X-ray and gamma-ray emission from millisecond pulsars

We present a self-consistent model to describe X-ray and (gamma)-ray emission from millisecond pulsars (MSPs). The Xrays of MSPs are produced by the backflow of primary charged particles from the outer gap and most likely consist of three components, two thermal components and one power law component if there is a strong multipole magnetic field on the stellar surface. The backflow of ultra-relativistic particles emits photons with energies about several tens of GeV via curvature radiation. These photons cause an electromagnetic cascade about 2-3 stellar radii above the polar cap. The synchrotron radiation of these cascade e^+- pairs produces hard X-rays with a power law index ~ 1.5 . Near 10^ 5 cm above the stellar surface, the primary charged particles encounter the strong surface magnetic field, which alters the local radius of curvature greatly, and they quickly loose more than half of their remaining energies to curvature radiation. These curvature photons heat up the polar cap area with a radius ~ 10^5 cm, which produce the softer thermal X-ray component. Finally, the primary charged particles deposit their remaining energies in a much smaller polar cap area, which corresponds to the footprints of outer gap and produce the medium hard X-ray component. (gamma)-rays are produced in the outer gap through synchrocurvature radiation. We have applied this model to the MSPs which emit pulsed X-rays and likely (gamma)-rays such as PSR J0437-4715, PSR J2124-3358, PSR J0218+4232 and PSR B1821-24. Our results give an agreement between predicted spectrum and the observed spectrum of MSP emission.