Astroparticle Physics

5.7 Energy Spectra of Primary Particles 75E n = E 0 (1 + ε) n . (5.31)To obtain the final energy E n = E, a number ofn = ln(E/E 0)ln(1 + ε)(5.32)cycles is required. Let us assume that the escape probabilityper cycle is P . The probability that particles still take part inthe acceleration mechanism after n cycles is (1 − P) n .Thisleads to the following number of particles with energies inexcess of E:N(> E) ∼∞∑(1 − P) m . (5.33)m=ncyclic energy gainBecause ofrewritten as∞∑m=0x m = 11 − xN(> E) ∼ (1 − P) n∞ ∑m=n∑∞= (1 − P) n (1 − P) m =m=0(for x < 1), (5.33) can be(1 − P) m−n(1 − P)nP, (5.34)where m − n has been renamed m. Equations (5.32) and(5.34) can be combined to form the integral energy spectrum integral primary spectraN(> E) ∼ 1 P( EE 0) −γ∼ E −γ , (5.35)where the spectral index γ is obtained from (5.34) and (5.35)with the help of (5.32) to(1 − P) n =( EE 0) −γ,n ln(1 − P) =−γ ln(E/E 0 ),n ln(1 − P) ln(1/(1 − P))γ =− = . (5.36)ln(E/E 0 ) ln(1 + ε)This simple consideration yields a power law of primarycosmic rays in agreement with observation.The energy gain per cycle surely is rather small (ε ≪ 1).If also the escape probability P is low (e.g., at reflectionsbetween two shock fronts), (5.36) is simplified to