Editing Brightness temperature

{{Short description|Measure of electromagnetic energy}}
'''Brightness temperature''' or '''radiance temperature''' is a measure of the intensity of electromagnetic energy coming from a source.<ref name="lewis">{{cite web |title=Brightness Units |url=https://www.lewiscenter.org/Old-Content/Global-Programs/Tools/Data-FAQs/Brightness-Units/index.html |website=Lewis Center for Educational Research |access-date=26 April 2023}}</ref> In particular, it is the temperature at which a [[black body]] would have to be in order to duplicate the observed [[Intensity (heat transfer)|intensity]] of a [[grey body]] object at a frequency <math>\nu</math>.<ref>{{cite web |url=https://www.astro.cf.ac.uk/observatory/radiotelescope/background/?page=brightness |title=Brightness Temperature |access-date=2015-09-29 |archive-url=https://web.archive.org/web/20170611230938/https://www.astro.cf.ac.uk/observatory/radiotelescope/background/?page=brightness |archive-date=2017-06-11 |url-status=dead }}</ref>
This concept is used in [[radio astronomy]],<ref name="keane">{{cite book |last1=Keane |first1=E.F. |title=The Transient Radio Sky |series=Springer Theses |url=https://link.springer.com/content/pdf/bbm:978-3-642-19627-0/1.pdf |publisher=Springer-Verlag Theses |access-date=26 April 2023 |location=Berlin Heidelberg |pages=171–174 |doi=10.1007/978-3-642-19627-0 |date=2011|isbn=978-3-642-19626-3 }}</ref> [[planetary science]],<ref>{{cite journal |last1=Maris |display-authors=etal |first1=M. |title=Revised planet brightness temperatures using the Planck/LFI 2018data release |journal=Astronomy & Astrophysics |date=2020 |url=https://www.researchgate.net/publication/346766315 |access-date=26 April 2023}}</ref> [[materials science]] and [[climatology]].<ref>{{cite web |title=AMSU Brightness Temperature-NOAA CDR |date=7 January 2021 |url=https://www.ncei.noaa.gov/products/climate-data-records/amsu-brightness-temperature-noaa |publisher=NOAA |access-date=26 April 2023}}</ref>

The brightness temperature provides "a more physically recognizable way to describe intensity".<ref>{{cite web |title = Emissivity, Energy Conservation, Brightness Temperature |url=http://profhorn.meteor.wisc.edu/wxwise/satmet/lesson2/emissivity.html |website=Satellite Meteorology |publisher=University of Wisconsin Madison |access-date=26 April 2023}}</ref>

When the electromagnetic radiation observed is [[thermal radiation]] emitted by an object simply by virtue of its temperature, then the actual temperature of the object will always be equal to or higher than the brightness temperature.<ref name="oxford">{{ cite web |title=brightness temperature |url=https://www.oxfordreference.com/display/10.1093/oi/authority.20110803095527476 |website=Oxford Reference |access-date=26 April 2023}}</ref> Since the [[emissivity]] is limited by 1, the brightness temperature is a lower bound of the object’s actual temperature.

For radiation emitted by a non-thermal source such as a pulsar, synchrotron, maser, or a laser, the brightness temperature may be far higher than the actual temperature of the source.<ref name="oxford"></ref> In this case, the brightness temperature is simply a measure of the intensity of the radiation as it would be measured at the origin of that radiation. 

In some applications, the brightness temperature of a surface is determined by an optical measurement, for example using a [[pyrometer]], with the intention of determining the real temperature. As detailed below, the real temperature of a surface can in some cases be calculated by dividing the brightness temperature by the [[emissivity]] of the surface. Since the emissivity is a value between 0 and 1, the real temperature will be greater than or equal to the brightness temperature. At high frequencies (short wavelengths) and low temperatures, the conversion must proceed through [[Planck's law]].

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).<ref name="RybickiLightman" /> 

Nonthermal sources can have very high brightness temperatures. In [[pulsar|pulsars]] the brightness temperature can reach 10<sup>30</sup>&nbsp;K.<ref name="blan92">{{cite journal |last1=Blandford |first1=R.D. |title=Pulsars and Physics |journal=Philosophical Transactions: Physical Sciences and Engineering |date=15 Oct 1992 |volume=341 |issue=1660 |pages=177–192 |jstor=53919 |url=https://www.jstor.org/stable/53919 |access-date=26 April 2023}}</ref> For the radiation of a [[helium–neon laser]] with a power of 1&nbsp;mW, a frequency spread Δf = 1 GHz, an output aperture of 1&nbsp;mm{{sup|2}}, and a beam dispersion half-angle of 0.56&nbsp;mrad, the brightness temperature would be {{val|1.5|e=10|u=K}}.<ref>{{cite web |title=Brightness Temperature of a Laser—C.E. Mungan, Spring 2010 |url=https://www.usna.edu/Users/physics/mungan/_files/documents/Scholarship/LaserBrightness.pdf |publisher=United States Naval Academy |access-date=26 April 2023}}</ref>

For a '''black body''', [[Planck's law]] gives:<ref name="RybickiLightman">Rybicki, George B., Lightman, Alan P., (2004) ''Radiative Processes in Astrophysics'', {{ISBN|978-0-471-82759-7}}</ref><ref name="BR" />
<math display="block">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 area]] 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 the [[Planck constant]]; <math>\nu</math> is [[frequency]]; <math>c</math> is the [[speed of light]]; and <math>k</math> is the [[Boltzmann 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 display="block">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|access-date=2013-08-24|archive-date=2018-03-07|archive-url=https://web.archive.org/web/20180307084011/http://www.cv.nrao.edu/course/astr534/BlackBodyRad.html|url-status=dead}}</ref>
<math display="block">I_{\nu} = \frac{2 \nu^2k T}{c^2}</math>
so that the brightness temperature can be simply written as:
<math display="block">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]] 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:<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}}{{dead link|date=July 2017 |bot=InternetArchiveBot |fix-attempted=yes }}</ref>
<math display="block">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 the Rayleigh–Jeans law:
<math display="block">T_b=\frac{I_{\nu}c^2}{2k\nu^2}</math>

For [[narrowband]] radiation with very low relative [[spectral linewidth]] <math>\Delta\nu \ll \nu</math> and known [[radiance]] <math>I</math> we can calculate the brightness temperature as:
<math display="block">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:
<math display="block">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 display="block">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 display="block">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 display="block">T_b = \frac{\pi I \lambda^2 L_c}{4kc \ln{2} }</math>

==In oceanography==

In oceanography, the microwave brightness temperature, as measured by satellites looking at the ocean surface, depends on salinity as well as on the temperature and roughness (e.g. from wind-driven waves) of the water.<ref>{{cite web |title=Can you explain "brightness temperature"? |url=https://aquarius.oceansciences.org/docs/aq_qa_webinar1_session1.pdf |publisher=NASA |access-date=26 April 2023 |archive-date=17 May 2023 |archive-url=https://web.archive.org/web/20230517192717/https://aquarius.oceansciences.org/docs/aq_qa_webinar1_session1.pdf |url-status=dead }}</ref>

== References ==
{{reflist}}

[[Category:Temperature]]
[[Category:Radio astronomy]]
[[Category:Planetary science]]