XMM-Newton and NuSTAR observations of the compact millisecond pulsar binary PSR~J1653$-$0158

We have presented the first joint {\it XMM-Newton} and {\it NuSTAR} analysis of the millisecond pulsar (MSP) binary PSR J1653$-$0158. The 75-minute orbital period inferred from optical and gamma-ray observations together with the 1.97-ms pulsation in the gamma-rays indicate that this system is the most compact Black Widow MSP system known to date. The orbital period was not detected in the {\it XMM-Newton} and {\it NuSTAR} data, probably due to insufficient photon counts obtained in the observations. Fitting the joint X-ray spectrum of PSR J1653$-$0158 with a power law gives a photon index $\Gamma = 1.71 \pm 0.09$. The X-ray luminosity of the source in the ($0.2-40$)~keV band is deduced to be $1.18 \times 10^{31}\;\!{\rm erg~s}^{-1}$, for an adopted distance of 0.84~kpc. We have shown that the broad-band X-ray spectrum can be explained by synchrotron radiation from electrons accelerated in the intra-binary shock, and the gamma-rays detected in the {\it Fermi} data are curvature radiations from electrons and positrons in the pulsar magnetosphere. Our kinematic analysis of the Tidarren systems PSR~J1653--0158 and PSR~J1311--3430 indicates that the two Tidarren systems are likely to have originated in the Galactic Disk.