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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:[1]

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 emitted per unit surface per unit time per unit solid angle and 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

References

  1. ^ Rybicki, George B., Lightman, Alan P., (2004) Radiative Processes in Astrophysics, ISBN 978-0-471-82759-7

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