2014MNRAS.437.2957J

Fermi Large Area Telescope (LAT) has detected pulsed gamma-ray emissions with high confidences from more than 40 millisecond pulsars (MSPs). Here we study the phase-averaged gamma-ray properties of MSPs by using revised version of a self-consistent outer gap model. In this model, a strong multipole magnetic field near the stellar surface for an MSP is assumed and such a field will be close to the surface magnetic fields ( ∼ 10¹¹-10¹² G) of young pulsars; the outer gap of an MSP is controlled by photon-photon pair production process, where the effects of magnetic inclination angle (α) and magnetic geometry have been taken into account. Therefore, the fractional size of the outer gap is a function of not only pulsar's period and magnetic field strength but also magnetic inclination angle and radial distance to the neutron star; the inner boundary of the outer gap can be estimated by the pair production process of the gamma-ray photons which are produced by the back-flowing particles through the null charge surface; inside the outer gap, a Gaussian distribution of the parallel electric field along the trans-field thickness is assumed, and the gamma-ray emission is represented by the emission from the average radial distance along the central field lines of the outer gap. Using this model, the phase-averaged gamma-ray spectra are calculated and compared with the observed spectra of 37 MSPs given by the second Fermi-LAT catalogue of gamma-ray pulsars; our results show that the Fermi-LAT results can be well explained by this model. The thermal X-ray emission properties from MSPs are also investigated.