On the Contribution of Local Sources to the Galactic Cosmic-Ray Spectrum: An Exact Series Solution for Two-Zone Diffusion

Technical Summary: Contribution of Local Sources to the Galactic Cosmic-Ray Spectrum via Two-Zone Diffusion

Problem Statement
Measurements of Galactic cosmic-ray (CR) proton and helium spectra below the "knee" reveal deviations from simple power laws, including structures in the multi-TeV range. While a potential explanation involves a dominant contribution from one or a few nearby sources, previous studies utilizing homogeneous diffusion models (HDM) have concluded that such a scenario is statistically unlikely. Specifically, Monte Carlo simulations suggest the probability of a nearby source significantly altering the local spectrum is very low (~0.4%). However, these conclusions rely on the assumption that CRs diffuse efficiently immediately upon leaving their sources. Recent observations of extended gamma-ray emission around Galactic accelerators (e.g., Cygnus X, pulsar halos) suggest that transport near sources may be inhibited, characterized by a suppressed diffusion coefficient. This paper investigates how the statistical probability of a dominant local source changes if CRs experience inefficient transport in a localized region surrounding their acceleration sites.

Methodology
To address the computational challenges of modeling CR propagation with a localized slow-diffusion region, the authors derived an exact analytical series solution (Green's function) for a spherical two-zone diffusion problem.

• Model Geometry: The model consists of an inner region ($0 < r < r_1$) with a diffusion coefficient $D_1$ and an outer region ($r_1 < r < r_2$) with a larger coefficient $D_2$. Continuity of density and diffusive flux is enforced at the interface $r_1$.
• Analytical Solution: The authors derived eigenmode expansions for both burst-like and continuous injection scenarios. This analytical kernel allows for rapid calculation of particle distributions, making large-scale Monte Carlo simulations feasible without the inefficiency of resolving small and large spatial scales simultaneously in numerical codes.
• Monte Carlo Simulations: Using the derived Green's function, the authors generated $10^5$ realizations of a Galactic supernova remnant (SNR) population following the "Jelly" model (adopted from Evoli et al.). They compared the flux contribution of the strongest local source (defined as $d < 0.5$ kpc and age $t < 1$ Myr) against the background flux from all other sources at 10 TeV.
• Candidate Analysis: The study also examined specific cataloged nearby candidates (pulsars and Loop I) to determine if known sources could reproduce observed spectral features under different transport assumptions (HDM, Two-Zone Diffusion Model [2ZDM], and Spatially Dependent Propagation [SDP]).

Key Contributions

1. Exact Series Solution: The derivation of a fast, analytical Green's function for a two-zone diffusion problem, enabling efficient Monte Carlo studies of source populations with local transport inhibition.
2. Quantitative Probability Shift: A rigorous re-evaluation of the statistical likelihood of a local source dominating the CR spectrum when near-source transport is inhibited.
3. Model Dependence Demonstration: A systematic comparison showing how the predicted contribution of specific nearby sources (e.g., Vela, Geminga, Monogem) varies drastically depending on the assumed transport model and injection spectrum parameters.

Results

• Probability Enhancement: In the standard HDM, the probability that the strongest local source contributes a flux comparable to the background at 10 TeV is approximately 0.4%. Under the 2ZDM, this probability increases to 1.7% for burst-like injection and 2.2% for continuous injection. While this represents a four-to-five-fold increase, the absolute probability remains low (percent level).
• Physical Mechanism: The inner slow-diffusion region acts as a temporary reservoir, delaying the escape of particles and redistributing them in time and energy. This effect is most pronounced for observers outside the inner region ($\xi > 1$) at timescales comparable to the diffusion time across the inner zone ($\tau \sim 1$).
• Candidate Source Analysis: When applying the 2ZDM to known nearby candidates (using pulsars as SNR tracers), a 10 TeV spectral feature can be reproduced only under specific additional assumptions:
  • Harder Injection Spectrum: The local sources must possess a significantly harder injection index ($\gamma \approx 2.08$) compared to the background population ($\gamma \approx 2.34$). If the local injection index matches the background, the contribution becomes negligible.
  • Transport Model Sensitivity: The predicted flux from a given source varies strongly between HDM, 2ZDM, and SDP models. For instance, under SDP, Geminga and Monogem show different relative contributions compared to 2ZDM, highlighting that the "importance" of a source is not intrinsic but dependent on the assumed magnetic environment and transport properties.

Significance and Claims
The paper concludes that while inhibited near-source transport significantly weakens the statistical argument against local source dominance, it does not eliminate the difficulty. The probability of a dominant local source remains low (~1–2%). Consequently, the interpretation of multi-TeV spectral structures as originating from specific local sources is highly model-dependent.

The authors assert that a nearby source can explain a spectral feature only if favorable conditions are met simultaneously: a specific source age/distance, a hard local injection spectrum, and a specific transport environment. Therefore, identifying a local source as the origin of a spectral feature based solely on spectral fitting is not robust. The paper argues that establishing a definitive local origin requires independent constraints on source injection history, particle transport mechanisms, and local interstellar turbulence, ideally through joint modeling of spectra, composition, anisotropy, and gamma-ray data.