HESS J1809$-$193: a halo of escaped electrons around a pulsar wind nebula?

Context. HESS J1809$-$193 is an unassociated very-high-energy $\gamma$-ray source located on the Galactic plane. While it has been connected to the nebula of the energetic pulsar PSR J1809$-$1917, supernova remnants and molecular clouds present in the vicinity also constitute possible associations. Recently, the detection of $\gamma$-ray emission up to energies of $\sim$100 TeV with the HAWC observatory has led to renewed interest in HESS J1809$-$193. Aims. We aim to understand the origin of the $\gamma$-ray emission of HESS J1809$-$193. Methods. We analysed 93.2 h of data taken on HESS J1809$-$193 above 0.27 TeV with the High Energy Stereoscopic System (H.E.S.S.), using a multi-component, three-dimensional likelihood analysis. In addition, we provide a new analysis of 12.5 yr of Fermi-LAT data above 1 GeV within the region of HESS J1809$-$193. The obtained results are interpreted in a time-dependent modelling framework. Results. For the first time, we were able to resolve the emission detected with H.E.S.S. into two components: an extended component that exhibits a spectral cut-off at $\sim$13 TeV, and a compact component that is located close to PSR J1809$-$1917 and shows no clear spectral cut-off. The Fermi-LAT analysis also revealed extended $\gamma$-ray emission, on scales similar to that of the extended H.E.S.S. component. Conclusions. Our modelling indicates that based on its spectrum and spatial extent, the extended H.E.S.S. component is likely caused by inverse Compton emission from old electrons that form a halo around the pulsar wind nebula. The compact component could be connected to either the pulsar wind nebula or the supernova remnant and molecular clouds. Due to its comparatively steep spectrum, modelling the Fermi-LAT emission together with the H.E.S.S. components is not straightforward. (abridged)