Gamma-rays from nebulae around binary systems containing energetic rotation powered pulsars

We consider nebulae which are created around binary systems containing rotation powered pulsars and companion stars with strong stellar winds. It is proposed that the stellar and pulsar winds have to mix at some distance from the binary system, defined by the orbital period of the companion stars and the velocity of the stellar wind. The mixed pulsar-stellar wind expands with a specific velocity determined by the pulsar power and the mass loss rate of the companion star. Relativistic particles, either from the inner pulsar magnetosphere and/or accelerated at the shocks between stellar and pulsar winds, are expected to be captured and isotropized in the reference frame of the mixed wind. Therefore, they can efficiently comptonize stellar radiation producing GeV-TeV $\gamma$-rays in the inverse Compton process. We calculate the $\gamma$-ray spectra expected in such scenario for the two example binary systems: J1816+4510 which is the redback type millisecond binary and LS 5039 which is supposed to contain energetic pulsar. It is concluded that the steady TeV $\gamma$-ray emission from J1816+4510 should be on the 100 hr sensitivity limit of the planned Cherenkov Telescope Array, provided that $\epsilon\sim 10%$ of the rotational energy lost by the pulsar is transferred to TeV electrons. On the other hand, the comparison of the predicted steady TeV $\gamma$-ray emission, expected from $\gamma$-ray binary LS 5039, with the observations of the TeV emission in a low state, reported by the H.E.S.S. Collaboration, allows us to put stringent upper limit on the product of the part of the hemisphere in which the mixed pulsar-stellar wind is confined, $\Delta_{\rm mix}$, and the energy conversion efficiency, $\epsilon$, from the supposed pulsar to the TeV electrons injected in this system, $\Delta_{\rm mix}\cdot \epsilon <1%$.