Consider the process + p p + 0 ( 0 photoproduction) with the proton

Consider the process γ + p → p + π⁰ (π⁰ photoproduction) with the proton at rest.

(a) Find the minimum energy of the photon E_γ. The Universe is filled by ‘background electromagnetic radiation’ at the temperature of T = 3 K, and photons with energy E_γ,_3K ≈ 1 meV.

(b) Find the minimum energy E_p of the cosmic-ray protons needed to induce π⁰ photoproduction

(c) If the cross-section, just above threshold, is σ = 0.6 mb and the background photon density is ρ ≈ 10⁸ m^-3, find the attenuation length. Is it small or large on the cosmological scale?