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Additional info for Astrophysics at Very High Energies: Saas-Fee Advanced Course 40. Swiss Society for Astrophysics and Astronomy
However, the rates are always limited because of the requirement to the source to be optically thin for gamma-rays regarding the • photon–photon pair production: γ + γ → e+ e− The cross-section of this process exceeds by orders of magnitude the cross-sections of the photomeson and pair-production cross-sections, therefore the condition τγγ ≤ 1 implies very low efficiency of photomeson and BHPP processes unless the protons are very effectively confined in the source. And, vice versa, the high efficiency of the photomeson and BHPP processes implies large optical depth of the source.
The parameter that characterises gamma-ray absorption at photon–photon interactions in a source of size R is the so-called optical depth τ (εγ ) = R 0 ω2 ω1 σ(εγ , ω)n ph (ω, r )dωdr, (44) where n ph (ω, r ) describes the spectral and spatial distribution of the target photon field in the source. Because of narrowness of the pair-production cross-section, for a large class of broad band target photon energy distributions n ph (ω), the optical depth at given gamma-ray energy εγ is essentially determined by a relatively narrow band of target photons with energy centred on ω∗ = 4/ γ .
The impact of such interference is very complex and quite sensitive to the choice of the principal parameters. Therefore each practical case is a subject to independent studies. Basic Definitions Calculations of emissivity of gamma rays resulting from interactions of accelerated dσ , particles require two functions—the differential cross section of the process, d EdΩ and the energy distribution of parent particles, f (E). g. diffuse shock acceleration or stochastic acceleration) and the local conditions of the ambient medium—the strength of magnetic field, the shock speeds, the level of turbulence, etc.