Atmospheric pressure

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:

$I_{\nu }={\frac {2h\nu ^{3}}{c^{2}}}{\frac {1}{e^{\frac {h\nu }{kT}}-1}}$

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:

$I_{\nu }={\frac {2\nu ^{2}kT}{c^{2}}}$

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.

Calculating by frequency

The brightness temperature of a source with known spectral radiance can be expressed as:

$T_{b}={\frac {h\nu }{k}}\ln ^{-1}\left(1+{\frac {2h\nu ^{3}}{I_{\nu }c^{2}}}\right)$

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:

$T_{b}={\frac {Ic^{2}}{2k\nu ^{2}\Delta \nu }}$

Calculating by wavelength

Spectral radiance of black body radiation is expressed by wavelength as:

$I_{\lambda }={\frac {2hc^{2}}{\lambda ^{5}}}{\frac {1}{e^{\frac {hc}{kT\lambda }}-1}}$

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 long-wave 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 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 Template:Val.

See also

• Compare with color temperature and effective temperature.

References

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