Brightness temperature

Brightness temperature is the temperature a black body in thermal equilibrium with its surroundings would have to be to duplicate the observed intensity of a grey body object at a frequency $\nu$. This concept is extensively used in radio astronomy and planetary science.

For a black body, Planck's law gives :

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

where

$I_\nu$ (the Intensity or Brightness) is the amount of energy per unit surface per unit time per unit solid angle emitted in the frequency range between $\nu$ and $\nu + d\nu$, $T$ is the temperature of the black body, $h$ is Planck's constant, $\nu$ is frequency, $c$ is the speed of light and $k$ is Boltzmann's constant.

For a grey body the spectral radiance is a portion of the black body radiance, determined by the emissivity $\epsilon$. That makes the reciprocal of the brightness temperature:

$T_b^{-1} = \frac{k}{h\nu}\, \text{ln}\left[1 + \frac{e^{\frac{h\nu}{kT}}-1}{\epsilon}\right]$

At low frequency and high temperatures, when $h\nu \ll kT$, we can use the Rayleigh–Jeans law:

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

so that the brightness temperature can be simply written as:

$T_b=\epsilon T\,$

In general, the brightness temperature is a function of $\nu$, and only in the case of blackbody radiation is it the same at all frequencies. The brightness temperature can be used to calculate the spectral index of a body, in the case of non-thermal radiation.

See also

Compare with color temperature and effective temperature.