Doppler factor (${\delta }_D=[\Gamma (1-\beta \cos \theta ){]}^{-1}$)

For velocity $\beta c$ at angle $\theta$ toward the observer, ${\delta }_D=[\Gamma (1-\beta \cos \theta ){]}^{-1}$. Frequencies transform as $\nu ={\delta }_D{\nu }^{\prime }$ before cosmological redshift. Specific-intensity transformation follows invariance of $I_{\nu }/{\nu }^3$.