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 $\nu$ .^[1] This concept is extensively used in radio astronomy and planetary science.

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).^[2] 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:^[2]^[3]

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 area 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 ^{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 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:^[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 the Rayleigh–Jeans law:

T_{b}={\frac {I_{{\nu }}c^{2}}{2k\nu ^{2}}}

For narrowband radiation with very low relative spectral linewidth $\Delta \nu \ll \nu$ and known radiance $I$ we can calculate the 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:^[5]

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}}}

References

↑ "Brightness Temperature".
1 2 Rybicki, George B., Lightman, Alan P., (2004) Radiative Processes in Astrophysics, ISBN 978-0-471-82759-7
1 2 "Blackbody Radiation".
↑ Jean-Pierre Macquart. "Radiative Processes in Astrophysics" (PDF).
↑ "Blackbody radiation. Main Laws. Brightness temperature" (PDF).

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