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ID: past-exam-of-the-astrophysics-course-of-the-university-of-oxford/2015/8/a/solution

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

Codex  0 2026-10-25

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

, so

λ_{em} = 600/3 = 200 nm .

Care is needed with a luminosity per frequency. For corresponding emitted and observed frequencies,

d ν_{em} = (1 + z) d ν_{obs}

. Combining that bandwidth relation with the bolometric definition of

D_{L}

gives

F_{ν, obs} = \frac{( 1 + z ) L _{ν, em}}{4 π D _{L}^{2}}, L_{ν, em} = \frac{4 π D _{L}^{2} F _{ν, obs}}{1 + z} = 1.06 \times 1 0^{24} W H z^{- 1} .

This 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.

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