J. R. Martinez, V. Bosch-Ramon
Gamma-ray binaries hosting a non-accreting neutron star and a massive star exhibit multi-wavelength emission on different spatial scales. The interaction between the neutron star and stellar winds mixes them and produces an outflow whose interaction with the medium on large scales can inflate a bubble or form a bow shock in systems that are old enough. The well-studied but still mysterious high-energy emitting binary LS 5039 shows extended (1 pc scale) X-rays that might arise from one of these two interaction regimes. We explain and predict the large-scale emission from LS 5039. We modelled the LS 5039 large-scale emission, including the thermal and non-thermal contributions. For the latter, we assumed particle acceleration at the termination shock of the mixed-wind outflow. We considered five scenarios: a very young and an intermediate-age bubble interacting with the interstellar medium, and a bubble evolving inside a supernova remnant, studied using a one-zone model; and two older bow shocks with a moderate and a high neutron star power, respectively, for which we used a multi-zone approach. We also investigated the particles escaping from these structures and their radiation. The large-scale X-rays surrounding LS 5039 are best explained as synchrotron radiation, as previously proposed. In some cases, very extended radio emission might be detectable, while particles escaping the region may be a minor but non-negligible steady contribution to the source gamma rays in the powerful bow-shock scenario. Additionally, escaping protons and nuclei with 0.1--1 PeV might inject up to ∼ 10^ 36 into the Galactic cosmic rays for optimistic injection luminosities. Gamma-ray binaries can be efficient accelerators at large scales and produce non-negligible amounts of broadband emission and 0.1--1 PeV cosmic rays. Our study can help future multi-wavelength observational campaigns to better characterise the large-scale medium interaction, age, and birthplace of LS 5039.