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 . This concept is extensively used in radio astronomy and planetary science.
For a black body, Planck's law gives :
where
(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 and, is the temperature of the black body, is Planck's constant, is frequency, is the speed of light and is Boltzmann's constant.
For a grey body the spectral radiance is a portion of the black body radiance, determined by the emissivity . That makes the reciprocal of the brightness temperature:
At low frequency and high temperatures, when, we can use the Rayleigh–Jeans law:
so that the brightness temperature can be simply written as:
In general, the brightness temperature is a function of, 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.
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—Blaise Cendrars (1887–1961)
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