Constraining the Properties of GRB Accreting Magnetar with $R/I$ Evolutionary Effects Using \emph{Swift}/XRT Data

A newly born millisecond magnetar has been proposed as one possible central engine of some long gamma-ray bursts (LGRBs) with X-ray plateau. In this work, we used a universal correlation between initial spin period ($P_0$) and surface magnetic field ($B_p$) of newborn magnetar based on an LGRB sample in \cite{Lan2025} to explore the propeller properties of accreting magnetars with $R/I$ evolutionary effects. We found that $B_p-P_0$ relation is approximately consistent with $B_p\propto P_{\rm eq}^{7/6}$. Here $P_{\rm eq}$ is equilibrium spin period in magnetic propeller model. The $B_p-P_0$ relation indicates that $P_0$ may not be true initial spin period of newborn magnetar but had reached an equilibrium spin period via fallback accretion in propeller model. The magnetar accretion rate in our LGRBs is in range of $\dot{M}\sim10^{-5}-10^{-2} M_{\odot} \rm s^{-1}$ 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. 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 \cite{Stratta2018} and \cite{Linweili2020}, which used constant $R/I/M_g$ scenario. We suggested that adopting a constant $R/I/M_g$ scenario for modeling propeller regime in accreting magnetar results in a higher mass accretion rate, which may impair our understanding of the physical nature and its surroundings of accreting magnetar, and low-metallicity progenitors can provide enough material to satisfy the accretion requirements of newborn accreting magnetar in LGRBs.