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.^{[1]}
For a black body, Planck's law gives:^{[2]}^{[3]}


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:^{[3]}


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 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 nonthermal radiation.
Contents

Calculating by frequency 1

Calculating by wavelength 2

See also 3

References 4
Calculating by frequency
The brightness temperature of a source with known spectral radiance can be expressed as:^{[4]}

T_b=\frac{h\nu}{k} \ln^{1}\left( 1 + \frac{2h\nu^3}{I_{\nu}c^2} \right)
When h\nu \ll kT we can use Rayleigh–Jeans law:

T_b=\frac{I_{\nu}c^2}{2k\nu^2}
For narrowband radiation with the very low relative spectral linewidth \Delta\nu \ll \nu and known radiance I we can calculate brightness temperature as:

T_b=\frac{I c^2}{2k\nu^2\Delta\nu}
Calculating by wavelength
Spectral radiance of blackbody radiation is expressed by wavelength as:^{[5]}

I_{\lambda}=\frac{2 hc^2}{\lambda^5}\frac{1}{ e^{\frac{hc}{kT \lambda}}  1}
So, the brightness temperature can be calculated as:

T_b=\frac{hc}{k\lambda} \ln^{1}\left(1 + \frac{2hc^2}{I_{\lambda}\lambda^5} \right)
For longwave radiation hc/\lambda \ll kT the brightness temperature is:

T_b=\frac{I_{\lambda}\lambda^4}{2kc}
For almost monochromatic radiation, the brightness temperature can be expressed by the radiance I and the coherence length L_c:

T_b=\frac{\pi I \lambda^2 L_c}{4kc \ln{2} }
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 10^{26} 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 14×10^{9} K.
See also
References

^ "Brightness temperature".

^ Rybicki, George B., Lightman, Alan P., (2004) Radiative Processes in Astrophysics, ISBN 9780471827597

^ ^{a} ^{b} "Blackbody Radiation".

^ JeanPierre Macquart. "Radiative Processes in Astrophysics".

^ "Blackbody radiation. Main Laws. Brightness temperature".
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