Astroparticle Physics

164 7 Secondary Cosmic Raysof these high-energy particles the gyroradii must be smallerthan the size of the source. Therefore, one can derive fromgalactic containmentmv 2R≤ evBa maximum value for the energy of a particle that can beaccelerated in the source,E max ≈ p max ≤ eBR (7.19)(v is a particle velocity, B is a magnetic field strength of thesource, R is the size of the source, m is the relativistic massof the particle). In units appropriate for astroparticle physicsthe maximum energy, which can be obtained by accelerationin the source, can be expressed in the following way:E max = 10 5 B RTeV3 × 10 −6 G 50 pc . (7.20)With a typical value of B = 3 µG for our Milky Way andthe very generous gyroradius of R = 5 kpc one obtainsE max = 10 7 TeV = 10 19 eV . (7.21)protonsThis equation implies that our Milky Way can hardly accelerateor store particles of these energies, so that for particleswith energies exceeding 10 20 eV one has to assume that theyare of extragalactic origin.For protons the Greisen–Zatsepin–Kuzmin cutoff (GZK)of photoproduction of pions off blackbody photons throughthe ∆ resonance takes an important influence on the propagation,γ + p → p + π 0 . (7.22)mean free path of protonsThe energy threshold for this process is at 6 × 10 19 eV (seeSect. 6.1). Protons exceeding this energy lose rapidly theirenergy by such photoproduction processes. The mean freepath for photoproduction is calculated to beλ γp = 1Nσ , (7.23)where N is the number density of blackbody photons andσ(γp → π 0 p) ≈ 100 µb the cross section at threshold.This leads toλ γp ≈ 10 Mpc . (7.24)