Unified model for X-ray and radio selected BL Lac objects

The existence of two populations of BL Lac sources, characterized by different spectral properties and number densities, has been confirmed by the recently obtained complete samples of radio- and X-ray-selected sources. In the framework of relativistic jet models, the different properties of the two populations can be accounted for if the X-ray radiation is less beamed than the radio one. We propose that this can be due to a larger opening angle of the flow velocity in the inner, X-ray-emitting part of a jet of constant bulk Lorentz factor. With these assumptions we compute the expected luminosity functions (LFs) in the radio and X-ray bands for beamed objects deriving from the same parent population but observed at different angles. We apply this formalism to X-ray-selected and radio-selected BL Lac objects, assuming FR I radio galaxies as the parent population. From the available data we derive here the radio LFs of X-ray-selected BL Lac objects and the X-ray LFs of radio-selected BL Lac objects in order to allow a statistically meaningful comparison of the two samples in the same band. We find the average bulk Lorentz factor and the opening angles of the X-ray- and radio-emitting parts of the jet. We conclude that FR I radio galaxies and X-ray- and radio-selected BL Lac objects can be the same phenomenon observed at decreasing angle to the jet axis.