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

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 ≡ 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 polarization degree is around (40\!-\!50)\,\, per\,cent for High-energy Polarimetry Detector (HPD) onboard POLAR-2 and is roughly (30\!-\!40)\,\, 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.