Contributions to the cosmic ray flux above the ankle: Clusters of galaxies

Motivated by the suggestion of Kang, Ryu & Jones that particles can be accelerated to high energies via diffusive shock acceleration processes at the accretion shocks formed by the infalling flow toward clusters of galaxies, we have calculated the expected particle flux from a cosmological ensemble of clusters. We use the observed temperature distribution of local clusters and assume a simple power-law evolutionary model for the comoving density of the clusters. The shock parameters, such as the shock radius and velocity, are deduced from the intracluster medium (ICM) temperature using the self-similar solutions for secondary infall on to the clusters. The magnetic field strength is assumed to be in equipartition with the postshock thermal energy behind the accretion shock. We also assume that the injected energy spectrum is a power law with the exponential cut-off at the maximum energy which is calculated from the condition that the energy gain rate for diffusive shock acceleration is balanced by the loss rate due to the interactions with the cosmic background radiation. In contrast to the earlier paper, we have adopted here the description of the cosmic ray diffusion by Jokipii which leads to considerably higher particle energies. Finally, the injected particle spectrum at the clusters is integrated over the cosmological distance to Earth by considering the energy loss due to the interactions with the cosmic background radiation. Our calculations show that the expected spectrum of high-energy protons from the cosmological ensemble of the cluster accretion shocks could match well the observed cosmic ray spectrum near 10(19) eV with reasonable parameters and models if about 10(-4) of the infalling kinetic energy can be injected into the intergalactic space as high-energy particles.