Spatial dependence of high energy electrons and their radiations in pulsar wind nebulae

Abstract

We investigate the spatial dependence of high energy electrons and their radiations in pulsar wind nebulae (PWNe). By assuming a time-dependent broken power-law injection and spatial dependence of convection velocity, magnetic field strength and diffusion coefficient on the radial distance of an expanding system, we numerically solve the Fokker-Planck transport equation including convection, diffusion, adiabatic loss and radiative loss in spherical coordinates, and investigate the effects of magnetic field, PWN age, maximum energy of electrons, and diffusion coefficient on electron spectra and non-thermal photon emissions. Our results indicate that (1) electron spectra and the corresponding photon spectra are a function of radial distance r of the expanding system; (2) for a given expansion velocity, the increase of the PWN age causes a slower decrease of the convection velocity (V ∝ r−β) and a more rapid decrease of the magnetic field strength (B ∝ r−1+β), but a more rapid increase of the diffusion coefficient (κ ∝ r1−β) because the index β decreases with the PWN age; and (3) the lower energy part of the electron spectra is dominated by convection and adiabatic loss, but the higher energy part is dominated by the competition between synchrotron loss and diffusion, and such a competition is a function of radial distance. Therefore the diffusion effect has an important role in the evolution of electron spectra as well as non-thermal photon spectra in a PWN.