  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.
