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

TL;DR

This work investigates GRB central engines powered by accreting millisecond magnetars undergoing magnetic propeller evolution, explicitly incorporating $R/I$ evolution and variable gravitational masses. Using a 105-object Swift/XRT LGRB plateau sample and four NS EoSs with two X-ray radiative efficiencies, the authors constrain magnetar parameters via MCMC and analyze the resulting $B_p$–$P_0$ distributions. They find a global $B_p \propto P_0^{1.30\pm0.16}$ trend, consistent with $B_p \propto P_{\rm eq}^{7/6}$, and infer mass accretion rates in the range $\dot{M} \sim 10^{-5}-10^{-2}\,M_\odot\,\mathrm{s}^{-1}$, with minimal EoS dependence but clear $\eta_{\rm X}$-dependence. After correcting for $R/I$ evolution, the results yield lower $\dot{M}$ than earlier constant-$R/I$ studies, suggest that most newborn magnetars survive to spin equilibrium with $M_{acc} \lesssim 0.5\,M_\odot$, and point to low-metallicity progenitors as viable sources of fallback material; these findings reinforce the magnetar engine scenario for LGRBs and highlight the role of NS physics in shaping GRB observables, with GW observations offering a potential avenue to further constrain the initial spin and EOS.

Abstract

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.

Figures (6)

• Figure 1: The distributions between the $B_p$ and $P_0$ for our LGRB sample in four EoSs with $M_b=2.0M_{\odot}$ and $\eta_{\rm X}=0.1,~0.5$. The different colors of these circles correspond to different $\epsilon$ values. The red solid lines show the best-fitting results, and green solid lines and blue solid lines represent the expected $B_p-P_{\rm eq}$ correlations at the lower and upper limits of the accretion rate derived from our LGRB sample, respectively.
• Figure 2: Comparisons of the derived magnetar accretion rate parameter $\dot{M}$ histograms in different EoSs and $\eta_X$ for our LGRB sample. The solid lines of different colors are the best Gaussian fits. The cumulative distributions corresponding to $\dot{M}$ in different EoSs and $\eta_X$ are also displayed in the inset.
• Figure 3: The accretion rate $\dot{M}$ vs. the lower limit of accretion timescale $t_{\rm ev}$ in different EoSs and $\eta_X$ for our LGRB sample. The red dotted lines correspond to $\dot{M}t_{\rm ev}=1M_{\odot}$.
• Figure 4: The distributions of the mass ratio parameter ${\cal R}=M_{\rm tot}/M_{\rm max}$ in different EoSs and $\eta_X$ for our LGRB sample. The red vertical dashed--dotted lines correspond to ${\cal R}=1$, which marks the boundary where an accreting magnetar collapses into a BH.
• Figure 5: The evolution of the fallback rate of progenitor materials with different masses and metallicities onto the accretion disk for different EoSs and $\eta_{\rm X}$. Lines of different colors and shapes represent different progenitor star masses and metallicities. The gray circles represent the fallback rate in our LGRB sample.