The thermalization of γ-rays in radioactive expanding ejecta: A simple model and its application for Kilonovae and Ia SNe

A semi-analytic approximation is derived for the time-dependent fraction f_γ(t) of the energy deposited by radioactive decay γ-rays in a homologously expanding plasma of general structure. An analytic approximation is given for spherically symmetric plasma distributions. Applied to Kilonovae (KNe) associated with neutron stars mergers and Type Ia supernovae, our semi-analytic and analytic approximations reproduce, with a few percent and 10 accuracy, respectively, the energy deposition rates, Q̇_dep, obtained in numeric Monte Carlo calculations. The time t_γ beyond which γ-ray deposition is inefficient is determined by an effective frequency-independent γ-ray opacity κ_γ,eff, t_γ = √(κ_γ,eff⟨Σ⟩ t^2), where ⟨Σ⟩∝ t^-2 is the average plasma column density. For β-decay dominated energy release, κ_γ,eff is typically close to the effective Compton scattering opacity, κ_γ,eff≈ 0.025 cm^2 g^-1 with a weak dependence on composition. For KNe, κ_γ,eff depends mainly on the initial electron fraction Y_e, κ_γ,eff≈ 0.03(0.05) cm^2 g^-1 for Y_e ≳ (≲) 0.25 (in contrast with earlier work that found κ_γ,eff larger by 1-2 orders of magnitude for low Y_e), and is insensitive to the (large) nuclear physics uncertainties. Determining t_γ from observations will therefore measure the ejecta ⟨Σ⟩ t^2, providing a stringent test of models. For ⟨Σ⟩ t^2=2×10^11  g cm^-2 s^2, a typical value expected for KNe, t_γ≈1 d.