Abstract

There is growing evidence from gamma-ray observations at high and very high energies that particle escape is a key aspect shaping the morphological properties of pulsar wind nebulae (PWNe) at various evolutionary stages. We aim to provide a simple model for the gamma-ray emission from these objects including the transport of particles across the different components of the system. We applied it to sources HESS J1809$-$193 and HESS J1825$-$137. We developed a multi-zone framework applicable to dynamically young PWNe, taking into account the diffusive escape of relativistic electron-positron pairs out of the nebula into the parent supernova remnant (SNR) and their confinement downstream of the magnetic barrier of the forward shock until an eventual release into the surrounding interstellar medium (ISM). For a wide range of turbulence properties in the nebula, the GeV-TeV inverse-Compton radiation from pairs that escaped into the remnant can be a significant if not dominant contribution to the emission from the system. It may dominate the pion-decay radiation from cosmic rays accelerated at the forward shock and advected downstream of it. In the TeV-PeV range, the contribution from particles escaped into the ISM can exceed by far that of the SNR+PWN components. Applied to HESS J1809$-$193 and HESS J1825$-$137, we found that spatially extended GeV-TeV emission components can be accounted for mostly from particles escaped into the ISM, while morphologically more compact components above $50-100$ are ascribed to the PWNe. In these two cases, the model suggests high turbulence in the nebula and a forward shock accelerating cosmic rays up to $ at most. The model provides the temporal and spectral properties of the flux of particles originally energized by the pulsar wind and ultimately released in the ISM. It can be used to constrain the transport of particles in the vicinity of pulsar-PWN-SNR systems from broadband gamma-ray observations, or in studies of the contribution of pulsar-related systems to the local positron flux.