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'''Brightness temperature''' is the temperature a [[black body]] in [[thermal equilibrium]] with its surroundings would have to be to duplicate the observed [[Intensity (heat transfer)|intensity]] of a [[grey body]] object at a frequency <math>\nu</math>.
This concept is extensively used in [[radio astronomy]] and [[planetary science]].<ref>{{cite web|url=http://www.princeton.edu/~achaney/tmve/wiki100k/docs/Brightness_temperature.html|title= Brightness temperature}}</ref>
 
For a '''black body''', [[Planck's law]] gives:<ref>Rybicki, George B., Lightman, Alan P., (2004) ''Radiative Processes in Astrophysics'', ISBN 978-0-471-82759-7</ref><ref name="BR" />
 
::<math>I_\nu = \frac{2 h\nu^{3}}{c^2}\frac{1}{e^{\frac{h\nu}{kT}}-1}</math>
 
where
 
<math>I_\nu</math> (the [[Intensity (physics)|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 <math>\nu</math> and <math>\nu + d\nu</math>; <math>T</math> is the [[temperature]] of the black body; <math>h</math> is [[Planck's constant]]; <math>\nu</math> is [[frequency]]; <math>c</math> is the [[speed of light]]; and <math>k</math> is [[Boltzmann's constant]].
 
For a '''grey body''' the  [[spectral radiance]] is a portion of the black body radiance, determined by the [[emissivity]] <math>\epsilon</math>.
That makes the reciprocal of the brightness temperature:
 
::<math>T_b^{-1} = \frac{k}{h\nu}\, \text{ln}\left[1 + \frac{e^{\frac{h\nu}{kT}}-1}{\epsilon}\right]</math>
 
At low frequency and high temperatures, when <math>h\nu \ll kT</math>, we can use the [[Rayleigh–Jeans law]]:<ref name="BR">{{cite web|url=http://www.cv.nrao.edu/course/astr534/BlackBodyRad.html|title= Blackbody Radiation}}</ref>
 
::<math>I_{\nu} = \frac{2 \nu^2k T}{c^2}</math>
 
so that the brightness temperature can be simply written as:
 
::<math>T_b=\epsilon T\,</math>
 
In general, the brightness temperature is a function of <math>\nu</math>, and only in the case of [[Black-body radiation|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:<ref>{{cite web|url=http://www.icrar.org/__data/assets/pdf_file/0006/819510/radiative.pdf|author= Jean-Pierre Macquart|title=Radiative Processes in Astrophysics}}</ref>
 
: <math>T_b=\frac{h\nu}{k} \ln^{-1}\left( 1 + \frac{2h\nu^3}{I_{\nu}c^2} \right)</math>
 
When <math>h\nu \ll kT</math> we can use Rayleigh–Jeans law:
: <math>T_b=\frac{I_{\nu}c^2}{2k\nu^2}</math>
 
For [[narrowband]] radiation with the very low relative [[spectral linewidth]] <math>\Delta\nu \ll \nu</math> and known [[radiance]] <math>I</math> we can calculate brightness temperature as:
: <math>T_b=\frac{I c^2}{2k\nu^2\Delta\nu}</math>
 
== Calculating by wavelength ==
Spectral radiance of black body radiation is expressed by wavelength as:<ref>{{cite web|url=http://www.heliosat3.de/e-learning/remote-sensing/Lec4.pdf|title=Blackbody radiation. Main Laws. Brightness temperature}}</ref>
: <math>I_{\lambda}=\frac{2 hc^2}{\lambda^5}\frac{1}{ e^{\frac{hc}{kT \lambda}} - 1}</math>
 
So, the brightness temperature can be calculated as:
: <math>T_b=\frac{hc}{k\lambda} \ln^{-1}\left(1 + \frac{2hc^2}{I_{\lambda}\lambda^5}  \right)</math>
 
For long-wave radiation <math>hc/\lambda \ll kT</math> the brightness temperature is:
: <math>T_b=\frac{I_{\lambda}\lambda^4}{2kc}</math>
 
For almost monochromatic radiation, the brightness temperature can be expressed by the [[radiance]] <math>I</math> and the [[coherence length]] <math>L_c</math>:
: <math>T_b=\frac{\pi I \lambda^2 L_c}{4kc \ln{2} }</math>
 
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 [[pulsar]]s it can reach 10<sup>26</sup>&nbsp;K. For the radiation of a typical [[helium-neon laser]] with a power of 60&nbsp;mW and a coherence length of 20&nbsp;cm, focused in a spot with a diameter of 10&nbsp;[[µm]], the brightness temperature will be nearly {{val|14|e=9|u=K}}.
 
<!-- Removed text that claims that brighness temperatures are only useful in Rayleigh–Jeans regime. That is not at all true -->
<!--
Brightness temperature is a useful diagnostic for temperature measurement if the astronomical source is a black body and we are in the Rayleigh–Jeans regime. It is not useful if the source is non-thermal and/or we are in the high frequency limit.
 
If the Planck distribution is reintroduced into the expression for brightness temperature we find:
 
<math>{T_b=\frac{h \nu}{k (\text{Exp}[h \nu /k T]-1)}}</math>
 
So, e.g. for the Sun, where the temperature may be estimated to be 6000&nbsp;K, we can plot the brightness temperature against wavelength and we find that when <math>h \nu /k T</math> is much less than one, the brightness temperature and physical temperature are the same, but at higher temperatures the brightness temperature is much lower.
-->
 
<!-- Image with unknown copyright status removed: [[Image:brightness_temp2.jpg]] -->
 
==See also==
* Compare with [[color temperature]] and [[effective temperature]].
 
==References==
{{Reflist}}
 
[[Category:Thermodynamics]]
[[Category:Radio astronomy]]
[[Category:Planetary science]]

Revision as of 06:34, 1 February 2014

Template:Multiple issues

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 ν. This concept is extensively used in radio astronomy and planetary science.[1]

For a black body, Planck's law gives:[2][3]

Iν=2hν3c21ehνkT1

where

Iν (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 ν and ν+dν; T is the temperature of the black body; h is Planck's constant; ν 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 ϵ. That makes the reciprocal of the brightness temperature:

Tb1=khνln[1+ehνkT1ϵ]

At low frequency and high temperatures, when hνkT, we can use the Rayleigh–Jeans law:[3]

Iν=2ν2kTc2

so that the brightness temperature can be simply written as:

Tb=ϵT

In general, the brightness temperature is a function of ν, 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:[4]

Tb=hνkln1(1+2hν3Iνc2)

When hνkT we can use Rayleigh–Jeans law:

Tb=Iνc22kν2

For narrowband radiation with the very low relative spectral linewidth Δνν and known radiance I we can calculate brightness temperature as:

Tb=Ic22kν2Δν

Calculating by wavelength

Spectral radiance of black body radiation is expressed by wavelength as:[5]

Iλ=2hc2λ51ehckTλ1

So, the brightness temperature can be calculated as:

Tb=hckλln1(1+2hc2Iλλ5)

For long-wave radiation hc/λkT the brightness temperature is:

Tb=Iλλ42kc

For almost monochromatic radiation, the brightness temperature can be expressed by the radiance I and the coherence length Lc:

Tb=πIλ2Lc4kcln2

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

References

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  1. Template:Cite web
  2. Rybicki, George B., Lightman, Alan P., (2004) Radiative Processes in Astrophysics, ISBN 978-0-471-82759-7
  3. 3.0 3.1 Template:Cite web
  4. Template:Cite web
  5. Template:Cite web