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.
(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 and ; is the temperature of the black body; is Planck's constant; is frequency; is the speed of light; and is Boltzmann's constant.
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 it is 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.
Calculating by frequency
The brightness temperature of a source with known spectral radiance can be expressed as:
When we can use Rayleigh–Jeans law:
Calculating by wavelength
Spectral radiance of black-body radiation is expressed by wavelength as:
So, the brightness temperature can be calculated as:
For long-wave radiation the brightness temperature is:
It should be noted that the brightness temperature is not a temperature in ordinary comprehension. It characterizes radiation, and depending on the mechanism of radiation can differ considerably from the physical temperature of a radiating body (though it is theoretically possible to construct a device which will heat up by a source of radiation with some brightness temperature to the actual temperature equal to brightness temperature). Not thermal sources can have very high brightness temperature. At pulsars it can reach 1026 K. For the radiation of a typical helium–neon laser with a power of 60 mW and a coherence length of 20 cm, focused in a spot with a diameter of 10 µm, the brightness temperature will be nearly 14×109 K.
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