2025ApJS..280...45L

A newly born millisecond magnetar has been proposed as one possible central engine of some gamma-ray bursts (GRBs) with X-ray plateau emission. In this work, we systematically analyzed the Swift/X-Ray Telescope data of long GRBs with plateau emission that were detected before 2023 December, estimated the physical parameters of the magnetar by considering the R/I evolutionary effects, and investigated possible relationships among these parameters and their relation to the GRB jet and magnetar wind radiation. We found that neglecting the evolutionary effects of R/I can lead to systematic overestimation or underestimation of magnetar physical parameters such as magnetic field strength (B_p), spin period (P₀), and ellipticity (ϵ) from 20% to 50%. We also found that the correlations among different magnetar parameters can be approximately expressed as $\epsilon \propto {P}_{0}^{1.57\pm 0.22}$, $\epsilon \propto {B}_{p}^{0.97\pm 0.13}$, and ${B}_{p}\propto {P}_{0}^{1.30\pm 0.16}$ for our selected equation of states (EoSs), with the 1σ deviation included. The correlations between the GRB jet emission and the magnetar wind emission can be approximately described as ${E}_{{{\rm{wind}}}}\propto {E}_{{{\rm{jet}}},{{\rm{iso}}}}^{0.83\pm 0.07}({E}_{{{\rm{jet}}}}^{0.76\pm 0.06})$, ${P}_{0}\propto {E}_{{{\rm{jet}}},{{\rm{iso}}}}^{-0.29\pm 0.03}({E}_{{{\rm{jet}}}}^{-0.26\pm 0.02})$, ${B}_{p}\propto {E}_{{{\rm{jet}}},{{\rm{iso}}}}^{-0.58\pm 0.06}({E}_{{{\rm{jet}}}}^{-0.55\pm 0.05})$, and $\epsilon \propto {E}_{{{\rm{jet}}},{{\rm{iso}}}}^{-0.55\pm 0.07}({E}_{{{\rm{jet}}}}^{-0.52\pm 0.06})$ for our selected EoSs. The universal correlations suggest that a nascent magnetar with the faster P₀, lower B_p, as well as lower ϵ are more inclined to power a more energetic GRB jet, and the ellipticity deformation and initial spin period of the newborn magnetar are likely to originate from the magnetically induced distortion mechanism and correspond to the equilibrium spin period as a result of interaction between the magnetar and its accretion disk, respectively. Finally, we found that the gravitational-wave (GW) signals from the remnants of those GW-dominated GRBs with redshift measurements cannot reach the sensitivity threshold of the current Advanced LIGO detector, and only two cases (GRB 150323A and 170607A) can reach the sensitivity threshold of the prospective ET detector. Future GW observations could not only offer the first smoking gun that a protomagnetar can serve as the central engine of GRBs but also play a crucial role in precisely constraining the neutron star EoS.