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Proton to helium ratio in cosmic rays: analysis of the PAMELA data

In our previous papers we proposed an anomalous diffusion model for solution of the knee problem in the primary cosmic-rays spectrum and explanation of different values of spectral exponent of protons and other nuclei at wide energy region. The anomaly results from large free paths (Levy flights) of particles between magnetic domains. The physical arguments and the calculations indicate that the bulk of observed cosmic rays with energy 108 – 1010 eV is formed by numerous distant (r > 1 kpc) sources. It means that the contribution of these sources to the observed flux may be evaluated in the framework of the steady-state approach (JG). The contribution of the nearby (r < 1 kpc) relatively young (t < 105 yrs) sources defines the spectrum in the high energy region and provides the knee (JL). Consequently the cosmic rays intensity from all galactic sources for some group of nuclei consists of local JL and global JG components. We show that in the framework of our anomalous diffusion model with two-component representation of CR intensity the behavior of proton to helium ratio observed by PAMELA can be reproduced.


Introduction
Experimental data of PAMELA [1] have shown that proton and helium spectra have different spectral indexes depending on rigidity. The result comes into conflict with predicted by the shock diffusion acceleration model and diffusive propagation in the Galaxy. It is interpreted as an indication of existence of different populations of cosmic ray sources.
The study is performed in frameworks of anomalous diffusion (AD) model of cosmic rays in fractal-like galactic medium, developed by the authors [2,3,4]. The AD model was shown to provide a description of the main feature of cosmic ray spectrum, "knee", at ∼ 3 · 10 6 GeV.
Possibility to explain of observable dependence proton to helium ratio by means of our model is investigated.

Proton to helium ratio
Determination of the key parameter of the model, exponents α and β, is based on the results of the investigation of particles diffusion in the cosmic and laboratory plasma and on interpretation of the data for magnetosphere [9,10].
The main parameters of the model (p, δ, α, β, D 0 ) were evaluated from experimental data. It has been shown that under condition p ≈ 2.85, δ ≈ 0.27, D 0 ≈ 1 · 10 −4 pc 1.1 y −0. 8  other nuclei, mass composition variations, the steepening of the all-particle spectrum in wide energy range [11]. In this paper the cosmic rays intensity from all galactic sources is presented as [3] J(⃗ r, t, E) = J L (⃗ r, t, E) Here J L is the contribution of nearby (r ≤ 1 kpc) young (t ≤ 10 6 yr) sources with p ≈ 2.85 calculated with (2) (T ∼ 10 4 yr) [3], and J G is the component determined by the multiple old (t ≥ 10 6 yr) distant (r ≥ 1 kpc) sources, calculated in stationary variant of theoretical AD spectrum (2) [12].
For analysis of the experimental data obtained in the Solar system, the intensity (3) was corrected for modulation effects. We use the spherically symmetric force model of [13] to describe the solar modulation. The influence of solar modulation on the particle flux is where T is the kinetic energy per nucleon, m p is the mass of a proton and J ISM is the interstellar flux (3). The potential energy Φ, describing the average energy loss of particle from interstellar space to 1 AU, is determined by solar modulation parameter ϕ : Φ = ϕZ/A, we use ϕ = 750M V in this paper.