Gamma-ray binaries allow us to study physical processes such as particle acceleration up to TeV energies and VHE gamma-ray emission and absorption with changing geometrical configurations on a periodic basis. These sources produce outflows of radio-emitting particles whose structure can be imaged with VLBI. LS 5039 is a gamma-ray binary that has shown variable VLBI structures in the past. We aim to characterise the radio morphological changes of LS 5039 and discriminate if they are either repeatable or erratic. We observed LS 5039 with the VLBA at 5 GHz during five consecutive days to cover the 3.9-day orbit and an extra day to disentangle between orbital or secular variability. We also compiled the available high-resolution radio observations of the source to study its morphological variability at different orbital phases. We used a simple model to interpret the obtained images. The new observations show that the morphology of LS 5039 up to projected distances of 10 milliarcseconds changes in 24 h. The observed radio morphological changes display a periodic orbital modulation. Multifrequency and multiepoch VLBI observations confirm that the morphological periodicity is stable on timescales of years. Using a simple model we show that the observed behaviour is compatible with the presence of a young non-accreting pulsar with an outflow behind it. The morphology is reproduced for inclinations of the orbit of 60-75 deg. For masses of the companion star in the range 20-50 Msun, this range of inclinations implies a mass of the compact object of 1.3-2.7 Msun. The periodic orbital modulation of the radio morphology of LS 5039 suggests that all gamma-ray binaries are expected to show a similar behaviour. The changes in the radio structure of LS 5039 are compatible with the presence of a young non-accreting neutron star, which suggests that the known gamma-ray binaries contain young pulsars.
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