A Model of the Spectral Evolution of Pulsar Wind Nebulae

Abstract

We study the spectral evolution of PWNe taking into account the energy injected when they are young. We model the evolution of the magnetic field inside a uniformly expanding PWN. Considering time dependent injection from the pulsar and coolings by radiative and adiabatic losses, we solve the evolution of the particle distribution function. The model is calibrated by fitting the calculated spectrum to the observations of the Crab Nebula at an age of a thousand years. The spectral evolution of the Crab Nebula in our model shows that the flux ratio of TeV gamma-rays to X-rays increases with time, which implies that old PWNe are faint in X-rays, but not in TeV gamma-rays. The increase of this ratio is because the magnetic field decreases with time and is not because the X-ray emitting particles are cooled more rapidly than the TeV gamma-ray emitting particles. Our spectral evolution model matches the observed rate of the radio flux decrease of the Crab Nebula. This result implies that our magnetic field evolution model is close to the reality. Finally, from the viewpoint of the spectral evolution, only a small fraction of the injected energy from the Crab Pulsar needs to go to the magnetic field, which is consistent with previous studies.