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
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The brightness temperature is not a temperature as ordinarily understood. 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). Nonthermal sources can have very high brightness temperatures. In pulsars the brightness temperature 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 7010140000000000000♠14×109 K.
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
At low frequency and high temperatures, when
so that the brightness temperature can be simply written as:
In general, the brightness temperature is a function of
Calculating by frequency
The brightness temperature of a source with known spectral radiance can be expressed as:
When
For narrowband radiation with very low relative spectral linewidth
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
For almost monochromatic radiation, the brightness temperature can be expressed by the radiance