Numerical simulations of internal shocks in spherical geometry: Hydrodynamics and prompt emission

¹ Astrophysics Research Center of the Open University (ARCO), The Open University of Israel, 1 University Road, PO Box 808, Raanana 4353701, Israel
² Centre de Recherche Astrophysique de Lyon (CRAL), ENS de Lyon, UMR 5574, Université Claude Bernard Lyon 1, CNRS, Lyon 69007, France
³ Department of Natural Sciences, The Open University of Israel, PO Box 808, Raanana 4353701, Israel
⁴ Department of Physics, The George Washington University, 725 21st Street NW, Washington, DC 20052, USA

^⋆ Corresponding author: arthur.charlet@ens-lyon.fr

Received: 9 December 2024
Accepted: 20 May 2025

Abstract

Context. Among the models used to explain the prompt emission of gamma-ray bursts (GRBs), internal shocks is a leading one. Its most basic ingredient is a collision between two cold shells of different Lorentz factors in an ultrarelativistic outflow, which forms a pair of shock fronts that accelerate electrons in their wake. In this model, key features of GRB prompt emission such as the doubly broken power-law spectral shape arise naturally from the optically thin synchrotron emission at both shock fronts.

Aims. We investigate the internal shocks model as a mechanism for prompt emission based on a full hydrodynamical analytic derivation in planar geometry, extending this approach to spherical geometry using hydrodynamic simulations.

Methods. We used the moving mesh relativistic hydrodynamics code GAMMA to study the collision of two ultrarelativistic cold shells of equal kinetic energy (and power). Using the built-in shock detection, we calculated the corresponding synchrotron emission by the relativistic electrons accelerated into a power-law energy distribution behind the shock in the fast-cooling regime.

Results. During the first dynamical time after the collision, the spherical effects cause the shock strength to decrease with radius. The observed peak frequency decreases faster than expected by other models in the rising part of the pulse and the peak flux is saturated even for moderately short pulses. This is likely caused by the very sharp edges of the shells in our model, while smoother edges would probably mitigate this effect. Our model traces the evolution of the peak frequency back to the source activity time scales.