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{{Short description|Thermal electromagnetic radiation}}
{{lead too long|date=January 2024}}
'''Black-body radiation''' is the [[thermal radiation|thermal]] [[electromagnetic radiation]] within, or surrounding, a body in [[thermodynamic equilibrium]] with its environment, emitted by a [[black body]] (an idealized opaque, non-reflective body). It has a specific, [[continuous spectrum]] of [[wavelength]]s, inversely related to intensity, that depend only on the body's [[temperature]], which is assumed, for the sake of calculations and theory, to be uniform and constant.<ref>{{harvnb|Loudon|2000}}, Chapter 1.</ref><ref>{{harvnb|Mandel|Wolf|1995}}, Chapter 13.</ref><ref>{{harvnb|Kondepudi|Prigogine|1998}}, Chapter 11.</ref><ref name=Landsberg>{{harvnb|Landsberg|1990}}, Chapter 13.</ref>[[File:Black body.svg|thumb|upright=1.4|As the temperature of a black body decreases, the emitted thermal radiation decreases in intensity and its maximum moves to longer wavelengths. Shown for comparison is the classical [[Rayleigh–Jeans law]] and its [[ultraviolet catastrophe]].]]
A perfectly insulated enclosure which is in thermal equilibrium internally contains blackbody radiation, and will emit it through a hole made in its wall, provided the hole is small enough to have a negligible effect upon the equilibrium. The thermal radiation spontaneously emitted by many ordinary objects can be approximated as blackbody radiation.
Of particular importance, although planets and stars (including the [[Earth]] and [[Sun]]) are neither in thermal equilibrium with their surroundings nor perfect black bodies, blackbody radiation is still a good first approximation for the energy they emit. The
A black body at room temperature ({{convert|23|C|K+F}}) radiates mostly in the [[infrared]] spectrum, which cannot be perceived by the human eye,<ref>[[J.R. Partington|Partington, J.R.]] (1949), p. 466.</ref> but can be sensed by some reptiles. As the object increases in temperature to about {{convert|500|C|K+F}}, the emission spectrum gets stronger and extends into the human visual range, and the object appears dull red. As its temperature increases further, it emits more and more orange, yellow, green, and blue light (and ultimately beyond violet, [[ultraviolet]]).
▲Of particular importance, although planets and stars (including the [[Earth]] and [[Sun]]) are neither in thermal equilibrium with their surroundings nor perfect black bodies, blackbody radiation is still a good first approximation for the energy they emit. The sun's radiation, after being filtered by the earth's atmosphere, thus characterises "daylight", which humans (also most other animals) have evolved to use for vision.<ref name=Morison>{{cite book |title=Introduction to Astronomy and Cosmology |author=Ian Morison |url=https://books.google.com/books?id=yrV8vvJzgWkC&pg=PA48 |page=48 |isbn=978-0-470-03333-3 |year=2008 |publisher=J Wiley & Sons}}</ref>
[[Tungsten filament]] lights have a continuous black body spectrum with a cooler colour temperature, around {{convert|2700|K|C+F}}, which also emits considerable energy in the infrared range. Modern-day [[fluorescent]] and [[LED]] lights, which are more efficient, do not have a continuous black body emission spectrum, rather emitting directly, or using combinations of phosphors that emit multiple narrow spectrums.
▲A black body at room temperature ({{convert|23|C|K+F}}) radiates mostly in the [[infrared]] spectrum, which cannot be perceived by the human eye,<ref>[[J.R. Partington|Partington, J.R.]] (1949), p. 466.</ref> but can be sensed by some reptiles. As the object increases in temperature to about {{convert|500|C|K+F}}, the emission spectrum gets stronger and extends into the human visual range, and the object appears dull red. As its temperature increases further, it emits more and more orange, yellow, green, and blue light (and ultimately beyond violet, [[ultraviolet]]).
▲[[Tungsten filament]] lights have a continuous black body spectrum with a cooler colour temperature, around {{convert|2700|K|C+F}}, which also emits considerable energy in the infrared range. Modern-day [[fluorescent]] and [[LED]] lights, which are more efficient, do not have a continuous black body emission spectrum, rather emitting directly, or using combinations of phosphors that emit multiple narrow spectrums.
[[Image:PlanckianLocus.png|thumb|303px|The color ([[chromaticity]]) of blackbody radiation scales inversely with the temperature of the black body; the [[Locus (mathematics)|locus]] of such colors, shown here in [[CIE 1931 color space|CIE 1931 ''x,y'' space]], is known as the [[Planckian locus]].]]
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===Spectrum===
[[File:Blacksmith at work.jpg|thumb|upright=1.5|Blacksmiths judge workpiece temperatures by the colour of the glow.<ref>{{cite web |author=Dustin |title=How Do Blacksmiths Measure The Temperature Of Their Forge And Steel?|url=https://blacksmithu.com/how-blacksmiths-measure-temperature/ |website=Blacksmith U|date=18 December 2018 }}</ref>]]
[[File:Gluehfarben no language horizontal.svg|thumb|upright=3|This blacksmith's colourchart stops at the melting temperature of steel]]
Black-body radiation has a characteristic, continuous [[spectral energy distribution|frequency spectrum]] that depends only on the body's temperature,<ref name=Kogure>{{cite book |chapter-url=https://books.google.com/books?id=qt5sueHmtR4C&pg=PA41 |page=41 |chapter=§2.3: Thermodynamic equilibrium and blackbody radiation |title=The astrophysics of emission-line stars |author1=Tomokazu Kogure |author2=Kam-Ching Leung |isbn=978-0-387-34500-0 |year=2007 |publisher=Springer}}</ref> called the Planck spectrum or [[Planck's law]]. The spectrum is peaked at a characteristic frequency that shifts to higher frequencies with increasing temperature, and at [[room temperature]] most of the emission is in the [[infrared]] region of the [[electromagnetic spectrum]].<ref>Wien, W. (1893). Eine neue Beziehung der Strahlung schwarzer Körper zum zweiten Hauptsatz der Wärmetheorie, ''Sitzungberichte der Königlich-Preußischen Akademie der Wissenschaften '' (Berlin), 1893, '''1''': 55–62.</ref><ref>Lummer, O., Pringsheim, E. (1899). Die Vertheilung der Energie im Spectrum des schwarzen Körpers, ''Verhandlungen der Deutschen Physikalischen Gessellschaft'' (Leipzig), 1899, '''1''': 23–41.</ref><ref name="Planck 1914">{{harvnb|Planck|1914}}</ref> As the temperature increases past about 500 degrees [[Celsius]], black bodies start to emit significant amounts of visible light. Viewed in the dark by the human eye, the first faint glow appears as a "ghostly" grey (the visible light is actually red, but low intensity light activates only the eye's grey-level sensors). With rising temperature, the glow becomes visible even when there is some background surrounding light: first as a dull red, then yellow, and eventually a "dazzling bluish-white" as the temperature rises.<ref>[[John William Draper|Draper, J.W.]] (1847). On the production of light by heat, ''London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science'', series 3, '''30''': 345–360. [https://archive.org/stream/londonedinburghp30lond#page/344/mode/2up]</ref><ref>{{harvnb|Partington|1949|pages = 466–467, 478}}
▲</ref> called the Planck spectrum or [[Planck's law]]. The spectrum is peaked at a characteristic frequency that shifts to higher frequencies with increasing temperature, and at [[room temperature]] most of the emission is in the [[infrared]] region of the [[electromagnetic spectrum]].<ref>Wien, W. (1893). Eine neue Beziehung der Strahlung schwarzer Körper zum zweiten Hauptsatz der Wärmetheorie, ''Sitzungberichte der Königlich-Preußischen Akademie der Wissenschaften '' (Berlin), 1893, '''1''': 55–62.</ref><ref>Lummer, O., Pringsheim, E. (1899). Die Vertheilung der Energie im Spectrum des schwarzen Körpers, ''Verhandlungen der Deutschen Physikalischen Gessellschaft'' (Leipzig), 1899, '''1''': 23–41.</ref><ref name="Planck 1914">{{harvnb|Planck|1914}}</ref> As the temperature increases past about 500 degrees [[Celsius]], black bodies start to emit significant amounts of visible light. Viewed in the dark by the human eye, the first faint glow appears as a "ghostly" grey (the visible light is actually red, but low intensity light activates only the eye's grey-level sensors). With rising temperature, the glow becomes visible even when there is some background surrounding light: first as a dull red, then yellow, and eventually a "dazzling bluish-white" as the temperature rises.<ref>[[John William Draper|Draper, J.W.]] (1847). On the production of light by heat, ''London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science'', series 3, '''30''': 345–360. [https://archive.org/stream/londonedinburghp30lond#page/344/mode/2up]</ref><ref>{{harvnb|Partington|1949|pages = 466–467, 478}}.</ref> When the body appears white, it is emitting a substantial fraction of its energy as [[ultraviolet radiation]]. The [[Sun]], with an [[effective temperature]] of approximately 5800 K,<ref>{{harvnb|Goody|Yung|1989|pages=482, 484}}</ref> is an approximate black body with an emission spectrum peaked in the central, yellow-green part of the [[visible spectrum]], but with significant power in the ultraviolet as well.
Blackbody radiation provides insight into the [[thermodynamic equilibrium]] state of cavity radiation.
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The concept of the black body is an idealization, as perfect black bodies do not exist in nature.<ref name="Planck 1914 42">{{harvnb|Planck|1914|page=42}}</ref> However, [[graphite]] and [[carbon black|lamp black]], with emissivities greater than 0.95, are good approximations to a black material. Experimentally, blackbody radiation may be established best as the ultimately stable steady state equilibrium radiation in a cavity in a rigid body, at a uniform temperature, that is entirely opaque and is only partly reflective.<ref name="Planck 1914 42"/> A closed box with walls of graphite at a constant temperature with a small hole on one side produces a good approximation to ideal blackbody radiation emanating from the opening.<ref>{{harvnb|Wien|1894}}</ref><ref>{{harvnb|Planck|1914|page=43}}</ref>
Blackbody radiation has the unique absolutely stable distribution of radiative intensity that can persist in thermodynamic equilibrium in a cavity.<ref name="Planck 1914 42"/> In equilibrium, for each frequency, the intensity of radiation which is emitted and reflected from a body relative to other frequencies (that is, the net amount of radiation leaving its surface, called the ''spectral radiance'') is determined solely by the equilibrium temperature and does not depend upon the shape, material or structure of the body.<ref name=Caniou>{{cite book |chapter-url=https://books.google.com/books?id=X-aFGcf6pOEC&pg=PA107 |page=107 |chapter=§4.2.2: Calculation of Planck's law |title=Passive infrared detection: theory and applications |author=Joseph Caniou |isbn=0-7923-8532-2 |year=1999 |publisher=Springer}}</ref> For a black body (a perfect absorber) there is no reflected radiation, and so the spectral radiance is entirely due to emission. In addition, a black body is a diffuse emitter (its emission is independent of direction). Consequently, blackbody radiation may be viewed as the radiation from a black body at thermal equilibrium.
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|page = 58
|url = https://books.google.com/books?id=y9zUEzA7iN0C&q=draper-point+red&pg=PA58
}}</ref> At {{val|1000|u=K}}, a small opening in the wall of a large uniformly heated opaque-walled cavity (such as an oven), viewed from outside, looks red; at {{val|6000|u=K}}, it looks white. No matter how the oven is constructed, or of what material, as long as it is built so that almost all light entering is absorbed by its walls, it will contain a good approximation to blackbody radiation. The spectrum, and therefore color, of the light that comes out will be a function of the cavity temperature alone. A graph of the
[[Image:Pahoehoe toe.jpg|thumb|left|250px|The temperature of a [[Lava#Pāhoehoe|Pāhoehoe]] lava flow can be estimated by observing its color. The result agrees well with other measurements of temperatures of lava flows at about {{Convert|1000|to|1200|C|F}}.]]
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Real objects never behave as full-ideal black bodies, and instead the emitted radiation at a given frequency is a fraction of what the ideal emission would be. The [[emissivity]] of a material specifies how well a real body radiates energy as compared with a black body. This emissivity depends on factors such as temperature, emission angle, and wavelength. However, it is typical in engineering to assume that a surface's spectral emissivity and absorptivity do not depend on wavelength so that the emissivity is a constant. This is known as the ''gray body'' assumption.
[[File:
With non-black surfaces, the deviations from ideal blackbody behavior are determined by both the surface structure, such as roughness or granularity, and the chemical composition. On a "per wavelength" basis, real objects in states of [[Thermodynamic equilibrium#Local and global equilibrium|local thermodynamic equilibrium]] still follow [[Kirchhoff's law (thermodynamics)|Kirchhoff's Law]]: emissivity equals absorptivity, so that an object that does not absorb all incident light will also emit less radiation than an ideal black body; the incomplete absorption can be due to some of the incident light being transmitted through the body or to some of it being reflected at the surface of the body.
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Thus for shorter wavelengths very few modes (having energy more than <math>h \nu</math>) were allowed, supporting the data that the energy emitted is reduced for wavelengths less than the wavelength of the observed peak of emission.
Notice that there are two factors responsible for the shape of the graph, which can be seen as working opposite to one another. Firstly,
Calculating the blackbody curve was a major challenge in [[theoretical physics]] during the late nineteenth century. The problem was solved in 1901 by [[Max Planck]] in the formalism now known as [[Planck's law]] of blackbody radiation.<ref>{{cite journal
|last = Planck
|first = Max
|author-link =
|title = Ueber das Gesetz der Energieverteilung im Normalspectrum
|trans-title = On the law of the distribution of energy in the normal spectrum
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|pages = 553–563
|year = 1901
|url = https://babel.hathitrust.org/cgi/pt?id=hvd.yl1bqm;view=1up;seq=583
|language = de
|doi=10.1002/andp.19013090310
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Planck's law states that<ref name="Rybicki 1979 22">{{harvnb|Rybicki|Lightman|1979|p=22}}</ref>
where
{{unbulleted list | style = padding-left: 1.5em
| <math>T</math> is the absolute [[temperature]] of the body.
}}
For a black body surface, the spectral radiance density (defined per unit of area normal to the propagation) is independent of the angle <math>\theta</math> of emission with respect to the normal. However, this means that, following [[Lambert's cosine law]], <math> B_\nu(T) \cos \theta</math> is the radiance density per unit area of emitting surface as the surface area involved in generating the radiance is increased by a factor <math> 1/\cos \theta</math> with respect to an area normal to the propagation direction. At oblique angles, the solid angle spans involved do get smaller, resulting in lower aggregate intensities.
The emitted energy flux density or irradiance <math>B_\nu(T,E)</math>, is related to the photon flux density <math>b_\nu(T,E)</math> through
▲:<math>B_\nu(T,E) = Eb_\nu(T,E)</math>
===Wien's displacement law===
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A consequence of Wien's displacement law is that the wavelength at which the intensity ''per unit wavelength'' of the radiation produced by a black body has a local maximum or peak, <math>\lambda_\text{peak}</math>, is a function only of the temperature:
where the constant ''b'', known as Wien's displacement constant, is equal to <math>\frac{hc}{k} \frac
|title=Wien wavelength displacement law constant
|url=http://physics.nist.gov/cgi-bin/cuu/Value?bwien
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|publisher=[[NIST]]
|access-date=July 8, 2023
}}</ref> <math>W_0</math> is the [[Lambert W function]]. So
Planck's law was also stated above as a function of frequency. The intensity maximum for this is given by<ref>
:<math>\nu_\text{peak} = T \times 5.879 \times 10^{10} \ \mathrm{Hz}/\mathrm{K}</math>.<ref>▼
{{cite web
| last = Nave
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Provides 5 variations of Wien's displacement law
</ref>
▲
In unitless form, the maximum occurs when {{nowrap|<math>e^x(1-x/3) = 1</math>,}} where {{nowrap|<math>x = h\nu / kT</math>.}} The approximate numerical solution is <math>x \approx 2.82</math>. At a typical room temperature of 293 K (20 °C), the maximum intensity is for {{nowrap|1=<math>\nu</math> = 17 THz}}.
===Stefan–Boltzmann law===
By integrating <math>B_\nu(T)\cos(\theta)</math> over the frequency the radiance <math>L</math> (units: power / [area
<math display="block"> L = \frac{2\pi^5}{15} \frac{k^4 T^4}{c^2 h^3} \frac{1}{\pi} = \sigma T^4 \frac{\cos(\theta)}{\pi}</math>
On a side note, at a distance d, the intensity <math>dI</math> per area <math>dA</math> of radiating surface is the useful expression
when the receiving surface is perpendicular to the radiation.
By subsequently integrating <math>L</math> over the solid angle <math>\Omega</math> for all azimuthal angle (0 to <math>2\pi</math>) and polar angle <math>\theta</math> from 0 to <math>\pi/2</math>, we arrive at the [[Stefan–Boltzmann law]]: the power {{math|''j''*}} emitted per unit area of the surface of a black body is directly proportional to the fourth power of its absolute temperature:
We used
==Applications==
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}}
The human body radiates energy as [[infrared]] light. The net power radiated is the difference between the power emitted and the power absorbed:
Applying the Stefan–Boltzmann law,
where
The total surface area of an adult is about {{val|2
| author=Infrared Services
| title=Emissivity Values for Common Materials
| url=http://infrared-thermography.com/material-1.htm
| access-date=2007-06-24
| archive-date=2007-06-25
| archive-url=https://web.archive.org/web/20070625060223/http://infrared-thermography.com/material-1.htm
| url-status=dead
}}</ref><ref>{{cite web
| author=Omega Engineering
| title=Emissivity of Common Materials
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| access-date=2007-06-24}}</ref> Skin temperature is about 33 °C,<ref>{{cite web
| last= Farzana|first= Abanty
| title=Temperature of a Healthy Human (Skin Temperature) | year=2001 | work=The Physics Factbook
| url=http://hypertextbook.com/facts/2001/AbantyFarzana.shtml
| access-date=2007-06-24}}</ref> but clothing reduces the surface temperature to about 28 °C when the ambient temperature is 20 °C.<ref>{{cite web
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| access-date=2007-06-24
}}</ref> Hence, the net radiative heat loss is about
The total energy radiated in one day is about 8 [[megajoule|MJ]], or 2000 kcal (food [[calorie]]s). [[Basal metabolic rate]] for a 40-year-old male is about 35 kcal/(m<sup>2</sup>·h),<ref name="Harris1918">{{cite journal|author = Harris J, Benedict F|title = A Biometric Study of Human Basal Metabolism|journal = Proc Natl Acad Sci USA| volume = 4|issue = 12| pages = 370–3|year = 1918|pmid = 16576330|doi = 10.1073/pnas.4.12.370 | pmc = 1091498 |bibcode = 1918PNAS....4..370H |last2 = Benedict|doi-access = free}}</ref> which is equivalent to 1700 kcal per day, assuming the same 2 m<sup>2</sup> area. However, the mean metabolic rate of sedentary adults is about 50% to 70% greater than their basal rate.<ref>{{cite journal|author=Levine, J|title=Nonexercise activity thermogenesis (NEAT): environment and biology|journal=Am J Physiol Endocrinol Metab | volume=286 | year=2004|pages=E675–E685|url=http://ajpendo.physiology.org/cgi/content/full/286/5/E675|doi=10.1152/ajpendo.00562.2003 | pmid=15102614 | issue=5}}</ref>
There are other important thermal loss mechanisms, including [[convection]] and [[evaporation]]. Conduction is negligible – the [[Nusselt number]] is much greater than unity. Evaporation by [[perspiration]] is only required if radiation and convection are insufficient to maintain a steady-state temperature (but evaporation from the lungs occurs regardless). Free-convection rates are comparable, albeit somewhat lower, than radiative rates.<ref>{{cite web
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Application of [[Wien's displacement law|Wien's law]] to human-body emission results in a peak wavelength of
For this reason, thermal imaging devices for human subjects are most sensitive in the 7–14 micrometer range.
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[[Image:Sun-Earth-Radiation.png|frame|The Earth only has an absorbing area equal to a two dimensional disk, rather than the surface of a sphere.]]
where {{unbulleted list | style = padding-left: 1.5em
The Sun emits that power equally in all directions. Because of this, the planet is hit with only a tiny fraction of it. The power from the Sun that strikes the planet (at the top of the atmosphere) is:
where {{unbulleted list | style = padding-left: 1.5em
Because of its high temperature, the Sun emits to a large extent in the ultraviolet and visible (UV-Vis) frequency range. In this frequency range, the planet reflects a fraction <math>\alpha</math> of this energy where <math>\alpha</math> is the [[albedo]] or reflectance of the planet in the UV-Vis range. In other words, the planet absorbs a fraction <math>1-\alpha</math> of the Sun's light, and reflects the rest. The power absorbed by the planet and its atmosphere is then:
Even though the planet only absorbs as a circular area <math>\pi R^2</math>, it emits in all directions; the spherical surface area being <math>4 \pi R^2</math>. If the planet were a perfect black body, it would emit according to the [[Stefan–Boltzmann law]]
where <math>T_{\rm E} </math> is the temperature of the planet. This temperature, calculated for the case of the planet acting as a black body by setting <math>P_{\rm abs} = P_{\rm emt\,bb}</math>, is known as the [[effective temperature]]. The actual temperature of the planet will likely be different, depending on its surface and atmospheric properties. Ignoring the atmosphere and greenhouse effect, the planet, since it is at a much lower temperature than the Sun, emits mostly in the infrared (IR) portion of the spectrum. In this frequency range, it emits <math>\overline{\epsilon}</math> of the radiation that a black body would emit where <math>\overline{\epsilon}</math> is the average emissivity in the IR range. The power emitted by the planet is then:
For a body in [[Radiative equilibrium#Definitions of radiative equilibrium#radiative exchange equilibrium|radiative exchange equilibrium]] with its surroundings, the rate at which it emits [[radiant energy]] is equal to the rate at which it absorbs it:<ref name="Prevost 1791">{{cite journal |last=Prevost |first=P. |year=1791 |title=Mémoire sur l'équilibre du feu |journal=Journal de Physique (Paris) |volume=38 |pages=314–322}}</ref><ref>Iribarne, J.V., Godson, W.L. (1981). ''Atmospheric Thermodynamics'', second edition, D. Reidel Publishing, Dordrecht, {{ISBN|90-277-1296-4}}, page 227.</ref>
Substituting the expressions for solar and planet power in equations 1–6 and simplifying yields the estimated temperature of the planet, ignoring greenhouse effect, {{math|''T''<sub>P</sub>}}:
In other words, given the assumptions made, the temperature of a planet depends only on the surface temperature of the Sun, the radius of the Sun, the distance between the planet and the Sun, the albedo and the IR emissivity of the planet.
Notice that a gray (flat spectrum) ball where <math> (
====Effective temperature of Earth====
Substituting the measured values for the Sun and Earth yields:
With the average emissivity <math> \overline{\varepsilon} </math> set to unity, the [[effective temperature]] of the Earth is:
or −18.8 °C.
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|title = Climate Change an Integrated Perspective
|author1=Willem Jozef Meine Martens |author2=Jan Rotmans
|year = 1999
|isbn = 978-0-7923-5996-8
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===Balfour Stewart===
In 1858, [[Balfour Stewart]] described his experiments on the thermal radiative emissive and absorptive powers of polished plates of various substances, compared with the powers of lamp-black surfaces, at the same temperature.<ref name="Stewart 1858">{{harvnb|Stewart|1858}}</ref> Stewart chose lamp-black surfaces as his reference because of various previous experimental findings, especially those of [[Pierre Prevost (physicist)|Pierre Prevost]] and of [[John Leslie (physicist)|John Leslie]]. He wrote, "Lamp-black, which absorbs all the rays that fall upon it, and therefore possesses the greatest possible absorbing power, will possess also the greatest possible radiating power." More an experimenter than a logician, Stewart failed to point out that his statement presupposed an abstract general principle: that there exist, either ideally in theory, or really in nature, bodies or surfaces that respectively have one and the same unique universal greatest possible absorbing power, likewise for radiating power, for every wavelength and equilibrium temperature.
Stewart measured radiated power with a [[thermopile]] and sensitive galvanometer read with a microscope. He was concerned with selective thermal radiation, which he investigated with plates of substances that radiated and absorbed selectively for different qualities of radiation rather than maximally for all qualities of radiation. He discussed the experiments in terms of rays which could be reflected and refracted, and which obeyed the Stokes-[[Helmholtz reciprocity]] principle (though he did not use an eponym for it). He did not in this paper mention that the qualities of the rays might be described by their wavelengths, nor did he use spectrally resolving apparatus such as prisms or diffraction gratings. His work was quantitative within these constraints. He made his measurements in a room temperature environment, and quickly so as to catch his bodies in a condition near the thermal equilibrium in which they had been prepared by heating to equilibrium with boiling water. His measurements confirmed that substances that emit and absorb selectively respect the principle of selective equality of emission and absorption at thermal equilibrium.
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The [[relativistic Doppler effect]] causes a shift in the frequency ''f'' of light originating from a source that is moving in relation to the observer, so that the wave is observed to have frequency ''f''':
where ''v'' is the velocity of the source in the observer's rest frame, ''θ'' is the angle between the velocity vector and the observer-source direction measured in the reference frame of the source, and ''c'' is the [[speed of light]].<ref>The Doppler Effect, T. P. Gill, Logos Press, 1965</ref> This can be simplified for the special cases of objects moving directly towards (''θ'' = π) or away (''θ'' = 0) from the observer, and for speeds much less than ''c''.
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For the case of a source moving directly towards or away from the observer, this reduces to
Here ''v'' > 0 indicates a receding source, and ''v'' < 0 indicates an approaching source.
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* [[Bolometer]]
* [[Color temperature]]
* [[Draper point]]
* [[Infrared thermometer]]
* [[Photon polarization]]
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* [[Pyrometer]]
* [[Rayleigh–Jeans law]]
* [[Sakuma–Hattori equation]]
* [[Terahertz radiation]]
* [[
* [[Wien approximation]]
{{div col end}}
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| last = Kirchhoff
| first = G.
| author-link = Gustav Kirchhoff
| title = Gessamelte Abhandlungen
| place = Leipzig
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|publisher=[[World Scientific]]
|isbn= 978-981-4449-53-3
}}
*{{cite book
|last1=Partington |first1=J.R.
| |||
