Modelling the Dynamics of Middle-Aged Pulsar Wind Nebulae in the Reverberation Phase

Abstract

Pulsar Wind Nebulae (PWNe) are among the most important sources emitting in the very-high-energy gamma-ray band. Predicting their long-term evolution is crucial for forthcoming high-energy observatories like ASTRI and CTA. In this work, We investigate the dynamical evolution of middle-aged PWNe - probably the major contributors to the Galactic TeV emission - and test the robustness of current approaches. To understand the diversity of these systems, we derive the Pulsar and Supernova Remnant (SNR) parameters governing PWN evolution. SNR evolution is set by supernova kinetic energy, ejecta mass, and ambient density, while pulsar energy injection powers the PWN expansion. Adopting standard distributions, we generate a synthetic PWN-SNR population and define a region of interest encompassing the majority of these objects. We use a semi-analytical framework for the early evolution and a 1D Lagrangian code to track their interaction with parent SNRs (the reverberation phase). Within our region of interest, we find large diversity in the late-stage evolution. Despite this, all systems converge toward a relaxed state consistent with the Sedov solution. To address 1D limitations, we perform 2D simulations - optimized to reduce computational cost while preserving physical accuracy - to study instability growth and long-term mixing. We find that instability growth depends on initial perturbations but does not significantly alter global dynamics. While effective volume evolution agrees with 1D predictions, multidimensional effects can increase the apparent size by up to 50%. For the first time, we investigated in detail the multidimensional evolution of middle-aged PWNe in the reverberation phase. By characterizing the late-time evolution across the population, our results confirm the robustness of 1D models, demonstrating that current predictions remain trustworthy.