Brightness temperature

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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 $ν$ . 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:

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

where

$I(\nu)d\nu \,$ 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;
$c \,$ is the speed of light; and
$k \,$ is Boltzmann's constant.

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

${I_{\nu }=\frac{2 \nu ^2k T}{c^2}}$

This can be rewritten to define the brightness temperature as:

${T_b=\frac{I_{\nu } c^2}{2 \nu ^2 k}}$

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:

${T_b=\frac{h \nu}{k (\text{Exp}[h \nu /k T]-1)}}$