Time-dependent modeling of pulsar wind nebulae: Study on the impact of the diffusion-loss approximations

TL;DR

This paper develops a comprehensive time-dependent model for pulsar wind nebulae, analyzing how different diffusion-loss approximations affect the predicted evolution and spectra, with a focus on the Crab nebula.

Contribution

It introduces a full time-energy dependent diffusion-loss model for PWNe and evaluates the impact of common approximations on their evolution and spectra.

Findings

Full model reveals deviations in multi-wavelength spectra.

Approximate models can significantly alter parameter estimates.

Escape and catastrophic loss approximations impact PWN evolution predictions.

Abstract

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 (SSC), 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 analyze 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 show which is the impact of these approaches in the determination of the PWN evolution. In particular, we find the time-dependent deviation of the multi-wavelength spectrum and the best-fit parameters obtained with the complete and the approximate models.