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{{For| the radiation from a black body in thermal equilibrium|Black-body radiation}}
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[[File:Black body.svg|thumb|303px|As the temperature of a black body decreases, its intensity also decreases and its peak moves to longer wavelengths. Shown for comparison is the classical [[Rayleigh-Jeans law]] and its [[ultraviolet catastrophe]].]]
 
A '''black body''' is an idealized [[physical body]] that absorbs all incident [[electromagnetic radiation]], regardless of frequency or angle of incidence.
 
A black body in [[thermal equilibrium]] (that is, at a constant temperature) emits electromagnetic radiation called [[black-body radiation]]. The radiation is emitted according to [[Planck's law]], meaning that it has a [[Frequency spectrum|spectrum]] that is determined by the [[temperature]] alone (see figure at right), not by the body's shape or composition.
 
A black body in thermal equilibrium has two notable properties:<ref name=Massoud/>
#It is an ideal emitter: it emits as much or more energy at every frequency than any other body at the same temperature.
#It is a diffuse emitter: the energy is radiated [[Isotropic radiator|isotropically]], independent of direction.
 
An approximate realization of a black surface is a hole in the wall of a large enclosure (see [[#Cavity with a hole|below]]). Any light entering the hole is reflected indefinitely or absorbed inside and is unlikely to re-emerge, making the hole a nearly perfect absorber. The radiation confined in such an enclosure may or may not be in thermal equilibrium, depending upon the nature of the walls and the other contents of the enclosure.<ref name=Landsberg/><ref name="Planck 1914 44">{{harvnb|Planck|1914|page=44, §52}}</ref>
 
Real materials emit energy at a fraction—called the [[emissivity]]—of black-body energy levels. By definition, a black body in thermal equilibrium has an emissivity of {{nowrap|''&epsilon;'' {{=}} 1.0}}. A source with lower emissivity independent of frequency often is referred to as a '''gray body'''.<ref name=emissivity/><ref name=gray_body/>
Construction of black bodies with emissivity as close to one as possible remains a topic of current interest.<ref name=Chun/>  A '''white body''' is one with a "rough surface [that] reflects all incident rays completely and uniformly in all directions."<ref>{{harvnb|Planck|1914|pages=9–10}}</ref>
 
In [[astronomy]], the radiation from [[star]]s and [[planet]]s is sometimes characterized in terms of an [[effective temperature]], the temperature of a black body that would emit the same total flux of electromagnetic energy.
 
==Definition==
The idea of a black body originally was introduced by [[Gustav Kirchhoff]] in 1860 as follows:
{|align="center" style="width:80%;"
|
...the supposition that bodies can be imagined which, for infinitely small thicknesses, completely absorb all incident rays, and neither reflect nor transmit any. I shall call such bodies ''perfectly black'', or, more briefly, ''black bodies''.<ref name=Kirchhoff/>
|}
A more modern definition drops the reference to "infinitely small thicknesses":<ref name=infinitesimal/>
{|align="center" style="width:80%;"
|
An ideal body is now defined, called a ''blackbody''. A ''blackbody'' allows ''all'' incident radiation to pass into it (no reflected energy) and internally absorbs ''all'' the incident radiation (no energy transmitted through the body). This is true of radiation for all wavelengths and for all angles of incidence. Hence the blackbody is ''a perfect absorber for all incident radiation.''
<ref name=Siegel/>
|}
 
==Idealizations==
This section describes some concepts developed in connection with black bodies.
 
[[File:Black-body realization.png|thumb|An approximate realization of a black body as a tiny hole in an insulated enclosure.]]
 
===Cavity with a hole===
A widely used model of a black surface is a small hole in a cavity with walls that are opaque to radiation.<ref name=Siegel/>  Radiation incident on the hole will pass into the cavity, and is very unlikely to be re-emitted if the cavity is large.  The hole is not quite a perfect black surface — in particular, if the wavelength of the incident radiation is longer than the diameter of the hole, part will be reflected.  Similarly, even in perfect thermal equilibrium, the radiation inside a finite-sized cavity will not have an ideal Planck spectrum for wavelengths comparable to or larger than the size of the cavity.<ref name=Zee/>
 
Suppose the cavity is held at a fixed temperature ''T'' and the radiation trapped inside the enclosure is at [[thermal equilibrium]] with the enclosure. The hole in the enclosure will allow some radiation to escape. If the hole is small, radiation passing in and out of the hole has negligible effect upon the equilibrium of the radiation inside the cavity. This escaping radiation will approximate [[black-body radiation]] that exhibits a distribution in energy characteristic of the temperature ''T'' and does not depend upon the properties of the cavity or the hole, at least for wavelengths smaller than the size of the hole.<ref name=Zee/> See the figure in the Introduction for the [[Frequency spectrum#Light|spectrum]] as a function of the [[Frequency spectrum|frequency]] of the radiation, which is related to the energy of the radiation by the equation ''E''=''hf'', with ''E'' = energy, ''h'' = [[Planck constant|Planck's constant]], ''f'' = frequency.
 
At any given time the radiation in the cavity may not be in thermal equilibrium, but [[laws of thermodynamics|the second law of thermodynamics]] states that if left undisturbed it will eventually reach equilibrium,<ref name=Adkins>
{{cite book |author=Clement John Adkins|chapter=§4.1 The function of the second law |title=Equilibrium thermodynamics |url=http://books.google.com/books?id=FW4Oz48TWwQC&pg=PA50 |page=50 |isbn=0-521-27456-7 |edition=3rd |publisher=Cambridge University Press |year=1983}}
</ref> although the time it takes to do so may be very long.<ref name=Batrouni/> Typically, equilibrium is reached by continual absorption and re-emission of radiation by material in the cavity or its walls.<ref name=Landsberg/><ref name="Planck 1914 44"/><ref>{{harvnb|Loudon|2000}}, Chapter 1</ref><ref>{{harvnb|Mandel|Wolf|1995}}, Chapter 13</ref> Radiation entering the cavity will be "[[H-theorem|thermalized]]"; by this mechanism: the energy will be redistributed until the ensemble of photons achieves a [[Planck's law|Planck distribution]]. The time taken for thermalization is much faster with condensed matter present than with rarefied matter such as a dilute gas.  At temperatures below  billions of Kelvin, direct [[Euler–Heisenberg Lagrangian|photon–photon interactions]]<ref name=Karplus>Robert Karplus* and Maurice Neuman ,"The Scattering of Light by Light", Phys. Rev. 83, 776–784 (1951)</ref> are usually negligible compared to interactions with matter.<ref name=Bergmann/> Photons are an example of an interacting [[boson]] gas,<ref name=boson/> and as described by the [[H-theorem]],<ref name=Tolman/> under very general conditions any interacting boson gas will approach thermal equilibrium.
 
===Transmission, absorption, and reflection===
A body's behavior with regard to thermal radiation is characterized by its transmission τ, absorption α, and reflection ρ.
The boundary of a body forms an interface with its surroundings, and this interface may be rough or smooth. A nonreflecting interface separating regions with different refractive indices must be rough, because the laws of reflection and refraction governed by the [[Fresnel equations]] for a smooth interface require a reflected ray when the refractive indices of the material and its surroundings differ.<ref name=Tipler/> A few idealized types of behavior are given particular names:
 
An opaque body is one that transmits none of the radiation that reaches it, although some may be reflected.<ref name=Kaviany/><ref name=Venkanna/> That is, τ=0 and α+ρ=1
 
A transparent body is one that transmits all the radiation that reaches it. That is, τ=1 and α=ρ=0.
 
A gray body is one where α, ρ and τ are uniform for all wavelengths. This term also is used to mean a body for which α is temperature and wavelength independent.
 
A white body is one for which all incident radiation is reflected uniformly in all directions: τ=0, α=0, and ρ=1.
 
For a black body, τ=0, α=1, and ρ=0. Planck offers a theoretical model for perfectly black bodies, which he noted do not exist in nature: besides their opaque interior, they have interfaces that are perfectly transmitting and non-reflective.<ref>{{harvnb|Planck|1914|page=10}}</ref>
 
===Kirchhoff's perfect black bodies===
 
Kirchhoff in 1860 introduced the theoretical concept of a perfect black body with a completely absorbing surface layer of infinitely small thickness, but Planck noted some severe restrictions upon this idea. Planck noted three requirements upon a black body: the body must (i) allow radiation to enter but not reflect; (ii) possess a minimum thickness adequate to absorb the incident radiation and prevent its re-emission; (iii) satisfy severe limitations upon scattering to prevent radiation from entering and bouncing back out. As a consequence, Kirchhoff's perfect black bodies that absorb all the radiation that falls on them, cannot be realized in an infinitely thin surface layer, and impose conditions upon scattering of the light within the black body that are difficult to satisfy.<ref>{{harvnb|Planck|1914|pages=9–10, §10}}</ref><ref name="Kirchhoff 1860c">{{harvnb|Kirchhoff|1860c}}</ref>
 
==Realizations==
A realization of a black body is a real world, physical embodiment. Here are a few.
 
===Cavity with a hole===
 
In 1898, [[Otto Lummer]] and [[Ferdinand Kurlbaum]] published an account of their cavity radiation source.<ref>{{harvnb|Lummer|Kurlbaum|1898}}</ref> Their design has been used largely unchanged for radiation measurements to the present day. It was a hole in the wall of a platinum box, divided by diaphragms, with its interior blackened with iron oxide. It was an important ingredient for the progressively improved measurements that led to the discovery of Planck's law.<ref name=Rechenberg/><ref>{{harvnb|Kangro|1976|page=159}}</ref> A version described in 1901 had its interior blackened with a mixture of chromium, nickel, and cobalt oxides.<ref>{{harvnb|Lummer|Kurlbaum|1901}}</ref>
 
===Near-black materials===
There is interest in blackbody-like materials for [[camouflage]] and [[radar-absorbent material]]s for radar invisibility.<ref name=Lewis/><ref name=Quinn/> They also have application as solar energy collectors, and infrared thermal detectors. As a perfect emitter of radiation, a hot material with black body behavior would create an efficient infrared heater, particularly in space or in a vacuum where convective heating is unavailable.<ref name=Mizuno/> They also are useful in telescopes and cameras as anti-reflection surfaces to reduce stray light, and to gather information about objects in high-contrast areas (for example, observation of planets in orbit around their stars), where blackbody-like materials absorb light that comes from the wrong sources.
 
It has long been known that a [[lamp-black]] coating will make a body nearly black. An improvement on lamp-black is found in manufactured carbon nanotubes. Nano-porous materials can achieve [[refractive indices]] nearly that of vacuum, in one case obtaining average reflectance of 0.045%.<ref name=Chun/><ref name=Yang/> In 2009, a team of Japanese scientists created a material close to an ideal black body, based on vertically aligned single-walled [[carbon nanotubes]]. This absorbs between 98% and 99% of the incoming light in the spectral range from the ultra-violet to the far-infrared regions.<ref name=Mizuno/>
 
Another example of a nearly perfect black material is [[super black]], made by chemically etching a [[nickel]]–[[phosphorus]] [[alloy]].<ref name=Brown/>
 
===Stars and planets===
{{For| more about the UBV color index|Photometric system}}
 
[[File:Idealized photosphere.png|thumb|An idealized view of the cross-section of a star. The [[photosphere]] contains [[photon]]s of light nearly in thermal equilibrium, and some escape into space as near-black-body radiation. ]]
 
A star or planet often is modeled as a black body, and electromagnetic radiation emitted from these bodies as [[black-body radiation]]. The figure shows a highly schematic cross-section to illustrate the idea. The [[photosphere]] of the star where the emitted light is generated is idealized as a layer within which the photons of light interact with the material in the photosphere and achieve a common temperature ''T'' that is maintained over a long period of time. Some photons escape and are emitted into space, but the energy they carry away is replaced by energy from within the star, so that the temperature of the photosphere is nearly steady. Changes in the core lead to changes in the supply of energy to the photosphere, but such changes are slow on the time scale of interest here. Assuming these circumstances can be realized, the outer layer of the star is somewhat analogous to the example of an enclosure with a small hole in it, with the hole replaced by the limited transmission into space at the outside of the photosphere. With all these assumptions in place, the star emits black-body radiation at the temperature of the photosphere.<ref name=Green/>
 
[[File:Effective temperature and color index.png|thumb|Effective temperature of a black body compared with the B-V and U-B color index of main sequence and super giant stars in what is called a [[color-color diagram]].<ref name=UBV/>]]
 
Using this model the [[effective temperature]] of stars is estimated, defined as the temperature of a black body that yields the same surface flux of energy as the star. If a star were a black body, the same effective temperature would result from any region of the spectrum. For example, comparisons in the ''B'' (blue) or ''V'' (visible) range lead to the so-called ''B-V'' [[color index]], which increases the redder the star,<ref name=Kelley/> with the Sun having an index of +0.648 ± 0.006.<ref name=Gray/> Combining the ''U'' (ultraviolet) and the ''B'' indices leads to the ''U-B'' index, which becomes more negative the hotter the star and the more the UV radiation. Assuming the Sun is a type G2 V star, its ''U-B'' index is +0.12.<ref name=Golay/> The two indices for two types of stars are compared in the figure with the effective surface temperature of the stars assuming they are black bodies. It can be seen that there is only a rough correlation. For example, for a given ''B-V'' index from the blue-visible region of the spectrum., the curves for both types of star lie below the corresponding black-body ''U-B'' index that includes the ultraviolet spectrum, showing that both types of star emit less ultraviolet light than a black body with the same ''B-V'' index. It is perhaps surprising that they fit a black body curve as well as they do, considering that stars have greatly different temperatures at different depths.<ref name=Aller/> For example, the Sun has an effective temperature of 5780 K,<ref name=Lang/> which can be compared to the temperature of the [[photosphere]] of the [[Sun]] (the region generating the light), which ranges from about 5000 K at its outer boundary with the [[chromosphere]] to about 9500 K at its inner boundary with the [[convection zone]] approximately 500&nbsp;km deep.<ref name=Bertotti/>
 
===Black holes===
{{See also|Hawking radiation}}
A [[black hole]] is a region of [[spacetime]] from which nothing escapes. Around a black hole there is a mathematically defined surface called an [[event horizon]] that marks the point of no return. It is called "black" because it absorbs all the light that hits the horizon, reflecting nothing, making it almost an ideal black body<ref name=Schutz/> (radiation with a wavelength equal to or larger than the radius of the hole may not be absorbed, so black holes are not perfect black bodies).<ref name=Davies/> Physicists believe that to an outside observer, black holes have a non-zero temperature and emit radiation with a nearly perfect [[black-body radiation|black-body]] spectrum, ultimately [[Black hole#Evaporation|evaporating]].<ref name=Wald/> The mechanism for this emission is related to [[vacuum fluctuations]] in which a [[virtual particles|virtual pair]] of particles is separated by the gravity of the hole, one member being sucked into the hole, and the other being emitted.<ref name=Carr/> The energy distribution of emission is described by [[Planck's law]] with a temperature ''T'':
:<math>T=\frac {\hbar c^3}{8\pi Gk_BM} \ ,</math>
where ''c'' is the [[speed of light]], ℏ is the reduced [[Planck constant]], ''k<sub>B</sub>'' is [[Boltzmann constant|Boltzmann's constant]], ''G'' is the [[gravitational constant]] and ''M'' is the mass of the black hole.<ref name=Frolov/> These predictions have not yet been tested either observationally or experimentally.<ref name=Wald2/>
 
===Cosmic microwave background radiation===
{{See also|Big Bang|Cosmic microwave background radiation}}
The big bang theory is based upon the [[cosmological principle]], which states that on large scales the Universe is homogeneous and isotropic.  According to theory, the Universe approximately a second after its formation was a near-ideal black body in thermal equilibrium at a temperature above 10<sup>10</sup> K. The temperature decreased as the Universe expanded and the matter and radiation in it cooled.  The cosmic microwave background radiation observed today is "the most perfect black body ever measured in nature".<ref name=White/> It has a nearly ideal Planck spectrum at a temperature of about 2.7K.  It departs from the perfect isotropy of true black-body radiation by an observed anisotropy that varies with angle on the sky only to about one part in 100,000.
 
==Radiative cooling==
{{See also|Radiative cooling|Radiosity (heat transfer)}}
The integration of [[Planck's law]] over all frequencies provides the total energy per unit of time per unit of surface area radiated by a black body maintained at a temperature ''T'', and is known as the [[Stefan–Boltzmann law]]:
:<math>P/A = \sigma T^4 \ , </math>
where ''&sigma;'' is the [[Stefan–Boltzmann constant]], {{nowrap|''&sigma;'' &asymp; 5.67 &times; 10<sup>&minus;8</sup> W/(m<sup>2</sup>K<sup>4</sup>).<ref name=NIST/>}} To remain in thermal equilibrium at constant temperature ''T'', the black body must absorb or internally generate this amount of [[power (physics)|power]] P over the given area A.
 
The cooling of a body due to thermal radiation is often approximated using the Stefan–Boltzmann law supplemented with a "gray body" [[emissivity]] {{nowrap|''&epsilon; ≤ 1''}}  {{nowrap|(''P/A'' <nowiki>=</nowiki> ''&epsilon;&sigma;T''<sup>4</sup>).}}  The rate of decrease of the temperature of the emitting body can be estimated from the power radiated and the body's [[heat capacity]].<ref name=Pearson/> This approach is a simplification that ignores details of the mechanisms behind heat redistribution (which may include changing composition, [[phase transition]]s or restructuring of the body) that occur within the body while it cools, and assumes that at each moment in time the body is characterized by a single temperature. It also ignores other possible complications, such as changes in the emissivity with temperature,<ref name=Vollmer/><ref name=Champion/> and the role of other accompanying forms of energy emission, for example, emission of particles like neutrinos.<ref name=Shifman/>
 
If a hot emitting body is assumed to follow the Stefan–Boltzmann law and its power emission ''P'' and temperature ''T'' is known, this law can be used to estimate the dimensions of the emitting object, because the total emitted power is proportional to the area of the emitting surface. In this way it was found that X-ray bursts observed by astronomers originated in neutron stars with a radius of about 10&nbsp;km, rather than black holes as originally conjectured.<ref name=Lewin/> It should be noted that an accurate estimate of size requires some knowledge of the emissivity, particularly its spectral and angular dependence.<ref name=Mason/>
 
==References==
 
===Citations===
{{Reflist|2|refs=
 
<ref name=Adkins>
{{cite book |author=Clement John Adkins |chapter=§4.1 The function of the second law |title=Equilibrium thermodynamics |url=http://books.google.com/books?id=FW4Oz48TWwQC&pg=PA50 |page=50 |isbn=0-521-27456-7 |edition=3rd |publisher=Cambridge University Press |year=1983}}
</ref>
 
<ref name=Aller>
{{cite book |title=Atoms, stars, and nebulae |url=http://books.google.com/books?id=HupvoeJDCGoC&pg=PA61 |page=61 |author=Lawrence Hugh Aller |publisher=Cambridge University Press |isbn=0-521-31040-7 |edition=3rd |year=1991}}
</ref>
 
<ref name=Batrouni>
In simple cases the approach to equilibrium is governed by a [[relaxation time]]. In others, the system may 'hang up' in a [[metastable state]], as stated by Adkins (1983) on page 10. For another example, see {{cite book |title=Equilibrium and non-equilibrium statistical thermodynamics |author=Michel Le Bellac, Fabrice Mortessagne, Ghassan George Batrouni |page=8 |url=http://books.google.com/books?id=LZcdi4uzaWIC&pg=PA8 |isbn= 0521821436 |year=2004 |publisher=Cambridge University Press}}
</ref>
 
<ref name=Bergmann>
{{cite book |title=Optics of waves and particles |author=Ludwig Bergmann, Clemens Schaefer, Heinz Niedrig |url=http://books.google.com/books?id=sJxv-yCSNEAC&pg=PA595 |page=595 |isbn=3-11-014318-6 |year=1999 |publisher=Walter de Gruyter |quote=Because the interaction of the photons with each other is negligible, a small amount of matter is necessary to establish thermodynamic equilibrium of heat radiation.}}
</ref>
 
<ref name=Bertotti>
{{cite book |author=B. Bertotti, Paolo Farinella, David Vokrouhlický |title=New Views of the Solar System |chapter=Figure 9.2: The temperature profile in the solar atmosphere |page=248 |url=http://books.google.com/books?id=i-YvHNPEqAIC&pg=PA248 |isbn=1-4020-1428-7 |year=2003 |publisher=Springer}}
</ref>
 
<ref name=boson>
The fundamental bosons are the [[photon]], the vector bosons of the [[weak interaction]], the [[gluon]], and the [[graviton]]. See {{cite book |title=Bose-Einstein condensation |url=http://books.google.com/books?id=suqJdr2pPIsC&pg=PA4 |page=4 |author=Allan Griffin, D. W. Snoke, S. Stringari |publisher=Cambridge University Press |year=1996 |isbn=0-521-58990-8}}
</ref>
 
<ref name=Brown>
See description of work by Richard Brown and his colleagues at the UK's National Physical Laboratory: {{cite journal
|url= http://www.newscientist.com/article/dn3356-mini-craters-key-to-blackest-ever-black.html
|title=Mini craters key to 'blackest ever black'
|date=6 February 2003
|author=Mick Hamer (correspondent)
|journal=New Scientist Magazine online
}}</ref>
 
<ref name=Carr>
{{cite book |title=Beyond Extreme Physics: Cutting-edge science |chapter=Chapter 6: Quantum black holes |author=Bernard J Carr and Steven B Giddings |url=http://books.google.com/books?id=N-K-q0_TexAC&pg=PA30 |page=30 |publisher=Rosen Publishing Group, Scientific American (COR) |isbn=1-4042-1402-X |year=2008}}
</ref>
 
<ref name=Champion>
{{cite book |title=MEMS and Microstructures in aerospace applications |author=Robert Osiander, M. Ann Garrison Darrin, John Champion |url=http://books.google.com/books?id=DaqsgZ2HtIwC&pg=PA187 |page=187 |isbn=0-8247-2637-5 |year=2006 |publisher=CRC Press}}
</ref>
 
<ref name=Chun>
{{cite journal |author=Ai Lin Chun |url=http://www.nature.com/nnano/reshigh/2008/0108/full/nnano.2008.29.html |title=Carbon nanotubes: Blacker than black  |journal=Nature Nanotechnology |date=25 Jan 2008 |doi=10.1038/nnano.2008.29}}
</ref>
 
<ref name=Davies>
{{cite journal |title=Thermodynamics of black holes |author=PCW Davies |journal=Rep Prog Phys |volume=41 |issue=8|pages=1313 ''ff'' |year=1978 |url=http://cosmos.asu.edu/publications/papers/ThermodynamicTheoryofBlackHoles%2034.pdf |doi=10.1088/0034-4885/41/8/004 |bibcode=1978RPPh...41.1313D}}
</ref>
 
<ref name=emissivity>
The emissivity of a surface in principle depends upon frequency, angle of view, and temperature. However,  by definition, the radiation from a ''gray body'' is simply proportional to that of a black body at the same temperature, so its emissivity does not depend upon frequency (or, equivalently, wavelength).  See {{cite book |title=Principles of heat transfer |author=Massoud Kaviany |chapter=Figure 4.3(b): Behaviors of a gray (no wavelength dependence), diffuse (no directional dependence) and opaque (no transmission) surface |page=381 |url=http://books.google.com/books?id=dKI4k-9jK88C&pg=PA381  |isbn=0-471-43463-9 |year=2002 |publisher=Wiley-IEEE}} and {{cite book |title=Encyclopedia of optical engineering, Volume 3  |author=Ronald G. Driggers |url=http://books.google.com/books?id=9ExHkgDv2z0C&pg=PA2303 |page=2303 |isbn=0-8247-4252-4 |publisher=CRC Press |year=2003}}
</ref>
 
<ref name=Frolov>
{{cite book |title=Introduction to Black Hole Physics |author=Valeri P. Frolov, Andrei Zelnikov |url=http://books.google.com/books?id=r_l5AK9DdXsC&pg=PA321 |page=321 |chapter=Equation 9.7.1 |isbn=0-19-969229-7 |year=2011 |publisher=Oxford University Press}}
</ref>
 
<ref name=Golay>
{{cite book |title=Introduction to astronomical photometry |author=M Golay |url=http://books.google.com/books?id=OmNuvvt31BkC&pg=PA82 |page=82 |chapter=Table IX: ''U-B'' Indices |isbn=90-277-0428-7 |year=1974 |publisher=Springer}}
</ref>
 
<ref name=Gray>
{{cite journal |author=David F Gray |url=http://adsabs.harvard.edu/full/1995PASP..107..120G |journal=Publications of the astronomical society of the Pacific |volume=107 |pages=120–123 |date=February 1995 |accessdate=2012-01-26 |title=Comparing the sun with other stars along the temperature coordinate|year=1992}}
</ref>
 
<ref name=gray_body>
Some authors describe sources of infrared radiation with emissivity greater than approximately 0.99 as a black body. See {{cite web |title=What is a Blackbody and Infrared Radiation? |url=http://www.electro-optical.com/eoi_page.asp?h=What%20is%20a%20Blackbody%20and%20Infrared%20Radiation? |work=Education/Reference tab |publisher=Electro Optical Industries, Inc. |year=2008}}
</ref>
 
<ref name=Green>
{{cite book |title=An introduction to the sun and stars |author=Simon F. Green, Mark H. Jones, S. Jocelyn Burnell |url=http://books.google.com/books?id=lb5owLGIQGsC&pg=PA53 |pages=21–22, 53 |quote=A source in which photons are much more likely to interact with the material within the source than to escape is a condition for the formation of a black-body spectrum |isbn=0-521-54622-2 |year=2004 |publisher=Cambridge University Press}}
</ref>
 
<ref name=infinitesimal>
The notion of an infinitely thin layer was dropped by Planck. See {{harvnb|Planck|1914|page=10, footnote 2}}, .
</ref>
 
<ref name=Kaviany>
{{cite book |title=Principles of heat transfer |author=Massoud Kaviany |chapter=Figure 4.3(b) Radiation properties of an opaque surface |page=381 |url=http://books.google.com/books?id=dKI4k-9jK88C&pg=PA381#v=onepage&q&f=false |publisher=Wiley-IEEE |isbn=0-471-43463-9 |year=2002}}
</ref>
 
<ref name=Kelley>
{{cite book |title=Exploring Ancient Skies: A Survey of Ancient and Cultural Astronomy |author= David H. Kelley, Eugene F. Milone, Anthony F. (FRW) Aveni |url=http://books.google.com/books?id=ILBuYcGASxcC&pg=PA52 |page=52  |isbn=1-4419-7623-X |publisher=Springer |year=2011 |edition=2nd}}
</ref>
 
<ref name=Kirchhoff>
Translated by F. Guthrie from ''Annalen der Physik'': '''109''', 275-301 (1860): {{cite journal |url=http://books.google.com/books?id=RVYEAAAAYAAJ&pg=PA1&lpg=PA1 |journal=The London, Edinburgh and Dublin philosophical magazine and journal of science |publisher=Taylor & Francis |year=1860|title=On the relation between the radiating and absorbing powers of different bodies for light and heat |author=G. Kirchhoff |volume=20 |issue=130 |date=July 1860}}
</ref>
 
<ref name=Landsberg>
The approach to thermal equilibrium of the radiation in the cavity can be catalyzed by adding a small piece of matter capable of radiating and absorbing at all frequencies. See {{cite book |title= Thermodynamics and statistical mechanics |author=Peter Theodore Landsberg |url=http://books.google.com/books?id=0gnWL7tmxm0C&pg=PA209 |page=209 |isbn=0-486-66493-7 |publisher=Courier Dover Publications |edition=Reprint of Oxford University Press 1978 }}
</ref>
 
<ref name=Lang>
{{cite book |title=Astrophysical formulae, Volume 1 |author=Kenneth R. Lang |url=http://books.google.com/books?id=HlGIXqzVEAgC&pg=PA23 |page=23 |isbn=3-540-29692-1 |edition=3rd |publisher=Birkhäuser |year=2006}}
</ref>
 
<ref name=Lewin>
{{cite book |url=http://books.google.com/books?id=ukItNVRRSp4C&pg=PA250#v=onepage&q&f=false |pages=251 ''ff'' |author=Walter Lewin with Warren Goldstein |title=For the love of physics |isbn=1-4391-0827-7 |year=2011 |publisher=Simon and Schuster |chapter=X-ray bursters!}}
</ref>
 
<ref name=Lewis>
{{cite journal |title=Materials keep a low profile |author=CF Lewis |date=June 1988 |journal=Mech. Eng. |pages=37–41 |url=http://www.cewriters.com/CEWritersSample_MaterialsFeature.pdf}}
</ref>
 
<ref name=Mason>
{{cite book |title=Astrophysics update, Volume 2 |author=TE Strohmayer |chapter=Neutron star structure and fundamental physics |editor=John W. Mason, ed |url=http://books.google.com/books?id=SOzEHo1RteMC&pg=PA41 |page=41 |isbn=3-540-30312-X |publisher=Birkhäuser |year=2006}}
</ref>
 
<ref name=Massoud>
{{cite book |url=http://books.google.com/books?id=9KIp_fmC9A0C&pg=PA568 |page=568 |author=Mahmoud Massoud |title=Engineering thermofluids: thermodynamics, fluid mechanics, and heat transfer |chapter=§2.1 Blackbody radiation |isbn=3-540-22292-8 |publisher=Springer |year=2005}}
</ref>
 
<ref name=Mizuno>
{{cite journal
|author=K. Mizuno ''et al.''
|title=A black body absorber from vertically aligned single-walled carbon nanotubes
|journal=[[Proceedings of the National Academy of Sciences]]
|volume=106 |format=free download|pages=6044–6077
|year=2009
|doi=10.1073/pnas.0900155106
|pmid=19339498
|issue=15
|pmc=2669394
|bibcode = 2009PNAS..106.6044M }}
</ref>
 
<ref name=NIST>
{{cite web |title=Stefan–Boltzmann constant  |url=http://physics.nist.gov/cgi-bin/cuu/Value?sigma |work=NIST reference on constants, units, and uncertainty |accessdate=2012-02-02}}
</ref>
 
<ref name=Pearson>
A simple example is provided by {{cite book |title=The Person Guide to Objective Physics for the IIT-JEE |author=Srivastava M. K. |chapter=Cooling by radiation |url=http://books.google.com/books?id=FTpK4a4FTLQC&pg=PA610#v=onepage&q&f=false |page=610 |publisher=Pearson Education India |year=2011 |isbn=81-317-5513-4}}
</ref>
 
<ref name=Quinn>
{{cite book |title=Textile Futures |author=Bradley Quinn |year=2010 |isbn=1-84520-807-2 |publisher=Berg |url=http://books.google.com/books?id=e_wnynE954MC&pg=PA68 |page=68}}
</ref>
 
<ref name=Rechenberg>
An extensive historical discussion is found in {{cite book |title=The historical development of quantum theory |author=Jagdish Mehra, Helmut Rechenberg |url=http://books.google.com/books?id=W5kyppVPyesC&pg=PA39 |pages=39 ''ff'' |isbn=0-387-95174-1 |year=2000 |publisher=Springer}}
</ref>
 
<ref name=Schutz>
{{cite book
| title = Gravity From the Group Up: An Introductory Guide to Gravity and General Relativity
| url = http://www.amazon.com/Gravity-Ground-Up-Introductory-Relativity/dp/0521455065
| isbn =  0-521-45506-5
| page = 304
| last1 = Schutz | first1 =  Bernard
|publisher=Cambridge University Press
|edition=1st
|year=2004
}}</ref>
 
<ref name=Shifman>
Neutrino emission is a mechanism of cooling in neutron stars, for example; see {{cite book |chapter= Cooling by neutrino emission |url=http://books.google.com/books?id=2yhBnW_CtLIC&pg=PA2135 |page=2135 |isbn=981-02-4969-1 |year=2001 |publisher=World Scientific |author=Mikhail A. Shifman |editor=B. L. Ioffe, Mikhail A. Shifman, eds}}
</ref>
 
<ref name=Siegel>
{{cite book
| title = Thermal Radiation Heat Transfer; Volume 1
| url = http://books.google.com/books?id=O389yQ0-fecC&pg=PA7
| isbn = 1-56032-839-8
| page = 7
| last1 = Siegel | first1 =  Robert
| last2 =  Howell | first2 =  John R.
|publisher=Taylor & Francis
|edition=4th
|year=2002
}}
 
</ref>
 
<ref name=Tipler>
{{cite book |title=Physics for Scientists and Engineers, Parts 1-35; Part 39 |url=http://books.google.com/books?id=Gp2tzUhbqjMC&pg=PA1044 |page=1044 |chapter=Relative intensity of reflected and transmitted light |isbn=0-7167-3821-X |year=1999 |edition=4th |publisher=Macmillan |author=Paul A. Tipler}}
</ref>
 
<ref name=Tolman>
{{cite book |author=[[Richard Chace Tolman]] |chapter=§103: Change of ''H'' with time as a result of collisions |title= The principles of statistical mechanics |pages= 455 ''ff'' |url=http://books.google.com/books?id=4TqQZo962s0C&printsec=frontcover |isbn=0-486-63896-0 |edition=Reprint of 1938 Oxford University Press |publisher=Dover Publications |year=2010 |quote=...we can define a suitable quantity ''H'' to characterize the condition of a gas which [will exhibit] a tendency to decrease with time as a result of collisions, unless the distribution of the molecules [is already that of] equilibrium. (p. 458)}}
</ref>
 
<ref name=UBV>
Figure modeled after {{cite book |title=Introduction to Stellar Astrophysics: Basic stellar observations and data |author=E. Böhm-Vitense |url=http://books.google.com/books?id=JWrtilsCycQC&pg=PA26 |page=26 |chapter=Figure 4.9 |publisher=Cambridge University Press |isbn=0-521-34869-2 |year=1989}}
</ref>
 
<ref name=Venkanna>
{{cite book |title=Fundamentals of heat and mass transfer |author=BA Venkanna |url=http://books.google.com/books?id=IIIVHRirRgEC&pg=PA386 |pages=385–386 |chapter=§10.3.4 Absorptivity, reflectivity, and transmissivity |isbn=81-203-4031-0 |year=2010 |publisher=PHI Learning Pvt. Ltd.}}
</ref>
 
<ref name=Vollmer>
{{cite book |title=Infrared Thermal Imaging: Fundamentals, Research and Applications |chapter=Figure 1.38: Some examples for temperature dependence of emissivity for different materials |url=http://books.google.com/books?id=nAIZb8-05IwC&pg=PA45 |page=45  |author=M Vollmer, K-P Mõllmann |isbn=3-527-63087-2 |year=2011 |publisher=John Wiley & Sons}}
</ref>
 
<ref name=Wald>
{{cite book |chapter=The thermodynamics of black holes |author=Robert M Wald |url=http://books.google.com/books?id=ZDtxnw-b1xEC&pg=PA1 |pages=1 ''ff'' |editor=Andrés Gomberoff, Donald Marolf, eds |title=Lectures on quantum gravity |isbn=0-387-23995-2 |year=2005|publisher=Springer}}
 
</ref>
<ref name=Wald2>
{{cite book  |author=Robert M Wald |url=http://books.google.com/books?id=ZDtxnw-b1xEC&pg=PA28 |chapter=The thermodynamics of black holes |quote= ... no results on black hole thermodynamics have been subject to any experimental or observational tests, ...|title=cited work |isbn= 0-387-23995-2 |page= 28}}
</ref>
 
<ref name=White>
{{cite journal|author=White, M. |year=1999 |title=Anisotropies in the CMB |journal= Proceedings of the Los Angeles Meeting, DPF 99. UCLA. |url=http://www.dpf99.library.ucla.edu/session9/white0910.pdf}} See also [http://arxiv.org/abs/astro-ph/9903232v1 arXive.org].
 
</ref>
 
<ref name=Yang>
{{cite journal |author=Zu-Po Yang ''et al.'' |url=http://pubs.acs.org/doi/abs/10.1021/nl072369t|title= Experimental observation of an extremely dark material made by a low-density nanotube array  |journal=Nano Letters  |volume=8 |pages= 446–451 |year=2008 |doi=10.1021/nl072369t |publisher=American Chemical Society|bibcode = 2008NanoL...8..446Y |pmid=18181658}}
</ref>
 
<ref name=Zee>
Corrections to the spectrum do arise related to boundary conditions at the walls, curvature, and topology, particularly for wavelengths comparable to the cavity dimensions; see {{cite book |title=Cavity-enhanced spectroscopies |url=http://books.google.com/books?id=OuLILePVdJQC&pg=PA202 |page=202 |author=Roger Dale Van Zee, J. Patrick Looney |isbn=0-12-475987-4 |year=2002 |publisher=Academic Press}}
</ref>
}}
 
=== Bibliography ===
{{refbegin}}
 
*{{cite book
|last1=Chandrasekhar |first1=S.
|author1-link=Subrahmanyan Chandrasekhar
|year=1950
|title=Radiative Transfer
|publisher=[[Oxford University Press]]
|ref=harv
}}
*{{cite book
|last1=Goody |first1=R. M.
|last2=Yung |first2=Y. L.
|author2-link=Yuk L. Yung
|year=1989
|title=Atmospheric Radiation: Theoretical Basis
|edition=2nd
|publisher=[[Oxford University Press]]
|isbn=978-0-19-510291-8
|ref=harv
}}
*{{Cite book
  |last1=Hermann |first1=A.
  |author1-link=Armin Hermann
  |others=Nash, C.W. (transl.)
  |year=1971
  |title=The Genesis of Quantum Theory
  |publisher=[[MIT Press]]
  |isbn=0-262-08047-8
  |ref=harv
}} a translation of ''Frühgeschichte der Quantentheorie (1899–1913)'', Physik Verlag, Mosbach/Baden.
*{{cite book
|last1=Kangro |first1=H.
|author1-link=Hans Kangro
|year=1976
|title=Early History of Planck's Radiation Law
|publisher=[[Taylor and Francis]]
|isbn=0-85066-063-7
|ref=harv
}}
*{{cite journal
|last1=Kirchhoff |first1=G.
|author1-link=Gustav Kirchhoff
|last2=[27 October 1859]
|year=1860a
|title=Über die Fraunhofer'schen Linien
|journal=[[Monatsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin]]
|pages=662–665
|doi=
|ref=harv
}}
*{{cite journal
|last1=Kirchhoff |first1=G.
|author1-link=Gustav Kirchhoff
|last2=[11 December 1859]
|date= 1860b
|title=Über den Zusammenhang zwischen Emission und Absorption von Licht und Wärme
|journal=[[Monatsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin]]
|pages=783–787
|doi=
|ref=harv
}}
*{{cite journal
|last1=Kirchhoff |first1=G.
|author1-link=Gustav Kirchhoff
|year=1860c
|title=Ueber das Verhältniss zwischen dem Emissionsvermögen und dem Absorptionsvermögen der Körper für Wärme and Licht
|journal=[[Annalen der Physik und Chemie]]
|volume=109
|pages=275–301
|url=http://books.google.com/books?id=RVYEAAAAYAAJ&pg=PA1&lpg=PA1
|doi=
|ref=harv
}} Translated by Guthrie, F. as {{cite journal
|last1=Kirchhoff |first1=G.
|year=1860
|title=On the relation between the radiating and absorbing powers of different bodies for light and heat
|journal=[[Philosophical Magazine]]
|volume=Series 4, volume 20 |pages=1–21
|url=http://books.google.com/books?id=RVYEAAAAYAAJ&pg=PA1&lpg=PA1
}}
*{{Citation
  | last = Kirchhoff
  | first = G.
  | author-link = Gustav Kirchhoff
  | title = Gessamelte Abhandlungen
  | place = Leipzig
  | publisher = Johann Ambrosius Barth
  | series =
  | volume =
  | origyear = 1862
  | year = 1882
  | month=
  | edition =
  | chapter = Ueber das Verhältniss zwischen dem Emissionsvermögen und dem Absorptionsvermögen der Körper für Wärme und Licht
  | chapterurl =
  | page =
  | pages = 571–598
  | ref = harv
}}
*{{cite book
|last1=Kondepudi |first1=D.
|author1-link=Dilip Kondepudi
|last2=Prigogine |first2=I.
|author2-link=Ilya Prigogine
|year=1998
|title=Modern Thermodynamics. From Heat Engines to Dissipative Structures
|publisher=[[John Wiley & Sons]]
|isbn=0-471-97393-9
|ref=harv
}}
*{{cite book
|last1=Kragh |first1=H.
|author1-link=Helge Kragh
|year=1999
|title=Quantum Generations: a History of Physics in the Twentieth Century
|publisher=[[Princeton University Press]]
|isbn=0-691-01206-7
|ref=harv
}}
*{{cite book
|last1=Kuhn |first1=T. S.
|author1-link=Thomas Kuhn
|year=1978
|title=Black–Body Theory and the Quantum Discontinuity
|publisher=[[Oxford University Press]]
|isbn=0-19-502383-8
|ref=harv
}}
*{{cite book
|last1=Loudon |first1=R.
|author1-link=Rodney Loudon
|year=2000
|origyear=1973
|title=The Quantum Theory of Light
|publisher=[[Cambridge University Press]]
|edition=third
|isbn=0-19-850177-3
|ref=harv
}}
*{{cite journal
|last1=Lummer |first1=O.
|author1-link=Otto Lummer
|last2=Kurlbaum |first2=F.
|author2-link=Ferdinand Kurlbaum
|year=1898
|title=Der electrisch geglühte "absolut schwarze" Körper und seine Temperaturmessung
|journal=[[Verhandlungen der Deutschen Physikalischen Gesellschaft]]
|volume=17
|pages=106–111
|ref=harv
}}
*{{cite journal
|last1=Lummer |first1=O.
|author1-link=Otto Lummer
|last2=Kurlbaum |first2=F.
|author2-link=Ferdinand Kurlbaum
|year=1901
|title=Der elektrisch geglühte "schwarze" Körper
|journal=[[Annalen der Physik]]
|volume=310
|issue=8
|pages=829–836
|doi=10.1002/andp.19013100809
|ref=harv
|bibcode = 1901AnP...310..829L }}
*{{cite book
|last1=Mandel |first1=L.
|author1-link=Leonard Mandel
|last2=Wolf |first2=E.
|author2-link=Emil Wolf
|year=1995
|title=Optical Coherence and Quantum Optics
|publisher=[[Cambridge University Press]]
|isbn=0-521-41711-2
|ref=harv
}}
*{{cite book
|last1=Mehra |first1=J.
|author1-link=Jagdish Mehra
|last2=Rechenberg |first2=H.
|author2-link=Hlemut Rechenberg
|year=1982
|title=The Historical Development of Quantum Theory
|volume=volume 1, part 1
|publisher=[[Springer-Verlag]]
|isbn=0-387-90642-8
|ref=harv
}}
*{{cite book
|last1=Mihalas |first1=D.
|author1-link=Dimitri Mihalas
|last2=Weibel-Mihalas |first2=B.
|year=1984
|title=Foundations of Radiation Hydrodynamics
|publisher=[[Oxford University Press]]
|isbn=0-19-503437-6
|ref=harv
}}
*{{cite journal
|last1=Milne |first1=E.A.
|author1-link=Edward Arthur Milne
|year=1930
|title=Thermodynamics of the Stars
|journal=[[Handbuch der Astrophysik]]
|volume=3, part 1
|pages=63–255
|doi=
|ref=harv
}}
*{{cite book
|last1=Planck |first1=M.
|author1-link=Max Planck
|others=Masius, M. (transl.)
|year=1914
|title=The Theory of Heat Radiation
|publisher=[[P. Blakiston's Son & Co.|P. Blakiston's Son & Co]]
|url=http://ia600201.us.archive.org/16/items/theheatradiation00planrich/theheatradiation00planrich.pdf
|edition=2nd
|ol=7154661M
|ref=harv
}}
*{{Cite book
|last1=Rybicki |first1=G. B.
|last2=Lightman |first2=A. P.
|author2-link=Alan Lightman
|year=1979
|title=Radiative Processes in Astrophysics
|url=http://books.google.com/books?id=LtdEjNABMlsC&printsec=frontcover
|publisher=[[John Wiley & Sons]]
|isbn=0-471-82759-2
|ref=harv
}}
*{{cite book
|last1=Schirrmacher |first1=A.
|author1-link=Arne Schirrmacher
|year=2001
|title=Experimenting theory: the proofs of Kirchhoff's radiation law before and after Planck
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|ref=harv
}}
*{{cite journal
|last1=Stewart |first1=B.
|author1-link=Balfour Stewart
|year=1858
|title=An account of some experiments on radiant heat
|journal=[[Transactions of the Royal Society of Edinburgh]]
|volume=22 |pages=1–20
|doi=
|ref=harv
}}
{{refend}}
 
==External links==
*{{cite web |title=Blacker than black |url=http://www.nasa.gov/topics/technology/features/new-nano.html |publisher=[[NASA]] |author=Lori J. Keesey |accessdate=2012-02-01}} A description and video of work done making "blacker-than-pitch" materials using carbon nanotubes at the NASA [[Goddard Space Flight Center]].
 
[[Category:Infrared]]
[[Category:Heat transfer]]
[[Category:Electromagnetic radiation]]
[[Category:Astrophysics]]

Latest revision as of 10:47, 6 November 2014

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