Photospheric Emission of an Accelerating/Decelerating Relativistic Jet

In gamma-ray bursts (GRBs), Compton scattering near the photosphere plays a crucial role in shaping the observed jet emission. Previous studies have considered the propagation of photons in a coasting relativistic jet. While previous studies have primarily focused on coasting relativistic jets, the behavior of photons in accelerating or decelerating jets has remained relatively unexplored. In this work, we investigate the photospheric emission of an accelerating/decelerating relativistic jet, with the Lorentz factor Γ described as Γ ∝ r δ , where r is the distance relative to the central engine of the GRB and δ is the index. We derive an analytical solution showing that when the optical depth is larger than unity, the comoving photon energy evolves as ϵ ′ ( r ) ∝ r − ( 2 + δ ) / 3 , which differs significantly from the coasting jet scenario. Using Monte Carlo radiation transfer (MCRaT) simulations, we numerically track the energy evolution of photons, which confirms our analytical results for τ ≫ 1 but reveals a deviation from the previously assumed power-law form near τ ≈ 1. Furthermore, we calculate the resulting radiation spectra and find that, compared to coasting jets, the spectrum significantly broadens in decelerating jets (δ = −1/3) while narrowing in accelerating jets (δ = 1/3). We find that this spectral broadening is primarily driven by the weakening of relativistic beaming in a decelerating jet.