General Relativity: An Introduction for Physicists · Problem 14.10

The Friedmann–Robertson–Walker geometry — Hobson, Efstathiou & Lasenby

Physics · surface-brightness-cosmological-objects

Problem

The observed surface brightness

\Sigma_{\mathrm{obs}}

of an extended object observed in the frequency range

[\nu_1, \nu_2]

is defined as the observed flux per unit solid angle. Thus, for a (small) circular object subtending an angular diameter

\Delta\theta

we have

\Sigma_{\mathrm{obs}} = \frac{4F_{\mathrm{obs}}(\nu_1, \nu_2)}{\pi(\Delta\theta)^2},

where

F_{\mathrm{obs}}(\nu_1, \nu_2)

is defined in Exercise 14.9. Show that

\Sigma_{\mathrm{obs}}

can be written as

\Sigma_{\mathrm{obs}} = \frac{4}{\pi}\frac{L_{\mathrm{em}}(\nu_1,\nu_2)}{4\pi\ell^2}\frac{K_z}{(1+z)^4},

where

\ell

is the physical (projected) diameter of the object,

L_{\mathrm{em}}(\nu_1, \nu_2)

is the intrinsic luminosity of the object in the frequency range

[\nu_1, \nu_2]

and

K_z

is the

K

-correction. Note: The above result is independent of cosmological parameters. Moreover, setting aside the K-correction, the

(1+z)^{-4}

-dependence means that the surface brightness of extended objects drops very rapidly with redshift, making the detection of high-

z

objects difficult.