Time-dependent modelling of pulsar wind nebulae: study on the impact of the diffusion-loss approximations

In this work, we present a leptonic, time-dependent model of pulsar wind nebulae (PWNe). The model seeks a solution for the lepton distribution function considering the full time-energy-dependent diffusion-loss equation. The time-dependent lepton population is balanced by injection, energy losses and escape. We include synchrotron, inverse-Compton (IC; with the cosmic-microwave background as well as with IR/optical photon fields), self-synchrotron Compton, and bremsstrahlung processes, all devoid of any radiative approximations. With this model in place we focus on the Crab nebula as an example and present its time-dependent evolution. Afterwards, we analyse the impact of different approximations made at the level of the diffusion-loss equation, as can be found in the literature. Whereas previous models ignored the escape term, e.g. with the diffusion-loss equation becoming advective, others approximated the losses as catastrophic, so that the equation has only time derivatives. Additional approximations are also described and computed. We study what the impact of these approaches is on the determination of the PWN evolution. In particular, we find the time-dependent deviation of the multi-wavelength spectrum and the best-fitting parameters obtained with the complete and the approximate models.