Altered luminosity functions of relativistically beamed jet populations

The intrinsic luminosity functions of extremely fast jets found in many active galaxies and gamma-ray bursts are difficult to measure since their apparent luminosities are strongly affected by relativistic beaming. Past studies have only provided analytical predictions for the beamed characteristics of populations in which all jets have the same Lorentz factor. However, the jets found in active galaxies are known to span a large range of speeds. Here we derive analytical expressions for the expected Doppler factor distributions and apparent (beamed) luminosity functions of randomly-oriented, two-sided jet populations which have bulk Lorentz factors distributed according to a simple power-law over a range [G1, G2]. We find that if a jet population has a uniform or inverted Lorentz factor distribution, its Doppler factor distribution and beamed luminosity function will be very similar to that of a jet population with Lorentz factors all equal to G2. In particular, the beamed and intrinsic luminosity functions will have the same slope at high luminosities. In the case of a steeply declining Lorentz factor distribution, the slopes at high luminosities will also be identical, but only up to an apparent luminosity equivalent to the upper cutoff of the intrinsic luminosity function. Also, at very high apparent luminosities, the slope will be proportional to the power law slope of the Lorentz factor distribution. At very low luminosities, the form of the apparent luminosity function is very sensitive to the lower cutoff and steepness of the Lorentz factor distribution. We discuss how it should be possible to recover useful information about the intrinsic luminosity functions of relativistic jets by studying flux-limited samples selected on the basis of beamed jet emission.