A self-regulated stochastic acceleration model of pulsar wind nebulae

Pulsar wind nebulae (PWNe) are clouds of the magnetized relativistic electron/positron plasma supplied from the central pulsar. However, the number of radio-emitting particles inside a PWN is larger than the expectation from the study of pulsar magnetospheres and then their origin is still unclear. A stochastic acceleration of externally injected particles by a turbulence inside the PWN is proposed by our previous studies. In this paper, the previous stochastic acceleration model of the PWN broadband spectra is improved by taking into account the time evolution of the turbulent energy and then the total energy balance inside a PWN is maintained. The turbulent energy supplied from the central pulsar is wasted by the backreaction from the stochastic particle acceleration and the adiabatic cooling according the PWN expansion. The model is applied to the Crab Nebula and reproduce the current broadband emission spectrum, especially the flat radio spectrum although time evolution of the turbulent energy (diffusion coefficient) is a bit complicated compared with our previous studies, where we assumed an exponential behavior of the diffusion coefficient.