2026ApJ...997..173L

A newly born millisecond magnetar has been proposed as one possible central engine of some long gamma-ray bursts (LGRBs) with X-ray plateau emission. In this work, we used a universal correlation between the initial spin period (P₀) and the surface magnetic field (B_p) of the newborn magnetar based on an LGRB sample in L. Lan et al. (2025) to explore the propeller properties of accreting magnetars with R/I evolutionary effects. We found that the B_p-P₀ relation is approximately consistent with ${B}_{p}\propto {P}_{{\rm{eq}}}^{7/6}$. Here P_eq is the equilibrium spin period in the magnetic propeller model, where the electromagnetic dipole radiation and the propeller mechanism jointly modulate the spin evolution of a newborn magnetar. The B_p-P₀ relation indicates that P₀ constrained by X-ray plateau data may not be the true initial spin period of a newborn magnetar but had reached an equilibrium spin period via fallback accretion in the propeller model. The magnetar accretion rate in our LGRBs is in the range of $\dot{M}\sim 1{0}^{-5}\,\mathrm{to}\,1{0}^{-2}\,{M}_{\odot }\,{{\rm{s}}}^{-1}$ by incorporating R/I evolutionary effects and using the transition relation between gravitational mass M_g and baryonic mass M_b in different equations of state and X-ray radiation efficiencies. Such accretion rates ensure that the accreting magnetars in our sample survive until reaching the equilibrium spin period, and the accretion rate is one order of magnitude lower compared to the statistical results in G. Stratta et al. (2018) and W. L. Lin et al. (2020), which used the constant R/I/M_g scenario. The fallback rate of progenitor envelope materials onto the magnetar accretion disk for our LGRBs is compatible with the theoretical mass fallback rate of some low-metallicity progenitors. We suggested that adopting a constant R/I/M_g scenario for modeling the propeller regime in accreting magnetars results in a higher mass accretion rate, which may impair our understanding of the physical nature of an accreting magnetar and its surroundings, and that the low-metallicity progenitors can provide enough material to satisfy the accretion requirements of the newborn accreting magnetar in LGRBs.