Modeling the three-dimensional pair cascade in binaries. Application to LS 5039

10 pages, 11 figures, accepted for publication in Astronomy and AstrophysicsLS 5039 is a Galactic binary system emitting high and very-high energy gamma rays. The gamma-ray flux is modulated on the orbital period and the TeV lightcurve shaped by photon-photon annihilation. The observed very-high energy modulation can be reproduced with a simple leptonic model but fails to explain the flux detected by HESS at superior conjunction, where gamma rays are fully absorbed. The contribution from an electron-positron pair cascade could be strong and prevail over the primary flux at superior conjunction. The created pairs can be isotropized by the magnetic field, resulting in a three-dimensional cascade. The aim of this article is to investigate the gamma ray radiation from this pair cascade in LS 5039. This additional component could account for HESS observations at superior conjunction in the system. A semi-analytical and a Monte Carlo method for computing three-dimensional cascade radiation are presented and applied in the context of binaries. Three-dimensional cascade radiation contributes significantly at every orbital phase in the TeV lightcurve, and dominates close to superior conjunction. The amplitude of the gamma-ray modulation is correctly reproduced for an inclination of the orbit of about 40 degrees. Primary pairs should be injected close to the compact object location, otherwise the shape of the modulation is not explained. In addition, synchrotron emission from the cascade in X-rays constrains the ambient magnetic field to below 10 G. The radiation from a three-dimensional pair cascade can account for the TeV flux detected by HESS at superior conjunction in LS 5039, but the very-high energy spectrum at low fluxes remains difficult to explain in this model