Exploration of Spin-down Rate of Neutron Star in High Mass X-ray   Binaries
  We use the evolutionary population synthesis method to investigate the statistical properties of the wind-fed neutron star (NS) compact ($P_{\rm orb}<10$ days) high-mass X-ray binaries (HMXBs) in our Galaxy, based on different spin-down models. We find that the spin-down rate in the supersonic propeller phase given \textbf{by assuming that the surrounding material is treated as forming a quasi-static atmosphere} or \textbf{by assuming that the characteristic velocity of matter and the typical Alfv$\acute{e}$n velocity of material in the magnetospheric boundary layer are comparable to the sound speed in the external medium} is too low to produce the observed number of compact HMXBs. We also find that the models suggested \textbf{by assuming that the infalling material is ejected with the corotation velocity at the magnetospheric radius when the magnetospheric radius is larger than the corotation radius} and \textbf{by simple integration of the magnetic torque over the magnetosphere} with a larger spin-down rate than that given by \citet{dav81} or \citet{is75} can predict a reasonable number of observed wind-fed NS compact HMXBs. Our calculated results indicate that subsonic propeller phase may not exist at all by comparing with the observed particular distributions of wind-fed NS compact HMXBs in the $P_s-P_{orb}$ diagram. However, the spin-down rate suggested by \citet{wan85,dai06,jia05} and that given by \citet{dav73} both seem reasonable to produce the observed distribution of wind-fed NS compact HMXBs in the $P_s-P_{orb}$ diagram. We cannot find which spin-down rate seems more reasonable from our calculations. 
Hai-Lang Dai, Xi-Wei Liu and Xiang-Dong Li
astro-ph.HE