p, He, and C to Fe cosmic-ray primary fluxes in diffusion models: Source and transport signatures on fluxes and ratios

The propagated fluxes of proton, helium, and heavier primary cosmic-ray species (up to Fe) are a means to indirectly access the source spectrum of cosmic rays. We check the compatibility of the primary fluxes with the transport parameters derived from the B/C analysis, but also if they bring further constraints. Proton data are well described in the simplest model defined by a power-law source spectrum and plain diffusion. They can also be accommodated by models with, e.g., convection and/or reacceleration. There is no need for breaks in the source spectral indices below $\sim 1$ TeV/n. Fits on the primary fluxes alone do not provide physical constraints on the transport parameters. If we let free the source spectrum $dQ/dE = q \beta^{\eta_S} {\cal R}^{-\alpha}$ and fix the diffusion coefficient $K(R)= K_0\beta^{\eta_T} {\cal R}^{\delta}$ such as to reproduce the B/C ratio, the MCMC analysis constrains the source spectral index $\alpha$ to be in the range $2.2-2.5$ for all primary species up to Fe, regardless of the value of the diffusion slope $\delta$. The $\eta_S$ low-energy shape of the source spectrum is degenerate with the low-energy shape $\eta_T$ of the diffusion coefficient: we find $\eta_S-\eta_T\approx 0$ for p and He data, but $\eta_S-\eta_T\approx 1$ for C to Fe primary species. This is consistent with the toy-model calculation in which the shape of the p/He and C/O to Fe/O data is reproduced if $\eta_S-\eta_T\approx 0-1$ (no need for different slopes $\alpha$). When plotted as a function of the kinetic energy per nucleon, the low-energy p/He ratio is shaped mostly by the modulation effect, whereas primary/O ratios are mostly shaped by their destruction rate.

💡 Research Summary

This paper investigates how the observed fluxes of primary cosmic‑ray nuclei—from protons (p) and helium (He) up to iron (Fe)—constrain both the source spectra and the transport parameters that govern their propagation through the Galaxy. The authors start from transport parameters (diffusion coefficient K(R)=K₀ β^{η_T} R^{δ}, convection speed V_c, re‑acceleration strength V_a) that have been tightly constrained by the boron‑to‑carbon (B/C) ratio using recent high‑precision data (AMS‑02, ACE/CRIS). These parameters are held fixed while the source term for each primary species is modeled as

 dQ/dE = q β^{η_S} R^{‑α},

where α is the source spectral index, η_S controls the low‑energy β‑dependence, and q is a normalization constant. By performing a joint Markov Chain Monte Carlo (MCMC) fit to the full set of primary fluxes (p, He, C, O, Ne, Mg, Si, Fe), the authors explore the allowed ranges of α and η_S and examine their degeneracies with the transport parameters.

The main findings are:

1. Universal source index – Across all primary species the spectral index α is tightly confined to the interval 2.2–2.5, independent of the diffusion slope δ (which varies between ≈0.3 and 0.9 in the B/C‑derived models). This range is consistent with the predictions of diffusive shock acceleration at supernova remnants.

2. Degeneracy between source and diffusion low‑energy shapes – The low‑energy β‑dependence of the source (η_S) and that of the diffusion coefficient (η_T) are strongly correlated. For protons and helium the data require η_S − η_T ≈ 0, meaning that both the source and diffusion behave as β⁰ at low energies. For heavier nuclei (C–Fe) the best‑fit relation is η_S − η_T ≈ 1, indicating that the source spectrum falls off more steeply than the diffusion coefficient in the sub‑GeV/n region. This difference is interpreted as a consequence of the larger nuclear destruction (spallation) cross‑sections for heavy nuclei, which suppress their low‑energy fluxes.

3. Energy dependence of elemental ratios – When plotted versus kinetic energy per nucleon, the p/He ratio at low energies is dominated by solar modulation (modeled with a force‑field potential), while ratios such as C/O, Fe/O are shaped primarily by the species‑dependent inelastic destruction rates. Consequently, the p/He ratio is a sensitive probe of heliospheric effects, whereas heavy‑to‑light ratios encode information about Galactic propagation and spallation.

4. No need for spectral breaks – The data up to ∼1 TeV/n are well described by a single power‑law source spectrum; introducing artificial breaks below this energy does not improve the fit. This suggests that the observed hardening of the all‑particle spectrum at a few hundred GV is not required for the primary nuclei considered here.

5. Limited impact of convection and re‑acceleration – Adding a convective wind or a diffusive re‑acceleration term (characterized by V_c and V_a) does not significantly improve the χ² of the fits. The simplest “plain diffusion” model already reproduces the observed fluxes within uncertainties, implying that current data do not demand more complex transport physics.

Overall, the study demonstrates that the transport parameters inferred from secondary‑to‑primary ratios (B/C) are fully compatible with the primary fluxes of a wide range of nuclei, provided that the source spectral index lies in the narrow range 2.2–2.5 and that the low‑energy β‑dependence of source and diffusion are appropriately correlated (η_S − η_T≈0–1). The work also clarifies which physical processes dominate different elemental ratios: solar modulation for light nuclei and nuclear destruction for heavier ones. These results lay a solid groundwork for future analyses that will incorporate even higher‑precision measurements from AMS‑02, DAMPE, CALET, and forthcoming missions, potentially allowing the detection of subtle deviations from the simple diffusion picture or the identification of new acceleration mechanisms at the highest energies.