Answer Happy

QUESTION 4. (4a) Show that the angles along which a particle is observed to move in one frame (unprimed) and in another frame (primed) that moves relative to the first frame with velocity v are related by the expression tan 0 = u' sin 0' (u' cos 0' + v) where w/ is the velocity of the particle in the primed frame. [8 marks] (4b) Apply this expression to the case of a photon, and use it to show that half of radiation emitted isotropically in the source frame will be concentrated in the observer's (unprimed) frame in a cone with half-angle approximately equal to 1/7, where y is the Lorentz factor for the source motion. [10 marks] (4c) The observed space densities of FR I radio galaxies and BL Lac objects are in the ratio of 20, 000/600. Estimate the typical jet Lorentz factors of these objects if they represent unbeamed and beamed members of a single population. Hint: The solid angle subtended by a cone of half-angle can be taken to be 0² when 0 is small. [7 marks] (4d) The observed flux density S is related to the flux density in the source rest frame S' as S = D².5 S", where D is the Doppler factor, D = [(1 - 3 cos 0-¹. The ratio of the fluxes of the jet and counter-jet of an AGN is observed to be R > 1500; use this information to derive limits for the angle the jet makes to the line of sight. [8 marks]