Past exam of the astrophysics course of the University of Oxford / 2015 / 8 / a / Solution

Use the supplied comoving distance $D_C=5421$ Mpc. In a flat background, $D_A=D_C/(1+z)=1807$ Mpc and $D_L=(1+z)D_C=16263$ Mpc. The observed wavelength is stretched by $1+z$, soCare is needed with a luminosity per frequency. For corresponding emitted and observed frequencies, $d{\nu }_{\mathrm{em}}=(1+z)d{\nu }_{\mathrm{obs}}$. Combining that bandwidth relation with the bolometric definition of $D_L$ givesThis is the spectral luminosity at the rest-frame frequency corresponding to 200 nm. A value at some different rest wavelength would require the quasar's spectral shape. The factor is different for a density per unit wavelength; interchanging these two conventions would give an incorrect redshift factor.