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Additional info for Astrophysics at Very High Energies: Saas-Fee Advanced Course 40. Swiss Society for Astrophysics and Astronomy
15). Another interesting feature is the shape of the gamma-ray spectrum in the cut-off region. The δ-function approximation which agrees quite well with the results of accurate calculations at low energies, fails to describe correctly the spectrum at highest energies around the cut-off (see Fig. 16). e. predicts exp[−(E γ /E γ,0 )] type cut-off term with E γ,0 = kπ E 0 the same shape as the proton spectrum but shifted by a factor of 6 towards low energies. 06E 0 . 01E 0 . Finally we note that for α = 2, the amplitude of the gamma-ray spectrum at energies E E 0 exceeds the level of the flux of muonic neutrinos.
Equation (16) allows one to derive the gamma-ray emissivity qγ (E γ ) for an arbitrary distribution of protons N p E p : qγ E γ = c n H = c nH ∞ Eγ 1 0 σinel E p N p E p Fγ σinel Eγ x Np Eγ x Eγ , Ep Ep dEp Ep Eγ x dx , x Fγ x, (17) where the inelastic pp cross section σinel E p is given by Eq. (15). Figure 16 shows the spectra of secondary gamma-ray, electrons and neutrinos obtained for the proton distribution given in the form N (E) = A E −α exp − E E0 β , (18) with α = 2, β = 1, and E 0 = 1000 TeV.
The cross-sections are normalised to one radiation length. The energies of primary electrons and gamma-ray ε0 (in units of m e c2 ) are indicated at the curves energies ε ≥ εcr . At lower energies electrons dissipate their energy by ionization rather than producing more high energy gamma-ray which would support further development of the electron–photon shower. In Fig. 13 the bremsstrahlung total cross-sections are shown calculated for 3 different values of minimum energy of emitted gamma-ray: εth = 2, εcr and εe /2.