2024ApJ...974..308B

Gamma-ray bursts (GRBs) are powered by ultrarelativistic jets. The launching sites of these jets are surrounded by dense media, which the jets must cross before they can accelerate and release high-energy emission. Interaction with the medium leads to the formation of a mildly relativistic sheath around the jet, resulting in an angular structure to the jet's asymptotic Lorentz factor and energy per solid angle, which modifies the afterglow emission. We build a semi-analytical tool to analyze the afterglow light curve and polarization signatures of jets observed from a wide range of viewing angles, and focus on ones with slowly declining energy profiles known as shallow jets. We find overall lower polarization compared to the classical top hat jet model. We provide an analytical expression for the peak polarization degree as a function of the energy profile power-law index, magnetic field configuration, and viewing angle, and show that it occurs near the light-curve break time for all viewers. When applying our tool to GRB 221009A, suspected to originate from a shallow jet, we find that the suggested jet structures for this event agree with the upper limits placed on the afterglow polarization in the optical and X-ray bands. We also find that at early times the polarization levels may be significantly higher, allowing for a potential distinction between different jet structure models and possibly constraining the magnetization in both forward and reverse shocks at that stage.