A detailed analysis of γ-γ annihilation absorption in γ-ray astronomy

The two-photon annihilation (γ-γ annihilation) is an important absorption mechanism in the theory of radiative transfer in γ-ray astronomy. In this paper, we try to change the known formulae of annihilation cross section given by QED theory in the frame of momemtum center into a new expression in the laboratory reference frame (the observerʹs frame) by means of relativistic transformation, which is more useful for practical application. Also, we derive a simplified direction-averaged expression for the annihilation cross section σ(ω,ω′) for a general isotropic ambient radiation field and obtain direction-averaged threshold condition of annihilation and the match condition, that have to be satisfied by the energies of two interacting photons, h̵ω and h̵ω′, to ensure the occurrence of annihilation and to achieve the maximum annihilation probability. Finally, we calculate the annihilation absorption coefficients kγγ(h̵ω) ∼ h̵ω for three typical ambient low-frequency radiation fields in astrophysics, i.e. the bremsstrahlung, the black body, and the non-thermal power law field. We further give some detailed physical discussions on the results of our calculation.