  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.
