High energy emission from the nebula around the Black Widow binary system containing millisecond pulsar B1957+20

The features of pulsed $\gamma$-ray emission from classical and millisecond pulsars indicate that the high energy radiation processes in their inner magnetospheres occur in a similar way. In the last decade several TeV $\gamma$-ray nebulae have been discovered around classical pulsars. The above facts suggest that $\gamma$-rays should be produced also in the surroundings of millisecond pulsars. We discuss a model for the bow shock nebula around the well known Black Widow binary system containing the millisecond pulsar B1957+20. This model predicts the existence of a synchrotron X-ray and inverse Compton $\gamma$-ray nebula around this system. We want to find out whether $\gamma$-ray emission from the nebula around B1957+20 could be detected by the future and present Cherenkov telescopes. Using the Monte Carlo method we followed the propagation of relativistic electrons in the vicinity of the pulsar. We calculated the very high energy radiation produced by them in the synchrotron process and the inverse Compton scattering of the Microwave Background Radiation and of the infrared radiation from the galactic disk. We also computed the X-ray emission produced by the electrons in the synchrotron process. We show that the hard X-ray tail emission observed from the vicinity of B1957+20 can be explained by our model. Moreover, we predict that the TeV $\gamma$-ray emission produced by the electrons in the inverse Compton process should be detectable by the future Cherenkov Telescope Array and possibly by the long term observations with the present Cherenkov arrays such as MAGIC and VERITAS. The $\gamma$-ray emission from B1957+20 is expected to be extended, inhomogeneous, and shifted from the present location of the binary system by a distance comparable to the radius of the nebula.