The Influence of Jet Geometry on the Statistics of Beamed Objects and its Implication for VHE Surveys

We investigate the role of a finite opening angle on the appearance of relativistic jets. We show that both apparent Doppler factors and superluminal motion are strongly affected by this effect. We simulate several configurations with various dependencies of the Lorentz factor gb as a function of the direction. For each model, the appearance of a moving shell is computed, taking into account relativistic motion and light travel time delays. The apparent velocity is computed by assuming that it is dominated by the brightest point in the sky, corresponding to the maximal possible Doppler factor. It is shown that the apparent velocity is strongly dependent on the smoothness of the variation of gb with the direction. A smooth variation limits the apparent velocity to the inverse of the typical angular range of variation, even for high Lorentz factors. The maximal Doppler factor depends on the precise distribution of gb. We discuss the implication of such results on the statistics of beamed sources and the possible luminosity function of CTA extragalactic sources.