The Universe Is Expanding — Question 6

Question 6

Luminosity distance and angular distance answer different questions

A source at z=1 has angular diameter distance $D_A=1.5$ Gpc. Assume photon conservation and $D_L=(1+z)^2D_A$. Its transverse size is 3.0 kpc.

1. Find luminosity distance and angular size in radians.

2. If one wrongly used DA instead of DL in the flux law $F=L/(4\pi D_L^2)$, by what factor would the predicted flux be too large?

Question 6 - Solution

Luminosity distance and angular distance answer different questions

(a) Method and calculation

$$\boxed{D_L=2^2(1.5)=6.0\ {\mathrm{Gpc}}}.$$$$\boxed{\theta=\ell/D_A=(3.0\times10^3)/(1.5\times10^9)=2.0\times10^{-6}\ {\mathrm{rad}}}.$$

(b) Interpretation and further calculation

$$F_{\mathrm{wrong}}/F_{\mathrm{true}}=D_L^2/D_A^2.$$$$\boxed{F_{\mathrm{wrong}}/F_{\mathrm{true}}=4^2=16}.$$

Physical check

The size is a proper transverse size at emission. Distance definitions converge nearby but differ substantially at high redshift.