Turbulent Diffusion of the Particles within Pulsar Wind Nebulae

Abstract

A turbulent diffusion model is presented to account for the energy and spatial diffusion of the particles within pulsar wind nebulae (PWNe), with the assumption that the energy injected from the pulsar into the nebula is split between the turbulence waves and electrons/positrons. In the model, the mutual interactions between the turbulence waves and particles are taken into account, representing the damping effect of the turbulence waves and the stochastic acceleration and spatial diffusion of the particles, respectively. The evolutions of the turbulence waves and particles are described with the coupled kinetic equations, in which the Kolmogorov- and Kraichnan-type turbulence are, respectively, considered. The model is applied to the Crab Nebula and shows that the spectral energy distribution of the PWN can be naturally explained. Our modeling results indicate that, for the Crab Nebula, the stochastic acceleration and spatial diffusion processes play a role in modifying the electron spectrum at the low energies of E e ≲ 1 TeV. The damping process seems more effective for modulating the turbulent spectrum in the Kraichnan-type turbulence, resulting in the nonlinear variations of the current energy and spatial diffusion coefficients with energy generated in the wave–particle systems. In the Kolmogorov-type turbulence, the diffusion coefficients are more consistent with the quasi-linear distributions, due to the energy cascade dominating over the damping effect.