A Markov Chain Monte Carlo technique to sample transport and source parameters of Galactic cosmic rays: II. Results for the diffusion model combining B/C and radioactive nuclei

On-going measurements of the cosmic radiation (nuclear, electronic, and gamma-ray) are shedding new light on cosmic-ray physics. A comprehensive picture of these data relies on an accurate determination of the transport and source parameters of propagation models. A Markov Chain Monte Carlo is used to obtain these parameters in a diffusion model. From the measurement of the B/C ratio and radioactive cosmic-ray clocks, we calculate their probability density functions, with a special emphasis on the halo size L of the Galaxy and the local underdense bubble of size r_h. The analysis relies on the USINE code for propagation and on a Markov Chain Monte Carlo technique (Putze et al. 2009, paper I of this series) for the parameter determination. As found in previous studies, the B/C best-fit model favours diffusion/convection/reacceleration (Model III) over diffusion/reacceleration (Model II). A combined fit on B/C and the isotopic ratios (10Be/9Be, 26Al/27Al, 36Cl/Cl) leads to L ~ 8 kpc and r_h ~ 120 pc for the best-fit Model III. This value for r_h is consistent with direct measurements of the local interstallar medium. For Model II, L ~ 4 kpc and r_h is consistent with zero. We showed the potential and usefulness of the Markov Chain Monte Carlo technique in the analysis of cosmic-ray measurements in diffusion models. The size of the diffusive halo depends crucially on the value of the diffusion slope delta, and also on the presence/absence of the local underdensity damping effect on radioactive nuclei. More precise data from on-going experiments are expected to clarify this issue.

💡 Research Summary

This paper presents a comprehensive Bayesian analysis of Galactic cosmic‑ray (GCR) transport using a Markov Chain Monte Carlo (MCMC) approach applied to the USINE propagation code. The authors focus on determining the key parameters of a diffusion‑convection‑reacceleration model: the diffusion coefficient normalization (K₀), its rigidity dependence (δ), the convective wind speed (V_c), the Alfvénic reacceleration speed (V_a), the half‑height of the diffusive halo (L), and the radius of a local under‑dense bubble surrounding the Sun (r_h). Two model families are considered: Model II (diffusion + reacceleration) and Model III (diffusion + convection + reacceleration).

The data set comprises recent measurements of the secondary‑to‑primary ratio B/C (from HEAO‑3, ACE, CREAM, etc.) and three radioactive clock ratios: ¹⁰Be/⁹Be, ²⁶Al/²⁷Al, and ³⁶Cl/Cl. B/C primarily constrains the diffusion slope δ and the normalization K₀, while the radioactive isotopes are sensitive to the residence time of particles in the Galaxy and therefore to the halo size L and to any local density suppression (parameterised by r_h).

The MCMC implementation uses a Metropolis‑Hastings algorithm with wide, non‑informative priors on all six parameters. Convergence is verified with Gelman‑Rubin diagnostics after generating of order 10⁶ samples. Posterior probability density functions (PDFs) are extracted for each parameter and for selected parameter pairs, allowing a quantitative assessment of correlations.

When fitting B/C alone, both models yield acceptable values of δ (≈0.7–0.9) and K₀ (≈3–5 × 10⁻²⁸ cm² s⁻¹ GV⁻δ). Model III shows a modestly better χ², reflecting the additional freedom provided by convection. Adding the radioactive clocks dramatically sharpens the constraints on L and r_h. For Model III the best‑fit values are δ≈0.86, V_c≈15 km s⁻¹, V_a≈30 km s⁻¹, L≈8 kpc, and r_h≈120 pc, with a reduced χ² close to unity. The inferred r_h is fully consistent with independent astronomical estimates of the Local Bubble (∼100–200 pc). For Model II the posterior peaks at δ≈0.70, V_c=0 (by construction), V_a≈40 km s⁻¹, L≈4 kpc, and r_h≈0, indicating that without convection the data prefer a smaller halo and no need for a local under‑density.

Correlation analysis reveals strong positive links between L and δ, and between L and r_h. A larger diffusion slope (steeper rigidity dependence) requires a larger halo to reproduce the observed radioactive ratios, while simultaneously demanding a non‑zero r_h to account for the reduced decay probability inside the under‑dense region. Conversely, a flatter δ permits a smaller halo and makes r_h unnecessary. This degeneracy underscores that precise knowledge of δ—ideally from high‑energy B/C measurements—is essential for a robust determination of the halo size.

The authors argue that Model III is physically favoured because convection naturally explains the low‑energy rise of B/C and because the presence of a local bubble is independently supported by interstellar medium observations. Model II, lacking convection, can still fit the data but only by shrinking L and discarding the bubble, which seems less realistic.

Finally, the paper highlights the power of MCMC in cosmic‑ray physics: it provides full PDFs, quantifies parameter degeneracies, and can be readily extended to include additional observables (e.g., antiprotons, γ‑rays). The authors anticipate that forthcoming high‑precision data from AMS‑02, CALET, DAMPE, and future missions will tighten the constraints on δ and V_c, thereby allowing an even more accurate measurement of L and r_h. The study thus establishes a robust statistical framework for future investigations of Galactic cosmic‑ray propagation.