X-ray timing studies of the low-field magnetar CXOU J164710.2-455216

We report results of X-ray timing analyses for the low-field magnetar CXOU~J164710.2$-$455216 which exhibited multiple outbursts. We use data taken with NICER, NuSTAR, Chandra, and Neil-Gehrels-Swift telescopes between 2017 and 2018 when the source was in an active state. We perform semi-phase-coherent timing analyses to measure the spin parameters and a spin-inferred magnetic-field strength ($B_s$) of the magnetar. Using a semi-phase-coherent method, we infer the magnetic field strengths to be $3-4times 10^{13}rm G$ at the observation period ($sim$MJD 58000), and by comparing with previous frequency measurements (MJD 54000) a long-term average value of $B_s$ is estimated to be $approx4times 10^{13}rm G$. So this analysis may add CXOU~J164710.2$-$455216 to the ranks of low-field magnetars. The inferred characteristic age ($tau_c$) is 1--2 Myr which is smaller than the age of Westerlund~1, so the magnetars association with the star cluster is still secure. For the low dipole field and the large age, recent multiple outbursts observed from the source are hard to explain unless it has strong magnetic multipole components. We also find timing anomalies around outburst epochs, which suggests that there may be spin-down torque applied to the magnetar near the epochs as was proposed in magnetar models.