Brightness temperature

Brightness temperature is the temperature at which a blackbody in thermal equilibrium with its surroundings would have to be in order to duplicate the observed intensity of an object at a frequency $u$.This is a useful concept only for radiation that obeys the Rayleigh–Jeans law, and it is extensively used in radio astronomy and planetary science.

For a blackbody, the Planck distribution gives:

::$I(\; u)\; =\; frac\{2\; h\; u^\{3\{c^2\}frac\{1\}\{e^\{frac\{h\; u\}\{kT-1\}$

where

:*$I(\; u)d\; u\; ,$ is the amount of energy per unit surface per unit time per unit solid angle emitted in the frequency range between ν and ν+dν;:*$T\; ,$ is the temperature of the black body;:*$h\; ,$ is Planck's constant;:*$u\; ,$ is Frequency.:*$c\; ,$ is the speed of light; and:*$k\; ,$ is Boltzmann's constant.

In the Rayleigh-Jeans limit of low frequency, we find:

$\{I\_\{\; u\; \}=frac\{2\; u\; ^2k\; T\}\{c^2$

This can be rewritten to define the brightness temperature as:

$\{T\_b=frac\{I\_\{\; u\; \}\; c^2\}\{2\; u\; ^2\; k$

Brightness temperature is a useful diagnostic for temperature measurement if the astronomical source is a blackbody and we are in the Rayleigh-Jeans regime. It is not useful if the source is non-thermal and/or we are in the high frequency limit.

If the Planck distribution is reintroduced into the expression for brightness temperature we find:

$\{T\_b=frac\{h\; u\}\{k\; (\; ext\{Exp\}\; [h\; u\; /k\; T]\; -1)$

So, e.g. for the Sun, where the temperature may be estimated to be 6000K, we can plot the brightness temperature against wavelength and we find that when $h\; u\; /k\; T$ is much less than one, the brightness temperature and physical temperature are the same, but at higher temperatures the brightness temperature is much lower.

ee also

Compare with color temperature and effective temperature.