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A Boundary-Consistent Two-Zone Electron Kernel for Distant Pulsar Contributions to Positron Flux and Anisotropy

This paper presents a semi-analytical two-zone diffusion kernel that resolves numerical instabilities to demonstrate how distant pulsars significantly contribute to the local GeV positron flux and anisotropy, confirming that Geminga-like slow-diffusion halos remain consistent with current AMS-02 data despite the dominance of distant sources in the fitted component.

Original authors: Yiwei Bao, Jie-Shuang Wang, Hao Zhou

Published 2026-06-26

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CC0 1.0

Original authors: Yiwei Bao, Jie-Shuang Wang, Hao Zhou

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: Who is Sending Us Positrons?

Imagine the Earth is sitting in a vast, dark ocean of space. High-energy particles called positrons (the antimatter twins of electrons) are constantly raining down on us. Scientists have been trying to figure out: Where are these particles coming from?

For a long time, there were two main theories:

The "Dark Matter" Theory: Invisible, mysterious particles colliding and creating positrons.
The "Pulsar" Theory: Dying, spinning stars (pulsars) acting like cosmic sprinklers, shooting out positrons.

This paper focuses on the Pulsar Theory. Specifically, it asks: Do these positrons come from the nearest stars to us, or do they travel from very far away?

The Problem: The "Slow Zone" vs. The "Fast Highway"

Usually, scientists assume particles travel through space at a steady, fast speed (like cars on a highway). However, recent observations suggest that right around a pulsar, there is a "bubble" where particles get stuck and move very slowly (like cars stuck in heavy traffic).

The authors built a new mathematical model to handle this. They call it a "Two-Zone Kernel."

Zone 1 (The Bubble): A small area right next to the pulsar where particles move slowly.
Zone 2 (The Interstellar Medium): The rest of space where particles move fast.

They wanted to see if this "slow traffic" near the source changes the answer to our big question: Are the positrons coming from nearby or far away?

The Analogy: The Coffee Shop and the Commute

Imagine you are a barista (the pulsar) making coffee (positrons) and throwing it out the door.

The "Slow Zone" (The Bubble): Just outside your door, there is a thick fog. It takes a long time for the coffee to get through the fog. While it's stuck in the fog, the coffee gets cold (this is cooling).
The "Fast Zone" (Space): Once the coffee gets past the fog, it flies through the air at high speed to reach you (Earth).

The Old Way of Thinking: Scientists used to assume the coffee flew instantly from the door to you. If you are far away, the coffee would arrive hot. If you are close, it arrives hot.

The New Way (This Paper): The authors realized that if the coffee has to wait in the fog first, it gets colder before it even starts its fast journey.

Nearby Pulsars: The coffee doesn't have to travel far after the fog, so it stays relatively hot.
Distant Pulsars: The coffee has to travel a long way after the fog. By the time it reaches you, it has cooled down a lot.

The Surprising Discovery: The "Distant Crowd" Wins

The authors did some complex math to count how many pulsars are nearby versus how many are far away.

The Logic: There are very few pulsars right next to us (in our "Local Bubble"). But there are thousands of pulsars further away.
The Result: Even though the distant coffee gets colder, there are so many distant sources that their combined contribution is actually huge.

The Numbers:

For medium-energy particles (10–100 GeV), the authors found that 37% to 47% of the positrons hitting Earth could be coming from pulsars more than 1,000 light-years away.
Even though the "slow bubble" near the pulsar makes the distant particles slightly weaker, the sheer number of distant stars means they still dominate the mix.

The "Local Bubble" Twist:
The authors also noted that our immediate neighborhood (within 100 light-years) is actually empty of young pulsars. It's a "bubble" left over from ancient supernovas. So, the closest pulsars aren't actually that close. This forces the model to rely even more on the "distant crowd."

The "Geminga" Check

There is a famous pulsar called Geminga that is relatively close to us. Some scientists thought Geminga might be the only source of the high-energy positrons we see.

The authors tested this. They asked: Can Geminga explain the data without breaking the rules?

The Rule: If Geminga were the only source, the positrons would be coming from one specific direction, creating a "wind" (anisotropy) that we should be able to measure.
The Finding: The data shows no strong wind. The positrons are coming from all directions equally.
Conclusion: Geminga can exist with a "slow bubble" around it, but it cannot be the only source. The "distant crowd" of other pulsars is necessary to smooth out the flow and hide the wind.

The Bottom Line

It's not just the neighbors: You don't need to look at the stars right next door to explain the positrons. A huge number of stars from far away (up to thousands of light-years) are contributing significantly.
The "Slow Bubble" doesn't help the distant ones: Having a slow zone near a pulsar actually makes the distant particles weaker (because they cool down while waiting in the fog). The fact that distant stars still contribute so much is purely because there are so many of them.
We can't measure the bubble size yet: While the model fits the data well, the authors admit that current measurements aren't precise enough to tell us exactly how big the "slow bubble" around a pulsar is. We need more data (like looking at the shape of the gamma-ray halos around these stars) to solve that specific puzzle.

In short: The positrons hitting us are a mix of a few nearby stars and a massive "crowd" of distant stars, all filtered through a "slow traffic" zone near their homes. The crowd wins, but we still need more clues to measure exactly how slow that traffic is.

Technical Summary: A Boundary-Consistent Two-Zone Electron Kernel for Distant Pulsar Contributions to Positron Flux and Anisotropy

Problem Statement
The local cosmic-ray electron and positron spectra serve as precision probes of nearby high-energy accelerators, particularly pulsars. While mature pulsars like Geminga and Monogem are natural candidates for the TeV-scale lepton flux, a critical question remains: must the local lepton flux be dominated by the nearest pulsars, or can more distant pulsars contribute collectively? This inquiry is complicated by two factors: (1) rapid radiative cooling (synchrotron and inverse-Compton) limits the propagation horizon of high-energy particles, and (2) anisotropy limits from Fermi-LAT and AMS-02 constrain models where a single nearby source dominates too broadly. Previous studies have suggested that distant sources can reproduce low-energy electrons, but a stable, semi-analytical framework to quantify the geometric source-count factor in the presence of near-source slow diffusion has been lacking. Furthermore, the innermost few hundred parsecs of the Galaxy (the Local Bubble) present a physical discontinuity where a smooth continuous pulsar density is suspect due to low molecular gas density and pulsar natal kicks.

Methodology
The authors develop a semi-analytical series solution for electron and positron propagation in a spherical two-zone diffusion model.

Two-Zone Kernel: The model defines a slow-diffusion region of radius $R_1$ surrounding the source, where the diffusion coefficient is $D_1(E)$ , and a standard interstellar region ( $R_1 < r < R_2$ ) with coefficient $D_2(E)$ . The coefficients share the same energy dependence ( $D \propto f_D(E)$ ), allowing for analytical eigenfunction solutions.
Boundary Conditions: The solution enforces continuity of density and diffusive flux at $R_1$ and an absorbing boundary at $R_2$ (set to 20 kpc). The formulation avoids the oscillatory cancellations inherent in direct integral evaluations of two-zone models, preserving sharp radiative cooling boundaries.
Cooling: Radiative losses (synchrotron and Klein–Nishina inverse-Compton on CMB, infrared, and optical fields) are treated via energy characteristics, linking injection energy to source age.
Population Modeling: The kernel is applied to a continuous disk population of pulsars. The authors impose an empty cavity for $R < 0.1$ kpc to account for the Local Bubble and pulsar proper motions.
Statistical Analysis: A "solver-in-the-loop" Markov Chain Monte Carlo (MCMC) analysis is performed to fit the model to AMS-02 positron flux data, CALET all-electron dipole anisotropy measurements, and Fermi-LAT anisotropy upper limits. The fit includes continuous spin-down injection histories, a broken-power-law pair injection spectrum, and profiles over non-negative linear amplitudes for a secondary-like positron basis and a smooth power-law background.

Key Contributions

Stable Series Solution: The paper provides a numerically stable, semi-analytical series solution for the two-zone diffusion problem with radiative cooling. This method is computationally efficient and avoids the numerical instabilities of brute-force finite-volume methods or oscillatory integrals, particularly in the middle-distance regime.
Geometric Quantification: The study quantifies how the "shell factor" (source count vs. distance) interacts with propagation kernels. It demonstrates that while a 3D uniform distribution is dominated by kiloparsec-scale shells at 10–100 GeV, a disk geometry ( $h=0.2$ kpc) shifts the contribution inward, yet sources beyond 1 kpc still supply 37–47% of the 10–100 GeV flux.
Role of the Slow Halo: The authors clarify that the inclusion of a slow-diffusion halo does not enhance the flux from distant sources relative to a one-zone model; rather, it introduces an energy-dependent cooling penalty that reduces the absolute flux from all distances, with a stronger suppression at higher energies.
Anisotropy Constraints: The work integrates anisotropy constraints directly into the population fit, using CALET dipole amplitudes and Fermi-LAT upper limits to test the viability of common-halo parameters.

Results

Flux Decomposition: In the fitted continuous pulsar population (excluding $R < 0.1$ kpc), the contribution from sources beyond 0.3 kpc dominates the local 0.1–0.3 kpc annulus at all energies from 10 GeV to the TeV scale. At 10 GeV, the $R > 0.3$ kpc contribution is roughly an order of magnitude larger than the local annulus.
Halo Parameters: The MCMC analysis yields a posterior distribution for the slow-halo radius $R_1$ centered at $0.0709^{+0.0510}_{-0.0407}$ kpc ( $\approx 71$ pc) and a diffusion coefficient $D_1(100 \text{ TeV}) \approx 2.08 \times 10^{28}$ cm $^2$ s $^{-1}$ . This result is consistent with TeV halo morphology studies of Geminga and Monogem.
Anisotropy Compatibility: The fitted model, with the derived halo parameters, remains compatible with current anisotropy limits (CALET and Fermi-LAT). The dipole amplitude predictions fall within the 95% upper limits.
Energy Dependence: While distant sources dominate the sub-TeV flux, radiative cooling causes the contribution to localize near the TeV scale. At 1 TeV, approximately 90% of the flux in the disk model comes from within 1 kpc.

Significance and Claims
The paper concludes that a collective contribution from distant pulsars (beyond the immediate Local Bubble neighborhood) is a plausible and dominant contributor to the positron and electron flux below a few hundred GeV. The fitted pulsar component is not dominated by the nearest sources but by a population extending beyond 0.3 kpc.

However, the authors maintain a modest stance regarding the uniqueness of their findings:

The flux and anisotropy data alone do not uniquely determine the halo size ( $R_1$ ) or diffusion coefficient ( $D_1$ ); degeneracies exist with injection shapes, background modeling, release histories, and catalog incompleteness.
The derived "Geminga-scale" halo size is a conditional constraint based on the specific population model and data set used, not a definitive measurement of physical sub-diffusion halos.
To move from a consistency test to a determination of the slow-diffusion region, additional observables are required, specifically TeV inverse-Compton surface-brightness profiles of individual halos, detailed pulsar population data (proper motions, birth sites), and full vector anisotropy likelihoods.

The work serves as a propagation-controlled demonstration that current data can test common-halo parameter space, providing a stable kernel for future population inferences while highlighting the necessity of multi-messenger inputs for a complete solution.