One-dimensional pair cascade emission in gamma-ray binaries

Context. In gamma-ray binaries such as LS 5039, a large number of electron-positron pairs are created by the annihilation of primary very high-energy (VHE) gamma rays with photons from the massive star. The radiation from these particles contributes to the total high-energy gamma-ray flux and can initiate a cascade, decreasing the effective gamma-ray opacity in the system. Aims. The aim of this paper is to model the cascade emission and investigate whether it can account for the VHE gamma-ray flux detected by HESS from LS 5039 at superior conjunction, where the primary gamma rays are expected to be fully absorbed. Methods. A one-dimensional cascade develops along the line-of-sight if the deflections of pairs induced by the surrounding magnetic field can be neglected. A semi-analytical approach can then be adopted, including the effects of the anisotropic seed radiation field from the companion star. Results. Cascade equations are numerically solved, yielding the density of pairs and photons. In LS 5039, the cascade contribution to the total flux is large and anti-correlated with the orbital modulation of the primary VHE gamma rays. The cascade emission dominates close to superior conjunction but is too strong to be compatible with HESS measurements. Positron annihilation does not produce detectable 511 keV emission. Conclusions. This study provides an upper limit to cascade emission in gamma-ray binaries at orbital phases where absorption is strong. The pairs are likely to be deflected or isotropized by the ambient magnetic field, which will reduce the resulting emission seen by the observer. Cascade emission remains a viable explanation for the detected gamma rays at superior conjunction in LS 5039