Revisiting the time-integrated polarizations of gamma-ray burst prompt phase

In the former studies, the time evolution information is missed in deducing the time-integrated polarizations of gamma-ray burst (GRB) 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≡θV/θⱼ. θV and θⱼ 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 PD is around $(40-50)%$ for High-energy Polarimetry Detector (HPD) on board POLAR-2 and is roughly $(30-40)%$ 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.