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| | Ship Surveyor Florencio Harig from Brompton, has pastimes such as knitting, tires for sale and russian dolls collecting. that included likely to Canaima National Park.<br><br>my web page [http://rz9.de/tiresforsale32603 buy tires online] |
| {{Refimprove|date=October 2008}}
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| {{more footnotes|date=August 2013}}
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| }}
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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>.
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| 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>
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| 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" />
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| ::<math>I_\nu = \frac{2 h\nu^{3}}{c^2}\frac{1}{e^{\frac{h\nu}{kT}}-1}</math>
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| where
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| <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]].
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| For a '''grey body''' the [[spectral radiance]] is a portion of the black body radiance, determined by the [[emissivity]] <math>\epsilon</math>.
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| That makes the reciprocal of the brightness temperature:
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| ::<math>T_b^{-1} = \frac{k}{h\nu}\, \text{ln}\left[1 + \frac{e^{\frac{h\nu}{kT}}-1}{\epsilon}\right]</math>
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| 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>
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| ::<math>I_{\nu} = \frac{2 \nu^2k T}{c^2}</math>
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| so that the brightness temperature can be simply written as:
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| ::<math>T_b=\epsilon T\,</math>
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| 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.
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| == Calculating by frequency ==
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| 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>
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| : <math>T_b=\frac{h\nu}{k} \ln^{-1}\left( 1 + \frac{2h\nu^3}{I_{\nu}c^2} \right)</math>
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| When <math>h\nu \ll kT</math> we can use Rayleigh–Jeans law:
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| : <math>T_b=\frac{I_{\nu}c^2}{2k\nu^2}</math>
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| 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:
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| : <math>T_b=\frac{I c^2}{2k\nu^2\Delta\nu}</math>
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| == Calculating by wavelength ==
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| 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>
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| : <math>I_{\lambda}=\frac{2 hc^2}{\lambda^5}\frac{1}{ e^{\frac{hc}{kT \lambda}} - 1}</math>
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| So, the brightness temperature can be calculated as:
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| : <math>T_b=\frac{hc}{k\lambda} \ln^{-1}\left(1 + \frac{2hc^2}{I_{\lambda}\lambda^5} \right)</math>
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| For long-wave radiation <math>hc/\lambda \ll kT</math> the brightness temperature is:
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| : <math>T_b=\frac{I_{\lambda}\lambda^4}{2kc}</math>
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| For almost monochromatic radiation, the brightness temperature can be expressed by the [[radiance]] <math>I</math> and the [[coherence length]] <math>L_c</math>:
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| : <math>T_b=\frac{\pi I \lambda^2 L_c}{4kc \ln{2} }</math>
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| 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> 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 {{val|14|e=9|u=K}}.
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| <!-- Removed text that claims that brighness temperatures are only useful in Rayleigh–Jeans regime. That is not at all true -->
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| <!--
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| 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.
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| If the Planck distribution is reintroduced into the expression for brightness temperature we find:
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| <math>{T_b=\frac{h \nu}{k (\text{Exp}[h \nu /k T]-1)}}</math>
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| So, e.g. for the Sun, where the temperature may be estimated to be 6000 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.
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| -->
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| <!-- Image with unknown copyright status removed: [[Image:brightness_temp2.jpg]] -->
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| ==See also==
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| * Compare with [[color temperature]] and [[effective temperature]].
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| ==References==
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| {{Reflist}}
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| [[Category:Thermodynamics]]
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| [[Category:Radio astronomy]]
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| [[Category:Planetary science]]
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Ship Surveyor Florencio Harig from Brompton, has pastimes such as knitting, tires for sale and russian dolls collecting. that included likely to Canaima National Park.
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