LIS proton data, and AMS-02 B/C ratios.
• Period: Focused on the "solar quiet period" (May 2015 – June 2022) to avoid complications from magnetic field reversals.
• Optimization: Used Markov Chain Monte Carlo (MCMC) to sample Galactic propagation parameters. Modulation parameters for each Bartels rotation were optimized as nuisance parameters within the likelihood calculation to reduce dimensionality.

3. Key Contributions

1. Unified Charge-Dependent Model: The paper presents the first unified framework that simultaneously fits proton and antiproton fluxes using a single set of physical parameters, without assigning arbitrary, distinct modulation potentials for each charge sign.
2. Realistic 3D Drift Implementation: By incorporating a realistic wavy HCS and 3D drift effects, the model naturally reproduces the observed charge-sign dependence (e.g., why antiprotons are more sensitive to the HCS tilt angle than protons).
3. Efficient Surrogate Modeling: The development of PropMat enables rapid, high-dimensional parameter scans that were previously computationally infeasible, setting a new standard for efficiency in cosmic ray modulation studies.
4. Self-Consistent LIS: The study derives the LIS for both primary (protons) and secondary (antiprotons) particles self-consistently from the same Galactic propagation model constrained by Voyager and AMS-02 data.

4. Key Results

• Flux Reproduction: The model successfully reproduces the time-resolved monthly fluxes of both protons and antiprotons during the solar minimum.
• Charge-Dependent Behavior:
  • Protons: During the $A>0$ polarity period studied, protons primarily enter via polar regions and are relatively insensitive to the HCS tilt angle ($\alpha$).
  • Antiprotons: Antiprotons are transported mainly via HCS drift. The model correctly predicts that a larger tilt angle extends the drift path, increasing particle losses and lowering the observed flux.
  • Comparison with FFA: The Force-Field Approximation fails to capture the correlation between proton and antiproton flux variations, requiring different potentials for each. The HELPROP model captures this correlation naturally.
• Parameter Constraints:
  • Galactic Propagation: The diffusion slope $\delta \approx 0.40$ (close to Kolmogorov $1/3$), diffusion halo size $z_h \approx 7.2$ kpc, and Alfvén speed $v_A \approx 10.6$ km/s are well-constrained.
  • Injection: A significant spectral break is identified in the injection spectrum at a rigidity of $\approx 2.45$ GV, with spectral indices changing from $\nu_1 \approx 2.02$ to $\nu_2 \approx 2.21$.
  • Modulation: The diffusion coefficient normalization $K_0$ and indices $a, b$ vary physically with solar activity, showing reasonable trends with the tilt angle and magnetic field strength.
• Goodness of Fit: The reduced $\chi^2$ is excellent (0.58) for the solar quiet period, confirming the model's validity under stable heliospheric conditions.

5. Significance

• Dark Matter Searches: By providing a precise, unified description of the background fluxes (protons and antiprotons) and their charge-dependent modulation, this model reduces systematic uncertainties in the search for anomalous signals (e.g., dark matter annihilation) in the GeV energy range.
• Methodological Advance: The integration of physics-based SDE solvers with deep learning surrogates offers a scalable blueprint for future multi-messenger astrophysics problems where complex transport equations must be solved repeatedly.
• Physical Insight: The results confirm that the wavy HCS drift is the dominant mechanism explaining the differences in proton and antiproton modulation, validating the necessity of 3D drift models over simplified 1D approximations.
• Future Applications: The framework is designed to be extended to electrons/positrons and to periods of complex solar activity (solar maximum), promising a comprehensive understanding of cosmic ray transport across the entire solar cycle.