2024MNRAS.529.4287S

In previous studies, the time evolution information is missed in deducing the time-integrated polarizations of gamma-ray burst prompt emission. Here, it is considered and the time-integrated polarizations is investigated through the summation of the time-resolved ones. The statistical properties of the distribution of the time-integrated polarization degree (Π) can be read from the q-Π curve, where q ≡ θ_V/θ_j. θ_V and θ_j are the observational and jet half-opening angles, respectively. Hence, only the q-Π curves are studied. In addition to a toroidal magnetic field in the radiation region, an aligned field is also discussed. We found the predicted time-integrated polarization degree is around $(40\!-\!50)\,\, \mathrm{ per\,cent}$ for High-energy Polarimetry Detector (HPD) onboard POLAR-2 and is roughly $(30\!-\!40)\,\, \mathrm{ per\,cent}$ for its Low-energy Polarimetry Detector (LPD). Therefore, Π value detected by the HPD will be larger than that of the LPD in statistics and the result of the former estimations will underestimate the value of Π in an ordered field. There are mainly two types of the q-Π curve profiles, corresponding to two ordered magnetic field configurations.