Non-linear propagation of cosmic rays around their sources

The cosmic ray spectrum observed at the Earth’s surface is the result of two concurring processes: particle acceleration at the source and transport from the sources to Earth. In this work we investigate a particular aspect of the transport process, which consists in the generation of hydromagnetic turbulence by the accelerated particles streaming outside the parent source. When these particles leave the acceleration site, they propagate in a magnetized environment where they interact with the background plasma. This interaction leads to the excitation of streaming instabilities. We show that in near source regions particle propagation is dominated by non-linear self-generation of the unstable waves. As a consequence, the scattering properties of the medium become dependent upon the spectrum and spatial distribution of the energetic particles. The enhancement of the magnetic turbulence forces particles to be self-confined in these near-source regions for a period of time, which can be non-negligible relative to the propagation time to the Earth. This self-confinement process can have important consequences from both the theoretical and observational points of view. In case of propagation around Galactic supernova remnants we calculate the amount of matter accumulated by self-confined accelerated protons. Then, we compare this quantity with the total amount of matter traversed by cosmic rays in the Galaxy, namely the cosmic ray grammage. The latter quantity is usually inferred from the measurement of the ratio of secondary-to-primary nuclei, as for instance the boron (B)/carbon (C) ratio, and used to estimate parameters of Galactic particle propagation, such as the residence time in the Galaxy. In this work we show that, depending on the level of ionization of the medium, the grammage accumulated in the source vicinity can be a non-negligible fraction of the cosmic ray grammage. Moreover, there is an irreducible grammage that cosmic rays traverse while trapped downstream of the shock that accelerated them, though this contribution is rather uncertain. We conclude that some caution should be used in inferring parameters of Galactic cosmic ray propagation from measurements of the B/C ratio. A possible signature of this selfconfinement process is the formation of extended halos of gamma ray emission from ⇡0-meson decay around sources. The sum of these halos over the distribution of supernova remnants located into the Galactic disc can be a non negligible contribution to the diffuse gamma-ray emission from the Galactic disc, measured by the FermiLAT telescope over 7 years of data acquisition. We estimate this contribution assuming that cosmic ray sources are supernova remnants with a rate of explosion of 1/30 yr-1. We find that the results strongly depend on the type of interstellar medium where the sources are located. In case of a fully ionized medium we show that the halos emission almost saturates the observed Galactic one. However, if the medium is partially ionized, we show that the effect of ion-neutral damping is to strongly reduce the confinement time and consequently the gamma-ray emission. In this case, the contribution to the Galactic emission is non negligible only for energies of -rays around 100 GeV. The effect of self-confinement appears to be important even in the proximity of Extra-Galactic sources of cosmic rays. In this context we show that the self-confinement process forces cosmic rays to be trapped in the source proximity. In addition, we find that particles with energies less than a critical value Ec are not able to reach the Earth, because the time of their confinement in the source proximity exceeds the Age of the Universe. As a consequence, the spectrum of cosmic rays leaving these sources and eventually reaching the Earth must have a low-energy cutoff at an energy Ec. In particular we show that, if the background magnetic field is extremely low, i.e. ⌧ nG, the value of this critical energy Ec depends only on the source luminosity LCR as / L2/3 CR . We consider as typical value of LCR about 1044erg/s and we obtain Ec ⇡ 107 GeV. For larger values of the background magnetic field we find that Ec depends also on the field strength B0 and on its coherence length lc and, considering as typical values of B0 ⇡ 0.1 nG and of lc ⇡ 10 Mpc, we obtain Ec ⇡ 2 ⇥ 108 GeV. In both these scenarios the cut-off energy Ec is in the energy range where the transition from Galactic to Extra-Galactic origin is assumed to take place.