X-Ray Afterglow Plateaus of Long Gamma-Ray Bursts: Further Evidence for Millisecond Magnetars

Many long-duration gamma-ray bursts (GRBs) were observed by {\it Swift}/XRT to have plateaus in their X-ray afterglow light curves. This plateau phase has been argued to be evidence for long-lasting activity of magnetar (ultra-strongly magnetized neutron stars) central engines. However, the emission efficiency of such magnetars in X-rays is still unknown. Here we collect 24 long GRB X-ray afterglows showing plateaus followed by steep decays. We extend the well-known relationship between the X-ray luminosity ${L_{\mathrm{X}}}$ and spin-down luminosity $L_{\mathrm{sd}}$ of pulsars to magnetar central engines, and find that the initial rotation period $P_{0}$ ranges from 1 ms to 10 ms and that the dipole magnetic field $B$ is centered around $10^{15}$ G. These constraints not only favor the suggestion that the central engines of some long GRBs are very likely to be rapidly rotating magnetars but also indicate that the magnetar plateau emission efficiency in X-rays is close to 100%.