# Brightness temperature

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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 ν. This concept is extensively used in radio astronomy and planetary science.

For a black body, Planck's law gives:

where

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

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

so that the brightness temperature can be simply written as: