p, He, and C to Fe cosmic-ray primary fluxes in diffusion models

Abstract

Context. The source spectrum of cosmic rays is not well determined by diffusive shock acceleration models. The propagated fluxes of proton, helium, and heavier primary cosmic-ray species (up to Fe) are a means to indirectly access it. But how robust are the constraints, and how degenerate are the source and transport parameters? Aims. We check the compatibility of the primary fluxes with the transport parameters derived from the B/C analysis, but also ask whether they add further constraints. We study whether the spectral shapes of these fluxes and their ratios are mostly driven by source or propagation effects. We then derive the source parameters (slope, abundance, and low-energy shape). Methods. Simple analytical formulae are used to address the issue of degeneracies between source/transport parameters, and to understand the shape of the p/He and C/O to Fe/O data. The full analysis relies on the USINE propagation package, the MINUIT minimisation routines (χ 2 analysis) and a Markov Chain Monte Carlo (MCMC) technique. Results. 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 ∼1 TeV/n. Fits to the primary fluxes alone do not provide physical constraints on the transport parameters. If we leave the source spectrum free, parametrised by the form dQ/dE = qβ η S R −α , and fix the diffusion coefficient K(R) = K 0 β η T R δ so as to reproduce the B/C ratio, the MCMC analysis constrains the source spectral index α to be in the range 2.2−2.5 for all primary species up to Fe, regardless of the value of the diffusion slope δ. The values of the parameter η S describing the low-energy shape of the source spectrum are degenerate with the parameter η T describing the low-energy shape of the diffusion coefficient: we find η S − η T ≈ 0 for p and He data, but η S − η T ≈ 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 η S − η T ≈ 0−1 (no need for different slopes α). When plotted as a function of the kinetic energy per nucleon, the low-energy p/He ratio is determined mostly by the modulation effect, whereas primary/O ratios are mostly determined by their destruction rate. Conclusions. Models based on fitting B/C are compatible with primary fluxes. The different spectral indices for the propagated primary fluxes up to a few TeV/n can be naturally ascribed to transport effects only, implying universality of elemental source spectra.

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