The evolution effects of radius and moment of inertia for rapidly rotating neutron stars

A newly born millisecond magnetar is thought to be the central engine of some gamma-ray bursts (GRBs), especially those that present long-lasting X-ray plateau emissions. By solving the field equations, we find that when the rotational speed of the magnetar is approaching the breakup limit, its radius $R$ and moment of inertia $I$ would undergo an obvious evolution as the magnetar spins down. Meanwhile, the values of $R$ and $I$ would sensitively depend on the adoption of neutron star (NS) equation of state (EoS) and the NS baryonic mass. With different EoSs and baryonic masses considered, the magnetic dipole radiation luminosity ($L_{\rm dip}$) could be variant within one to two orders of magnitude. We thus suggest that when using the X-ray plateau data of GRBs to diagnose the properties of the nascent NSs, EoS and NS mass information should be invoked as simultaneously constrained parameters. On the other hand, due to the evolution of $R$ and $I$, the temporal behavior of $L_{\rm dip}$ would become more complicated. For instance, if the spin-down process is dominated by gravitational wave emission due to the NS asymmetry caused by magnetic field distortion ($\epsilon\propto B_{p}^{2}$), the segment $L_{\rm dip}\propto t^{0}$ could be followed by $L_{\rm dip}\propto t^{-\gamma}$ with $\gamma$ larger than 3. This case could naturally interpret the so-called internal X-ray plateau feature shown in some GRB afterglows, which means the sharp decay following the plateau is unnecessarily corresponding to the NS collapsing. This may explain why some internal X-ray plateaus are followed by late time central engine activity, manifested through flares and second shallow plateaus.