X-rays from Radio Millisecond Pulsars: Comptonized Thermal Radiation
  X-ray emission from many rotation-powered millisecond pulsars (MSPs) is observed to be of predominantly thermal nature. In PSR J0437--4715, the nearest MSP known, an additional faint power-law tail is observed above 2.5 keV, commonly attributed to non-thermal magnetospheric radiation. We propose that the hard emission in this and other similar MSPs is instead due to weak Comptonization of the thermal (blackbody or hydrogen atmosphere) polar cap emission by energetic electrons/positrons of small optical depth in the pulsar magnetosphere. This spectral model implies that all soft X-rays are of purely thermal origin, which has profound implications in the study of neutron star structure and fundamental pulsar physics. 
Slavko Bogdanov, Jonathan E. Grindlay, and George B. Rybicki