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 ($\Pi$) can be read from the $q-\Pi$ curve, where $q\equiv\theta_V/\theta_j$. $\theta_V$ and $\theta_j$ are the observational and jet half-opening angles, respectively. Hence, only the $q-\Pi$ 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, $\Pi$ 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 $\Pi$ in an ordered field. There are mainly two types of the $q-\Pi$ curve profiles, corresponding to two ordered magnetic field configurations.