Opacity buildup in impulsive relativistic sources

Opacity effects in relativistic sources of high-energy gamma-rays, such as gamma-ray bursts (GRBs) or blazars, can probe the Lorentz factor of the outflow as well as the distance of the emission site from the source and, thus, help constrain the composition of the outflow (protons, pairs, magnetic field) and the emission mechanism. Most previous works consider the opacity in steady state. Here we study time-dependent effects of the opacity to pair production (γγ → e⁺e^-) in impulsive relativistic sources. We present a simple, yet rich, semianalytic model for the time and energy dependence of the optical depth, τ_γγ, in which a thin spherical shell expands ultrarelativistically and emits isotropically in its own rest frame over a finite range of radii, R₀ ≤ R ≤ R₀ + ΔR. This is particularly relevant for GRB internal shocks. We find that for impulsive sources (ΔR ≲ R₀), while the instantaneous spectrum has an exponential cutoff above the photon energy ε₁ (T) where τ_γγ(ε₁) = 1, the time-integrated spectrum has a power-law high-energy tail above the photon energy ε₁* ∼ ε₁(ΔT) where ΔT is the duration of the emission episode. Furthermore, photons with ε > ε₁* should arrive mainly near the onset of the spike or flare corresponding to the short emission episode, since in impulsive sources it takes time to build up the (target) photon field, and thus, τ_γγ(ε) initially increases with time and ε₁(T) correspondingly decreases with time, so that photons of energy ε > ε₁* are able to escape the source mainly very early on while ε₁(T) > ε. As the source approaches a quasi-steady state (ΔR ≫ R₀), the time-integrated spectrum develops an exponential cutoff, while the power-law tail becomes increasingly suppressed.