  Several gamma-ray binaries have been recently detected by the High-Energy
Stereoscopy Array (H.E.S.S.) and the Major Atmospheric Imaging Cerenkov (MAGIC)
telescope. In two cases, their nature is unknown, since a final observational
feature for a black hole or a pulsar compact object companion is still missing.
One such system is LS 5039. Here we present results from a model (it includes a
detailed account of the system geometry, the angular dependence of processes
such as Klein-Nishina inverse Compton and gamma-gamma absorption, and a Monte
Carlo simulation of cascading) of the high energy phenomenology of LS 5039 in
which it is assumed that the companion object is a pulsar rotating around an
O6.5V star in the 3.9 days orbit. We show that the H.E.S.S. phenomenology at
all scales (spectra along the orbit in both broad and short phase-bins and
lightcurve) is described within this model. We focus on presenting predictions
for photons with lower energies (for E>1 GeV), subject to test in the
forthcoming months with the GLAST satellite, and we also present predictions
for future observations with high-energy arrays, such as H.E.S.S. II. Both set
of predictions go beyond the description of the current data, and could provide
a high-energy determination of the system's composition.
