A black body or blackbody is an idealized physical body that absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence. (It does not only absorb radiation, but can also emit radiation. The name 'black body' is given because it absorbs radiation in all frequencies, not because it only absorbs.) A white body is one with a 'rough surface that reflects all incident rays completely and uniformly in all directions.'^[1]

Fairly precise measurements of the energy distribution in this cavity radiation, or as we shall call it black body radiation. In 1895, at the University of Berlin, Wien and Lummer punched a small hole in the side of an otherwise completely closed oven, and began to measure the radiation coming out.

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 spectrum that is determined by the temperature alone (see figure at right), not by the body's shape or composition.

An ideal black body in thermal equilibrium has two notable properties:^[2]

1. It is an ideal emitter: at every frequency, it emits as much or more thermal radiative energy as any other body at the same temperature.
2. It is a diffuse emitter: the energy is radiated isotropically, independent of direction.

An approximate realization of a black body is a hole in the wall of a large insulated enclosure (an oven, for example). Any light entering the hole is reflected or absorbed at the internal surfaces of the body and is unlikely to re-emerge, making the hole a nearly perfect absorber. When the radiation confined in such an enclosure is in thermal equilibrium, the radiation emitted from the hole will be as great as from any body at that equilibrium temperature.

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 ε = 1.0. A source with lower emissivity independent of frequency often is referred to as a gray body.^[3][4]Construction of black bodies with emissivity as close to one as possible remains a topic of current interest.^[5]

In astronomy, the radiation from stars and planets 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.

• 2Idealizations
• 3Realizations
• 6References

Definition[edit]

The idea of a black body originally was introduced by Gustav Kirchhoff in 1860 as follows:

...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.^[6]

A more modern definition drops the reference to 'infinitely small thicknesses':^[7]

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 for radiation of all wavelengths and for all angles of incidence. Hence the blackbody is a perfect absorber for all incident radiation.^[8]

Idealizations[edit]

This section describes some concepts developed in connection with black bodies.

Cavity with a hole[edit]

A widely used model of a black surface is a small hole in a cavity with walls that are opaque to radiation.^[8] 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 greater 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.^[9]

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.^[9] See the figure in the Introduction for the spectrum as a function of the frequency of the radiation, which is related to the energy of the radiation by the equation E=hf, with E = energy, h = Planck's constant, f = frequency.

At any given time the radiation in the cavity may not be in thermal equilibrium, but the second law of thermodynamics states that if left undisturbed it will eventually reach equilibrium,^[10] although the time it takes to do so may be very long.^[11] Typically, equilibrium is reached by continual absorption and emission of radiation by material in the cavity or its walls.^{[12][13][14][15]} Radiation entering the cavity will be 'thermalized' by this mechanism: the energy will be redistributed until the ensemble of photons achieves a 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 photon–photon interactions^[16] are usually negligible compared to interactions with matter.^[17] Photons are an example of an interacting boson gas,^[18] and as described by the H-theorem,^[19] under very general conditions any interacting boson gas will approach thermal equilibrium.

Transmission, absorption, and reflection[edit]

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.^[20] 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.^[21][22] 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 grey 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.^[23]

Kirchhoff's perfect black bodies[edit]

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.^[24][25]

Realizations[edit]

A realization of a black body is a real world, physical embodiment. Here are a few.

Cavity with a hole[edit]

In 1898, Otto Lummer and Ferdinand Kurlbaum published an account of their cavity radiation source.^[26] 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.^[27][28] A version described in 1901 had its interior blackened with a mixture of chromium, nickel, and cobalt oxides.^[29] See also Hohlraum.

Near-black materials[edit]

There is interest in blackbody-like materials for camouflage and radar-absorbent materials for radar invisibility.^[30][31] 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.^[32] They are also 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%.^[5][33] In 2009, a team of Japanese scientists created a material called nanoblack which is 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.^[32]

Other examples of nearly perfect black materials are super black, prepared by chemically etching a nickel–phosphorusalloy,^[34] and vantablack made of carbon nanotubes; both absorb 99.9% of light or more.

Stars and planets[edit]

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.^[35]

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-Vcolor index, which increases the redder the star,^[37] with the Sun having an index of +0.648 ± 0.006.^[38] 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.^[39] The two indices for two types of most common star sequences are compared in the figure (diagram) with the effective surface temperature of the stars if they were perfect black bodies. There is a rough correlation. For example, for a given B-V index measurement, the curves of both most common sequences of star (the main sequence and the supergiants) lie below the corresponding black-body U-B index that includes the ultraviolet spectrum, showing that both groupings 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.^[40] For example, the Sun has an effective temperature of 5780 K,^[41] which can be compared to the temperature of its photosphere (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 km (310 mi) deep.^[42]

Black holes[edit]

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^[43] (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).^[44] Physicists believe that to an outside observer, black holes have a non-zero temperature and emit radiation with a nearly perfect black-body spectrum, ultimately evaporating.^[45] The mechanism for this emission is related to vacuum fluctuations in which a virtual pair of particles is separated by the gravity of the hole, one member being sucked into the hole, and the other being emitted.^[46] The energy distribution of emission is described by Planck's law with a temperature T:

${\displaystyle T=\frac{\hslash c^3}{8\pi G{k}_{B}M},}$

where c is the speed of light, ℏ is the reduced Planck constant, k_B is Boltzmann's constant, G is the gravitational constant and M is the mass of the black hole.^[47] These predictions have not yet been tested either observationally or experimentally.^[48]

Cosmic microwave background radiation[edit]

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 10¹⁰ 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'.^[49] It has a nearly ideal Planck spectrum at a temperature of about 2.7 K. 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[edit]

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:

${\displaystyle P/A=\sigma T^4,}$

where σ is the Stefan–Boltzmann constant, σ ≈ 5.67 × 10⁻⁸ W/(m²K⁴).^[50] To remain in thermal equilibrium at constant temperature T, the black body must absorb or internally generate this amount of 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' emissivityε ≤ 1(P/A = εσT⁴). The rate of decrease of the temperature of the emitting body can be estimated from the power radiated and the body's heat capacity.^[51] This approach is a simplification that ignores details of the mechanisms behind heat redistribution (which may include changing composition, phase transitions 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,^[52][53] and the role of other accompanying forms of energy emission, for example, emission of particles like neutrinos.^[54]

If a hot emitting body is assumed to follow the Stefan–Boltzmann law and its power emission P and temperature T are 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 km, rather than black holes as originally conjectured.^[55] An accurate estimate of size requires some knowledge of the emissivity, particularly its spectral and angular dependence.^[56]

See also[edit]

• Vantablack, a substance produced in 2014 and the blackest known
• Planckian locus, black body incandescence in a given chromaticity space

References[edit]

Citations[edit]

1. ^Planck 1914, pp. 9–10
2. ^Mahmoud Massoud (2005). '§2.1 Blackbody radiation'. Engineering thermofluids: thermodynamics, fluid mechanics, and heat transfer. Springer. p. 568. ISBN978-3-540-22292-7.
3. ^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 Massoud Kaviany (2002). 'Figure 4.3(b): Behaviors of a gray (no wavelength dependence), diffuse (no directional dependence) and opaque (no transmission) surface'. Principles of heat transfer. Wiley-IEEE. p. 381. ISBN978-0-471-43463-4. and Ronald G. Driggers (2003). Encyclopedia of optical engineering, Volume 3. CRC Press. p. 2303. ISBN978-0-8247-4252-2.
4. ^Some authors describe sources of infrared radiation with emissivity greater than approximately 0.99 as a black body. See 'What is a Blackbody and Infrared Radiation?'. Education/Reference tab. Electro Optical Industries, Inc. 2008. Archived from the original on 2016-03-07. Retrieved 2019-06-10.
5. ^ ^abAi Lin Chun (25 Jan 2008). 'Carbon nanotubes: Blacker than black'. Nature Nanotechnology. doi:10.1038/nnano.2008.29.
6. ^Translated by F. Guthrie from Annalen der Physik: 109, 275-301 (1860): G. Kirchhoff (July 1860). 'On the relation between the radiating and absorbing powers of different bodies for light and heat'. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. 20 (130).
7. ^The notion of an infinitely thin layer was dropped by Planck. See Planck 1914, p. 10, footnote 2, .
8. ^ ^abSiegel, Robert; Howell, John R. (2002). Thermal Radiation Heat Transfer; Volume 1 (4th ed.). Taylor & Francis. p. 7. ISBN978-1-56032-839-1.
9. ^ ^abCorrections to the spectrum do arise related to boundary conditions at the walls, curvature, and topology, particularly for wavelengths comparable to the cavity dimensions; see Roger Dale Van Zee; J. Patrick Looney (2002). Cavity-enhanced spectroscopies. Academic Press. p. 202. ISBN978-0-12-475987-9.
10. ^Clement John Adkins (1983). '§4.1 The function of the second law'. Equilibrium thermodynamics (3rd ed.). Cambridge University Press. p. 50. ISBN978-0-521-27456-2.
11. ^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 Michel Le Bellac; Fabrice Mortessagne; Ghassan George Batrouni (2004). Equilibrium and non-equilibrium statistical thermodynamics. Cambridge University Press. p. 8. ISBN978-0521821438.
12. ^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 Peter Theodore Landsberg (1990). Thermodynamics and statistical mechanics (Reprint of Oxford University Press 1978 ed.). Courier Dover Publications. p. 209. ISBN978-0-486-66493-4.
13. ^Planck 1914, p. 44, §52
14. ^Loudon 2000, Chapter 1
15. ^Mandel & Wolf 1995, Chapter 13
16. ^Robert Karplus* and Maurice Neuman ,'The Scattering of Light by Light', Phys. Rev. 83, 776–784 (1951)
17. ^Ludwig Bergmann; Clemens Schaefer; Heinz Niedrig (1999). Optics of waves and particles. Walter de Gruyter. p. 595. ISBN978-3-11-014318-8. 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.
18. ^The fundamental bosons are the photon, the vector bosons of the weak interaction, the gluon, and the graviton. See Allan Griffin; D. W. Snoke; S. Stringari (1996). Bose-Einstein condensation. Cambridge University Press. p. 4. ISBN978-0-521-58990-1.
19. ^Richard Chace Tolman (2010). '§103: Change of H with time as a result of collisions'. The principles of statistical mechanics (Reprint of 1938 Oxford University Press ed.). Dover Publications. pp. 455 ff. ISBN978-0-486-63896-6. ...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)
20. ^Paul A. Tipler (1999). 'Relative intensity of reflected and transmitted light'. Physics for Scientists and Engineers, Parts 1-35; Part 39 (4th ed.). Macmillan. p. 1044. ISBN978-0-7167-3821-3.
21. ^Massoud Kaviany (2002). 'Figure 4.3(b) Radiation properties of an opaque surface'. Principles of heat transfer. Wiley-IEEE. p. 381. ISBN978-0-471-43463-4.
22. ^BA Venkanna (2010). '§10.3.4 Absorptivity, reflectivity, and transmissivity'. Fundamentals of heat and mass transfer. PHI Learning Pvt. Ltd. pp. 385–386. ISBN978-81-203-4031-2.
23. ^Planck 1914, p. 10
24. ^Planck 1914, pp. 9–10, §10
25. ^Kirchhoff 1860c
26. ^Lummer & Kurlbaum 1898
27. ^An extensive historical discussion is found in Jagdish Mehra; Helmut Rechenberg (2000). The historical development of quantum theory. Springer. pp. 39 ff. ISBN978-0-387-95174-4.
28. ^Kangro 1976, p. 159
29. ^Lummer & Kurlbaum 1901
30. ^CF Lewis (June 1988). 'Materials keep a low profile'(PDF). Mech. Eng.: 37–41.^{[permanent dead link]}
31. ^Bradley Quinn (2010). Textile Futures. Berg. p. 68. ISBN978-1-84520-807-3.
32. ^ ^abK. Mizuno; et al. (2009). 'A black body absorber from vertically aligned single-walled carbon nanotubes'. Proceedings of the National Academy of Sciences. 106 (15): 6044–6077. Bibcode:2009PNAS..106.6044M. doi:10.1073/pnas.0900155106. PMC2669394. PMID19339498.
33. ^Zu-Po Yang; et al. (2008). 'Experimental observation of an extremely dark material made by a low-density nanotube array'. Nano Letters. 8 (2): 446–451. Bibcode:2008NanoL...8..446Y. doi:10.1021/nl072369t. PMID18181658.
34. ^See description of work by Richard Brown and his colleagues at the UK's National Physical Laboratory: Mick Hamer (correspondent) (6 February 2003). 'Mini craters key to 'blackest ever black''. New Scientist Magazine Online.
35. ^Simon F. Green; Mark H. Jones; S. Jocelyn Burnell (2004). An introduction to the sun and stars. Cambridge University Press. pp. 21–22, 53. ISBN978-0-521-54622-5. 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
36. ^Figure modeled after E. Böhm-Vitense (1989). 'Figure 4.9'. Introduction to Stellar Astrophysics: Basic stellar observations and data. Cambridge University Press. p. 26. ISBN978-0-521-34869-0.
37. ^David H. Kelley; Eugene F. Milone; Anthony F. (FRW) Aveni (2011). Exploring Ancient Skies: A Survey of Ancient and Cultural Astronomy (2nd ed.). Springer. p. 52. ISBN978-1-4419-7623-9.
38. ^David F Gray (February 1995). 'Comparing the sun with other stars along the temperature coordinate'. Publications of the Astronomical Society of the Pacific. 107: 120–123. Bibcode:1995PASP..107..120G. doi:10.1086/133525.
39. ^M Golay (1974). 'Table IX: U-B Indices'. Introduction to astronomical photometry. Springer. p. 82. ISBN978-90-277-0428-3.
40. ^Lawrence Hugh Aller (1991). Atoms, stars, and nebulae (3rd ed.). Cambridge University Press. p. 61. ISBN978-0-521-31040-6.
41. ^Kenneth R. Lang (2006). Astrophysical formulae, Volume 1 (3rd ed.). Birkhäuser. p. 23. ISBN978-3-540-29692-8.
42. ^B. Bertotti; Paolo Farinella; David Vokrouhlický (2003). 'Figure 9.2: The temperature profile in the solar atmosphere'. New Views of the Solar System. Springer. p. 248. ISBN978-1-4020-1428-4.
43. ^Schutz, Bernard (2004). Gravity From the Group Up: An Introductory Guide to Gravity and General Relativity (1st ed.). Cambridge University Press. p. 304. ISBN978-0-521-45506-0.
44. ^PCW Davies (1978). 'Thermodynamics of black holes'(PDF). Rep Prog Phys. 41 (8): 1313–1355. Bibcode:1978RPPh...41.1313D. doi:10.1088/0034-4885/41/8/004. Archived from the original(PDF) on 2013-05-10.
45. ^Robert M Wald (2005). 'The thermodynamics of black holes'. In Andrés Gomberoff; Donald Marolf (eds.). Lectures on quantum gravity. Springer Science & Business Media. pp. 1–38. ISBN978-0-387-23995-8.
46. ^Bernard J Carr & Steven B Giddings (2008). 'Chapter 6: Quantum black holes'. Beyond Extreme Physics: Cutting-edge science. Rosen Publishing Group, Scientific American (COR). p. 30. ISBN978-1-4042-1402-6.
47. ^Valeri P. Frolov; Andrei Zelnikov (2011). 'Equation 9.7.1'. Introduction to Black Hole Physics. Oxford University Press. p. 321. ISBN978-0-19-969229-3.
48. ^Robert M Wald (2005). 'The thermodynamics of black holes (pp. 1–38)'. In Andrés Gomberoff; Donald Marolf (eds.). Lectures on Quantum Gravity. Springer Science & Business Media. p. 28. ISBN978-0-387-23995-8. ... no results on black hole thermodynamics have been subject to any experimental or observational tests ...
49. ^White, M. (1999). 'Anisotropies in the CMB'(PDF). Proceedings of the Los Angeles Meeting, DPF 99. UCLA. See also arXive.org.
50. ^'Stefan–Boltzmann constant'. NIST reference on constants, units, and uncertainty. Retrieved 2012-02-02.
51. ^A simple example is provided by Srivastava M. K. (2011). 'Cooling by radiation'. The Person Guide to Objective Physics for the IIT-JEE. Pearson Education India. p. 610. ISBN978-81-317-5513-6.
52. ^M Vollmer; K-P Mõllmann (2011). 'Figure 1.38: Some examples for temperature dependence of emissivity for different materials'. Infrared Thermal Imaging: Fundamentals, Research and Applications. John Wiley & Sons. p. 45. ISBN978-3-527-63087-5.
53. ^Robert Osiander; M. Ann Garrison Darrin; John Champion (2006). MEMS and Microstructures in aerospace applications. CRC Press. p. 187. ISBN978-0-8247-2637-9.
54. ^Krishna Rajagopal; Frank Wilczek (2001). '6.2 Coling by Neutrino Emissions (pp. 2135-2136) – The Condensed Matter Physics of QCD'. In Mikhail A. Shifman (ed.). At The Frontier of Particle Physics: Handbook of QCD (On the occasion of the 75th birthday of Professor Boris Ioffe). 3. Singapore: World Scientific. pp. 2061–2151. arXiv:hep-ph/0011333v2. CiteSeerX10.1.1.344.2269. doi:10.1142/9789812810458_0043. ISBN978-981-02-4969-4. For the first 10^5–6 years of its life, the cooling of a neutron star is governed by the balance between heat capacity and the loss of heat by neutrino emission. ... Both the specific heat C_V and the neutrino emission rate L_ν are dominated by physics within T of the Fermi surface. ... The star will cool rapidly until its interior temperature is T < T_c ∼ ∆, at which time the quark matter core will become inert and the further cooling history will be dominated by neutrino emission from the nuclear matter fraction of the star.
55. ^Walter Lewin; Warren Goldstein (2011). 'X-ray bursters!'. For the love of physics. Simon and Schuster. pp. 251 ff. ISBN978-1-4391-0827-7.
56. ^TE Strohmayer (2006). 'Neutron star structure and fundamental physics'. In John W. Mason (ed.). Astrophysics update, Volume 2. Birkhäuser. p. 41. ISBN978-3-540-30312-1.

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• Mihalas, D.; Weibel-Mihalas, B. (1984). Foundations of Radiation Hydrodynamics. Oxford University Press. ISBN978-0-19-503437-0.
• Milne, E.A. (1930). 'Thermodynamics of the Stars'. Handbuch der Astrophysik. 3, part 1: 63–255.
• Planck, M. (1914). The Theory of Heat Radiation. Masius, M. (transl.) (2nd ed.). P. Blakiston's Son & Co. OL7154661M.
• Rybicki, G. B.; Lightman, A. P. (1979). Radiative Processes in Astrophysics. John Wiley & Sons. ISBN978-0-471-82759-7.
• Schirrmacher, A. (2001). Experimenting theory: the proofs of Kirchhoff's radiation law before and after Planck. Münchner Zentrum für Wissenschafts und Technikgeschichte.
• Stewart, B. (1858). 'An account of some experiments on radiant heat'. Transactions of the Royal Society of Edinburgh. 22: 1–20. doi:10.1017/S0080456800031288.

External links[edit]

• Keesey, Lori J. (Dec 12, 2010). 'Blacker than black'. NASA. Engineers now developing a blacker-than pitch material that will help scientists gather hard-to-obtain scientific measurements... nanotech-based material now being developed by a team of 10 technologists at the NASA Goddard Space Flight Center

Black-body radiation is the thermalelectromagnetic 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 spectrum and reverse intensity that depends only on the body's temperature, which is assumed for the sake of calculations and theory to be uniform and constant.^[1][2][3][4]

The thermal radiation spontaneously emitted by many ordinary objects can be approximated as black-body radiation. A perfectly insulated enclosure that is in thermal equilibrium internally contains black-body radiation and will emit it through a hole made in its wall, provided the hole is small enough to have negligible effect upon the equilibrium.

A black body at room temperature appears black, as most of the energy it radiates is in the infrared spectrum and cannot be perceived by the human eye. Since, by definition, the human eye cannot perceive light waves below the visible frequency, a black body, viewed in the dark at the lowest just faintly visible temperature, subjectively appears grey, even though its objective physical spectrum peak is in the infrared range.^[5] When it becomes a little hotter, it appears dull red. As its temperature increases further it becomes yellow, white, and ultimately blue-white.

Although planets and stars are neither in thermal equilibrium with their surroundings nor perfect black bodies, black-body radiation is used as a first approximation for the energy they emit.^[6]Black holes are near-perfect black bodies, in the sense that they absorb all the radiation that falls on them. It has been proposed that they emit black-body radiation (called Hawking radiation), with a temperature that depends on the mass of the black hole.^[7]

The term black body was introduced by Gustav Kirchhoff in 1860.^[8] Black-body radiation is also called thermal radiation, cavity radiation, complete radiation or temperature radiation.

• 1Theory
• 2Equations
• 3Applications
  • 3.2Temperature relation between a planet and its star
• 4History
• 7References

Theory[edit]

Spectrum[edit]

Black-body radiation has a characteristic, continuous frequency spectrum that depends only on the body's temperature,^[9] 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.^[10][11][12] 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.^[13][14] 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,^[15] 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.

Black-body radiation provides insight into the thermodynamic equilibrium state of cavity radiation.

Black body[edit]

All normal (baryonic) matter emits electromagnetic radiation when it has a temperature above absolute zero. The radiation represents a conversion of a body's internal energy into electromagnetic energy, and is therefore called thermal radiation. It is a spontaneous process of radiative distribution of entropy.

Conversely all normal matter absorbs electromagnetic radiation to some degree. An object that absorbs all radiation falling on it, at all wavelengths, is called a black body. When a black body is at a uniform temperature, its emission has a characteristic frequency distribution that depends on the temperature. Its emission is called black-body radiation.

The concept of the black body is an idealization, as perfect black bodies do not exist in nature.^[16]Graphite and lamp black, with emissivities greater than 0.95, however, are good approximations to a black material. Experimentally, black-body 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.^[16] A closed box of graphite walls at a constant temperature with a small hole on one side produces a good approximation to ideal black-body radiation emanating from the opening.^[17][18]

Black-body radiation has the unique absolutely stable distribution of radiative intensity that can persist in thermodynamic equilibrium in a cavity.^[16] In equilibrium, for each frequency the total intensity of radiation that is emitted and reflected from a body (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.^[19] 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, black-body radiation may be viewed as the radiation from a black body at thermal equilibrium.

Black-body radiation becomes a visible glow of light if the temperature of the object is high enough. The Draper point is the temperature at which all solids glow a dim red, about 798 K.^[20] At 1000 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 6000 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 black-body 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 amount of energy inside the oven per unit volume and per unit frequency interval plotted versus frequency, is called the black-body curve. Different curves are obtained by varying the temperature.

Two bodies that are at the same temperature stay in mutual thermal equilibrium, so a body at temperature T surrounded by a cloud of light at temperature T on average will emit as much light into the cloud as it absorbs, following Prevost's exchange principle, which refers to radiative equilibrium. The principle of detailed balance says that in thermodynamic equilibrium every elementary process works equally in its forward and backward sense.^[21][22] Prevost also showed that the emission from a body is logically determined solely by its own internal state. The causal effect of thermodynamic absorption on thermodynamic (spontaneous) emission is not direct, but is only indirect as it affects the internal state of the body. This means that at thermodynamic equilibrium the amount of every wavelength in every direction of thermal radiation emitted by a body at temperature T, black or not, is equal to the corresponding amount that the body absorbs because it is surrounded by light at temperature T.^[23]

When the body is black, the absorption is obvious: the amount of light absorbed is all the light that hits the surface. For a black body much bigger than the wavelength, the light energy absorbed at any wavelength λ per unit time is strictly proportional to the black-body curve. This means that the black-body curve is the amount of light energy emitted by a black body, which justifies the name. This is the condition for the applicability of Kirchhoff's law of thermal radiation: the black-body curve is characteristic of thermal light, which depends only on the temperature of the walls of the cavity, provided that the walls of the cavity are completely opaque and are not very reflective, and that the cavity is in thermodynamic equilibrium.^[24] When the black body is small, so that its size is comparable to the wavelength of light, the absorption is modified, because a small object is not an efficient absorber of light of long wavelength, but the principle of strict equality of emission and absorption is always upheld in a condition of thermodynamic equilibrium.

In the laboratory, black-body radiation is approximated by the radiation from a small hole in a large cavity, a hohlraum, in an entirely opaque body that is only partly reflective, that is maintained at a constant temperature. (This technique leads to the alternative term cavity radiation.) Any light entering the hole would have to reflect off the walls of the cavity multiple times before it escaped, in which process it is nearly certain to be absorbed. Absorption occurs regardless of the wavelength of the radiation entering (as long as it is small compared to the hole). The hole, then, is a close approximation of a theoretical black body and, if the cavity is heated, the spectrum of the hole's radiation (i.e., the amount of light emitted from the hole at each wavelength) will be continuous, and will depend only on the temperature and the fact that the walls are opaque and at least partly absorptive, but not on the particular material of which they are built nor on the material in the cavity (compare with emission spectrum).

The radiance or observed intensity is not a function of direction. Therefore, a black body is a perfect Lambertian radiator.

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.

With non-black surfaces, the deviations from ideal black-body 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 local thermodynamic equilibrium still follow 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.

In astronomy, objects such as stars are frequently regarded as black bodies, though this is often a poor approximation. An almost perfect black-body spectrum is exhibited by the cosmic microwave background radiation. Hawking radiation is the hypothetical black-body radiation emitted by black holes, at a temperature that depends on the mass, charge, and spin of the hole. If this prediction is correct, black holes will very gradually shrink and evaporate over time as they lose mass by the emission of photons and other particles.

A black body radiates energy at all frequencies, but its intensity rapidly tends to zero at high frequencies (short wavelengths). For example, a black body at room temperature (300 K) with one square meter of surface area will emit a photon in the visible range (390–750 nm) at an average rate of one photon every 41 seconds, meaning that for most practical purposes, such a black body does not emit in the visible range.^{[citation needed]}

The study of the laws of black bodies and the failure of classical physics to describe them helped establish the foundations of quantum mechanics.

Further explanation[edit]

According to the Classical Theory of Radiation, if each Fourier mode of the equilibrium radiation ( in an otherwise empty cavity with perfectly reflective walls) is considered as a degree of freedom capable of exchanging energy, then, according to the equipartition theorem of classical physics, there would be an equal amount of energy in each mode. Since there are an infinite number of modes, this would imply infinite heat capacity , as well as an nonphysical spectrum of emitted radiation that grows without bound with increasing frequency, a problem known as the ultraviolet catastrophe.

In the longer wavelengths this deviation is not so noticeable, as ${\displaystyle h\nu }$ and ${\displaystyle nh\nu }$ are very small. In the shorter wavelengths of the ultraviolet range, however, classical theory predicts the energy emitted tends to infinity, hence the ultraviolet catastrophe. As all possible vibrational modes (including those whose energy less than ${\displaystyle h\nu }$-The quantum of energy), the energy summed to infinity. The theory even predicted that all bodies would emit most of their energy in the ultraviolet range, clearly contradicted by the experimental data which showed a different peak wavelength at different temperatures. (see also wiens law)

Instead, in the quantum treatment of this problem, the numbers of the energy modes are quantized, attenuating the spectrum at high frequency in agreement with experimental observation and resolving the catastrophe. The modes that had more energy than the thermal energy of the substance itself were not considered, and by quantization - modes having infinitesimally little energy were excluded.

Thus for shorter wavelengths very few modes(having energy more than ${\displaystyle h\nu }$) 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. Firstly, longer wavelengths have a larger number of modes associated with them. Secondly, shorter wavelengths have more energy associated per mode. The two factors combined give the characteristic maximum wavelength .

Calculating the black-body 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 black-body radiation.^[27] By making changes to Wien's radiation law (not to be confused with Wien's displacement law) consistent with thermodynamics and electromagnetism, he found a mathematical expression fitting the experimental data satisfactorily. Planck had to assume that the energy of the oscillators in the cavity was quantized, i.e., it existed in integer multiples of some quantity. Einstein built on this idea and proposed the quantization of electromagnetic radiation itself in 1905 to explain the photoelectric effect. These theoretical advances eventually resulted in the superseding of classical electromagnetism by quantum electrodynamics. These quanta were called photons and the black-body cavity was thought of as containing a gas of photons. In addition, it led to the development of quantum probability distributions, called Fermi–Dirac statistics and Bose–Einstein statistics, each applicable to a different class of particles, fermions and bosons.

The wavelength at which the radiation is strongest is given by Wien's displacement law, and the overall power emitted per unit area is given by the Stefan–Boltzmann law. So, as temperature increases, the glow color changes from red to yellow to white to blue. Even as the peak wavelength moves into the ultra-violet, enough radiation continues to be emitted in the blue wavelengths that the body will continue to appear blue. It will never become invisible—indeed, the radiation of visible light increases monotonically with temperature.^[28] The Stefan–Boltzmann law also says that the total radiant heat energy emitted from a surface is proportional to the fourth power of its absolute temperature. The law was formulated by Josef Stefan in 1879 and later derived by Ludwig Boltzmann. The formula E = σT⁴ is given, where E is the radiant heat emitted from a unit of area per unit time, T is the absolute temperature, and σ = 5.670367×10⁻⁸ W·m⁻²⋅K⁻⁴ is the Stefan–Boltzmann constant.^[29]

Equations[edit]

Planck's law of black-body radiation[edit]

Planck's law states that^[30]

${\displaystyle B_{\nu }(\nu ,T)=\frac{2h{\nu }^3}{c^2}\frac{1}{e^{h\nu /kT}-1},}$

where

B_ν(T) is the spectral radiance (the power per unit solid angle and per unit of area normal to the propagation) density of frequency ν radiation per unit frequency at thermal equilibrium at temperature T.

h is the Planck constant;

c is the speed of light in a vacuum;

k is the Boltzmann constant;

ν is the frequency of the electromagnetic radiation;

T 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 ${\displaystyle \theta }$ of emission with respect to the normal. However, this means that, following Lambert's cosine law, ${\displaystyle B_{\nu }(T)\cos \theta }$ is the radiance density per unit area of emitting surface as the surface area involved in generating the radiance is increased by a factor ${\displaystyle 1/\cos \theta }$ 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.

Wien's displacement law[edit]

Wien's displacement law shows how the spectrum of black-body radiation at any temperature is related to the spectrum at any other temperature. If we know the shape of the spectrum at one temperature, we can calculate the shape at any other temperature. Spectral intensity can be expressed as a function of wavelength or of frequency.

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 is at a maximum, ${\displaystyle {\lambda }_{max}}$, is a function only of the temperature:

${\displaystyle {\lambda }_{max}=\frac{b}{T},}$

where the constant b, known as Wien's displacement constant, is equal to 2.897771955×10⁻³ m K.^[31]

Planck's law was also stated above as a function of frequency. The intensity maximum for this is given by

${\displaystyle {\nu }_{max}=T\times 5.879\times {10}^{10}\mathrm{H}\mathrm{z}/\mathrm{K}}$.^[32]

In unitless form, the maximum occurs when ${\displaystyle e^x(1-x/3)=1}$, where ${\displaystyle x=h\nu /kT}$. The approximate numerical solution is ${\displaystyle x\approx 2.82}$. For example, at a typical room temperature of 293 K (20 °C), the maximum intensity is for ${\displaystyle \nu }$ = 17 THz, or a wavelength of 17 microns (far infrared).

Stefan–Boltzmann law[edit]

By integrating ${\displaystyle B_{\nu }(T)}$ over the frequency the integrated radiance ${\displaystyle L}$ is

${\displaystyle L=\frac{2{\pi }^5}{15}\frac{k^4{T}^4}{c^2{h}^3}\frac{1}{\pi }=\sigma T^4\frac{1}{\pi }}$

by using ${\displaystyle {\int }_0^{\mathrm{\infty }}dx\frac{x^3}{e^x-1}=\frac{{\pi }^4}{15}}$ with ${\displaystyle x\equiv \frac{h\nu }{kT}}$ and with ${\displaystyle \sigma \equiv \frac{2{\pi }^5}{15}\frac{k^4}{c^2{h}^3}=5.670373\times {10}^{-8}\frac{W}{m^2{K}^4}}$ being the Stefan–Boltzmann constant. The radiance ${\displaystyle L}$ is then

${\displaystyle \sigma T^4\frac{\cos \theta }{\pi }}$

per unit of emitting surface.

On a side note, at a distance d, the intensity ${\displaystyle dI}$ per area ${\displaystyle dA}$ of radiating surface is the useful expression

${\displaystyle dI=\sigma T^4\frac{\cos \theta }{\pi d^2}dA}$

when the receiving surface is perpendicular to the radiation.

By subsequently integrating over the solid angle ${\displaystyle \mathrm{\Omega }}$ (where ) the Stefan–Boltzmann law is calculated, stating that the power j* emitted per unit area of the surface of a black body is directly proportional to the fourth power of its absolute temperature:

${\displaystyle j^{\star }=\sigma T^4,}$

by using

${\displaystyle \int \cos \theta d\mathrm{\Omega }={\int }_0^{2\pi }{\int }_0^{\pi /2}\cos \theta \sin \theta d\theta d\varphi =\pi .}$

Applications[edit]

Human-body emission[edit]

The human body radiates energy as infrared light. The net power radiated is the difference between the power emitted and the power absorbed:

${\displaystyle P_{\text{net}}=P_{\text{emit}}-P_{\text{absorb}}.}$

Applying the Stefan–Boltzmann law,

${\displaystyle P_{\text{net}}=A\sigma \epsilon \left(T^4-T_0^4\right),}$

where A and T are the body surface area and temperature, ${\displaystyle \epsilon }$ is the emissivity, and T₀ is the ambient temperature.

The total surface area of an adult is about 2 m², and the mid- and far-infrared emissivity of skin and most clothing is near unity, as it is for most nonmetallic surfaces.^[33][34] Skin temperature is about 33 °C,^[35] but clothing reduces the surface temperature to about 28 °C when the ambient temperature is 20 °C.^[36] Hence, the net radiative heat loss is about

${\displaystyle P_{\text{net}}=100\text{W}.}$

The total energy radiated in one day is about 8 MJ, or 2000 kcal (food calories). Basal metabolic rate for a 40-year-old male is about 35 kcal/(m²·h),^[37] which is equivalent to 1700 kcal per day, assuming the same 2 m² area. However, the mean metabolic rate of sedentary adults is about 50% to 70% greater than their basal rate.^[38]

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.^[39] Thus, radiation accounts for about two-thirds of thermal energy loss in cool, still air. Given the approximate nature of many of the assumptions, this can only be taken as a crude estimate. Ambient air motion, causing forced convection, or evaporation reduces the relative importance of radiation as a thermal-loss mechanism.

Application of Wien's law to human-body emission results in a peak wavelength of

${\displaystyle {\lambda }_{\text{peak}}=\frac{2.898\times {10}^{-3}\text{K}\cdot \text{m}}{305\text{K}}=9.50\mu \text{m}.}$

For this reason, thermal imaging devices for human subjects are most sensitive in the 7–14 micrometer range.

Temperature relation between a planet and its star[edit]

The black-body law may be used to estimate the temperature of a planet orbiting the Sun.

The temperature of a planet depends on several factors:

• Incident radiation from its star
• Emitted radiation of the planet, e.g., Earth's infrared glow
• The albedo effect causing a fraction of light to be reflected by the planet
• The greenhouse effect for planets with an atmosphere
• Energy generated internally by a planet itself due to radioactive decay, tidal heating, and adiabatic contraction due by cooling.

The analysis only considers the Sun's heat for a planet in a Solar System.

The Stefan–Boltzmann law gives the total power (energy/second) the Sun is emitting:

${\displaystyle P_{\mathrm{S}\mathrm{e}\mathrm{m}\mathrm{t}}=4\pi R_{\mathrm{S}}^2\sigma T_{\mathrm{S}}^4(1)}$

where

${\displaystyle \sigma }$ is the Stefan–Boltzmann constant,

${\displaystyle T_{\mathrm{S}}}$ is the effective temperature of the Sun, and

${\displaystyle R_{\mathrm{S}}}$ is the radius of the Sun.

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:

${\displaystyle P_{\mathrm{S}\mathrm{E}}=P_{\mathrm{S}\mathrm{e}\mathrm{m}\mathrm{t}}\left(\frac{\pi R_{\mathrm{E}}^2}{4\pi D^2}\right)(2)}$

where

${\displaystyle R_{\mathrm{E}}}$ is the radius of the planet and

${\displaystyle D}$ is the distance between the Sun and the planet.

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 ${\displaystyle \alpha }$ of this energy where ${\displaystyle \alpha }$ is the albedo or reflectance of the planet in the UV-Vis range. In other words, the planet absorbs a fraction ${\displaystyle 1-\alpha }$ of the Sun's light, and reflects the rest. The power absorbed by the planet and its atmosphere is then:

${\displaystyle P_{\mathrm{a}\mathrm{b}\mathrm{s}}=(1-\alpha )P_{\mathrm{S}\mathrm{E}}(3)}$

Even though the planet only absorbs as a circular area ${\displaystyle \pi R^2}$, it emits equally in all directions as a sphere. If the planet were a perfect black body, it would emit according to the Stefan–Boltzmann law

${\displaystyle P_{\mathrm{e}\mathrm{m}\mathrm{t}\mathrm{b}\mathrm{b}}=4\pi R_{\mathrm{E}}^2\sigma T_{\mathrm{E}}^4(4)}$

where ${\displaystyle T_{\mathrm{E}}}$ is the temperature of the planet. This temperature, calculated for the case of the planet acting as a black body by setting ${\displaystyle P_{\mathrm{a}\mathrm{b}\mathrm{s}}=P_{\mathrm{e}\mathrm{m}\mathrm{t}\mathrm{b}\mathrm{b}}}$, 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 ${\displaystyle \overline{ϵ}}$ of the radiation that a black body would emit where ${\displaystyle \overline{ϵ}}$ is the average emissivity in the IR range. The power emitted by the planet is then:

${\displaystyle P_{\mathrm{e}\mathrm{m}\mathrm{t}}=\overline{ϵ}{P}_{\mathrm{e}\mathrm{m}\mathrm{t}\mathrm{b}\mathrm{b}}(5)}$

For a body in radiative exchange equilibrium with its surroundings, the rate at which it emits radiant energy is equal to the rate at which it absorbs it:^[40][41]

${\displaystyle P_{\mathrm{a}\mathrm{b}\mathrm{s}}=P_{\mathrm{e}\mathrm{m}\mathrm{t}}(6)}$

Substituting the expressions for solar and planet power in equations 1–6 and simplifying yields the estimated temperature of the planet, ignoring greenhouse effect, T_P:

${\displaystyle T_P=T_S\sqrt{\frac{R_S\sqrt{\frac{1-\alpha }{\overline{\epsilon }}}}{2D}}(7)}$

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 ${\displaystyle (1-\alpha )=\overline{\epsilon }}$ comes to the same temperature as a black body no matter how dark or light gray .

Effective temperature of Earth[edit]

Substituting the measured values for the Sun and Earth yields:

${\displaystyle T_{\mathrm{S}}=5778\mathrm{K},}$^[42]

${\displaystyle R_{\mathrm{S}}=6.96\times {10}^8\mathrm{m},}$^[42]

${\displaystyle D=1.496\times {10}^{11}\mathrm{m},}$^[42]

${\displaystyle \alpha =0.306}$^[43]

With the average emissivity ${\displaystyle \overline{\epsilon }}$ set to unity, the effective temperature of the Earth is:

${\displaystyle T_{\mathrm{E}}=254.356\mathrm{K}}$

or −18.8 °C.

This is the temperature of the Earth if it radiated as a perfect black body in the infrared, assuming an unchanging albedo and ignoring greenhouse effects (which can raise the surface temperature of a body above what it would be if it were a perfect black body in all spectrums^[44]). The Earth in fact radiates not quite as a perfect black body in the infrared which will raise the estimated temperature a few degrees above the effective temperature. If we wish to estimate what the temperature of the Earth would be if it had no atmosphere, then we could take the albedo and emissivity of the Moon as a good estimate. The albedo and emissivity of the Moon are about 0.1054^[45] and 0.95^[46] respectively, yielding an estimated temperature of about 1.36 °C.

Estimates of the Earth's average albedo vary in the range 0.3–0.4, resulting in different estimated effective temperatures. Estimates are often based on the solar constant (total insolation power density) rather than the temperature, size, and distance of the Sun. For example, using 0.4 for albedo, and an insolation of 1400 W m⁻², one obtains an effective temperature of about 245 K.^[47]Similarly using albedo 0.3 and solar constant of 1372 W m⁻², one obtains an effective temperature of 255 K.^[48][49][50]

Cosmology[edit]

The cosmic microwave background radiation observed today is the most perfect black-body radiation ever observed in nature, with a temperature of about 2.7 K.^[51] It is a 'snapshot' of the radiation at the time of decoupling between matter and radiation in the early universe. Prior to this time, most matter in the universe was in the form of an ionized plasma in thermal, though not full thermodynamic, equilibrium with radiation.

According to Kondepudi and Prigogine, at very high temperatures (above 10¹⁰ K; such temperatures existed in the very early universe), where the thermal motion separates protons and neutrons in spite of the strong nuclear forces, electron-positron pairs appear and disappear spontaneously and are in thermal equilibrium with electromagnetic radiation. These particles form a part of the black body spectrum, in addition to the electromagnetic radiation.^[52]

History[edit]

In his first memoir, Augustin-Jean Fresnel (1788–1827) responded to a view he extracted from a French translation of Isaac Newton's Optics. He says that Newton imagined particles of light traversing space uninhibited by the caloric medium filling it, and refutes this view (never actually held by Newton) by saying that a black body under illumination would increase indefinitely in heat.^[53]

Balfour Stewart[edit]

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.^[23] Stewart chose lamp-black surfaces as his reference because of various previous experimental findings, especially those of Pierre Prevost and of 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 thermo-pile 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.

Stewart offered a theoretical proof that this should be the case separately for every selected quality of thermal radiation, but his mathematics was not rigorously valid.^[54] He made no mention of thermodynamics in this paper, though he did refer to conservation of vis viva. He proposed that his measurements implied that radiation was both absorbed and emitted by particles of matter throughout depths of the media in which it propagated. He applied the Helmholtz reciprocity principle to account for the material interface processes as distinct from the processes in the interior material. He did not postulate unrealizable perfectly black surfaces. He concluded that his experiments showed that in a cavity in thermal equilibrium, the heat radiated from any part of the interior bounding surface, no matter of what material it might be composed, was the same as would have been emitted from a surface of the same shape and position that would have been composed of lamp-black. He did not state explicitly that the lamp-black-coated bodies that he used as reference must have had a unique common spectral emittance function that depended on temperature in a unique way.

Gustav Kirchhoff[edit]

In 1859, not knowing of Stewart's work, Gustav Robert Kirchhoff reported the coincidence of the wavelengths of spectrally resolved lines of absorption and of emission of visible light. Importantly for thermal physics, he also observed that bright lines or dark lines were apparent depending on the temperature difference between emitter and absorber.^[55]

Kirchhoff then went on to consider some bodies that emit and absorb heat radiation, in an opaque enclosure or cavity, in equilibrium at temperature T.

Here is used a notation different from Kirchhoff's. Here, the emitting power E(T, i) denotes a dimensioned quantity, the total radiation emitted by a body labeled by index i at temperature T. The total absorption ratio a(T, i) of that body is dimensionless, the ratio of absorbed to incident radiation in the cavity at temperature T . (In contrast with Balfour Stewart's, Kirchhoff's definition of his absorption ratio did not refer in particular to a lamp-black surface as the source of the incident radiation.) Thus the ratio E(T, i) / a(T, i) of emitting power to absorption ratio is a dimensioned quantity, with the dimensions of emitting power, because a(T, i) is dimensionless. Also here the wavelength-specific emitting power of the body at temperature T is denoted by E(λ, T, i) and the wavelength-specific absorption ratio by a(λ, T, i) . Again, the ratio E(λ, T, i) / a(λ, T, i) of emitting power to absorption ratio is a dimensioned quantity, with the dimensions of emitting power.

In a second report made in 1859, Kirchhoff announced a new general principle or law for which he offered a theoretical and mathematical proof, though he did not offer quantitative measurements of radiation powers.^[56] His theoretical proof was and still is considered by some writers to be invalid.^[54][57] His principle, however, has endured: it was that for heat rays of the same wavelength, in equilibrium at a given temperature, the wavelength-specific ratio of emitting power to absorption ratio has one and the same common value for all bodies that emit and absorb at that wavelength. In symbols, the law stated that the wavelength-specific ratio E(λ, T, i) / a(λ, T, i) has one and the same value for all bodies, that is for all values of index i . In this report there was no mention of black bodies.

In 1860, still not knowing of Stewart's measurements for selected qualities of radiation, Kirchhoff pointed out that it was long established experimentally that for total heat radiation, of unselected quality, emitted and absorbed by a body in equilibrium, the dimensioned total radiation ratio E(T, i) / a(T, i), has one and the same value common to all bodies, that is, for every value of the material index i.^[58] Again without measurements of radiative powers or other new experimental data, Kirchhoff then offered a fresh theoretical proof of his new principle of the universality of the value of the wavelength-specific ratio E(λ, T, i) / a(λ, T, i) at thermal equilibrium. His fresh theoretical proof was and still is considered by some writers to be invalid.^[54][57]

But more importantly, it relied on a new theoretical postulate of 'perfectly black bodies,' which is the reason why one speaks of Kirchhoff's law. Such black bodies showed complete absorption in their infinitely thin most superficial surface. They correspond to Balfour Stewart's reference bodies, with internal radiation, coated with lamp-black. They were not the more realistic perfectly black bodies later considered by Planck. Planck's black bodies radiated and absorbed only by the material in their interiors; their interfaces with contiguous media were only mathematical surfaces, capable neither of absorption nor emission, but only of reflecting and transmitting with refraction.^[59]

Kirchhoff's proof considered an arbitrary non-ideal body labeled i as well as various perfect black bodies labeled BB . It required that the bodies be kept in a cavity in thermal equilibrium at temperature T . His proof intended to show that the ratio E(λ, T, i) / a(λ, T, i) was independent of the nature i of the non-ideal body, however partly transparent or partly reflective it was.

His proof first argued that for wavelength λ and at temperature T, at thermal equilibrium, all perfectly black bodies of the same size and shape have the one and the same common value of emissive power E(λ, T, BB), with the dimensions of power. His proof noted that the dimensionless wavelength-specific absorption ratio a(λ, T, BB) of a perfectly black body is by definition exactly 1. Then for a perfectly black body, the wavelength-specific ratio of emissive power to absorption ratio E(λ, T, BB) / a(λ, T, BB) is again just E(λ, T, BB), with the dimensions of power. Kirchhoff considered, successively, thermal equilibrium with the arbitrary non-ideal body, and with a perfectly black body of the same size and shape, in place in his cavity in equilibrium at temperature T . He argued that the flows of heat radiation must be the same in each case. Thus he argued that at thermal equilibrium the ratio E(λ, T, i) / a(λ, T, i) was equal to E(λ, T, BB), which may now be denoted B_λ (λ, T), a continuous function, dependent only on λ at fixed temperature T, and an increasing function of T at fixed wavelength λ, at low temperatures vanishing for visible but not for longer wavelengths, with positive values for visible wavelengths at higher temperatures, which does not depend on the nature i of the arbitrary non-ideal body. (Geometrical factors, taken into detailed account by Kirchhoff, have been ignored in the foregoing.)

Thus Kirchhoff's law of thermal radiation can be stated: For any material at all, radiating and absorbing in thermodynamic equilibrium at any given temperature T, for every wavelength λ, the ratio of emissive power to absorptive ratio has one universal value, which is characteristic of a perfect black body, and is an emissive power which we here represent by B_λ (λ, T) . (For our notation B_λ (λ, T), Kirchhoff's original notation was simply e.)^{[58][60][61][62][63][64]}

Kirchhoff announced that the determination of the function B_λ (λ, T) was a problem of the highest importance, though he recognized that there would be experimental difficulties to be overcome. He supposed that like other functions that do not depend on the properties of individual bodies, it would be a simple function. Occasionally by historians that function B_λ (λ, T) has been called 'Kirchhoff's (emission, universal) function,'^{[65][66][67][68]} though its precise mathematical form would not be known for another forty years, till it was discovered by Planck in 1900. The theoretical proof for Kirchhoff's universality principle was worked on and debated by various physicists over the same time, and later.^[57] Kirchhoff stated later in 1860 that his theoretical proof was better than Balfour Stewart's, and in some respects it was so.^[54] Kirchhoff's 1860 paper did not mention the second law of thermodynamics, and of course did not mention the concept of entropy which had not at that time been established. In a more considered account in a book in 1862, Kirchhoff mentioned the connection of his law with Carnot's principle, which is a form of the second law.^[69]

According to Helge Kragh, 'Quantum theory owes its origin to the study of thermal radiation, in particular to the 'black-body' radiation that Robert Kirchhoff had first defined in 1859–1860.'^[70]

Doppler effect[edit]

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':

Black Body Radiation Definition Pdf

Black Body Radiation Formula Pdf

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.^[71] 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.

Through Planck's law the temperature spectrum of a black body is proportionally related to the frequency of light and one may substitute the temperature (T) for the frequency in this equation.

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.

This is an important effect in astronomy, where the velocities of stars and galaxies can reach significant fractions of c. An example is found in the cosmic microwave background radiation, which exhibits a dipole anisotropy from the Earth's motion relative to this black-body radiation field.

See also[edit]

Black Body Radiation Pdf File

References[edit]

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2. ^Mandel & Wolf 1995, Chapter 13.
3. ^Kondepudi & Prigogine 1998, Chapter 11.
4. ^Landsberg 1990, Chapter 13.
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7. ^Alessandro Fabbri; José Navarro-Salas (2005). 'Chapter 1: Introduction'. Modeling black hole evaporation. Imperial College Press. ISBN1-86094-527-9.
8. ^From (Kirchhoff, 1860) (Annalen der Physik und Chemie), p. 277: 'Der Beweis, welcher für die ausgesprochene Behauptung hier gegeben werden soll, … vollkommen schwarze, oder kürzer schwarze, nennen.' (The proof, which shall be given here for the proposition stated [above], rests on the assumption that bodies are conceivable which in the case of infinitely small thicknesses, completely absorb all rays that fall on them, thus [they] neither reflect nor transmit rays. I will call such bodies 'completely black [bodies]' or more briefly 'black [bodies]'.) See also (Kirchhoff, 1860) (Philosophical Magazine), p. 2.
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12. ^Planck 1914
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14. ^Partington 1949, pp. 466–467, 478.
15. ^Goody & Yung 1989, pp. 482, 484
16. ^ ^abcPlanck 1914, p. 42
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23. ^ ^abStewart 1858
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42. ^ ^abcNASA Sun Fact Sheet
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44. ^Principles of Planetary Climate by Raymond T. Peirrehumbert, Cambridge University Press (2011), p. 146. From Chapter 3 which is available online hereArchived March 28, 2012, at the Wayback Machine, p. 12 mentions that Venus' black-body temperature would be 330 K 'in the zero albedo case', but that due to atmospheric warming, its actual surface temperature is 740 K.
45. ^Saari, J. M.; Shorthill, R. W. (1972). 'The Sunlit Lunar Surface. I. Albedo Studies and Full Moon'. The Moon. 5 (1–2): 161–178. Bibcode:1972Moon....5..161S. doi:10.1007/BF00562111.
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54. ^ ^abcdSiegel 1976
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60. ^Chandrasekhar 1950, p. 8
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63. ^Mihalas & Weibel-Mihalas 1984, p. 328
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66. ^Hermann 1971, p. 7
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69. ^Kirchhoff & 1862/1882, p. 573
70. ^Kragh 1999, p. 58
71. ^The Doppler Effect, T. P. Gill, Logos Press, 1965

Bibliography[edit]

• Chandrasekhar, S. (1950). Radiative Transfer. Oxford University Press.
• Goody, R. M.; Yung, Y. L. (1989). Atmospheric Radiation: Theoretical Basis (2nd ed.). Oxford University Press. ISBN978-0-19-510291-8.
• Hermann, A. (1971). The Genesis of Quantum Theory. Nash, C.W. (transl.). MIT Press. ISBN0-262-08047-8. a translation of Frühgeschichte der Quantentheorie (1899–1913), Physik Verlag, Mosbach/Baden.
• Kirchhoff, G.; [27 October 1859] (1860a). 'Über die Fraunhofer'schen Linien' [On Fraunhofer's lines]. Monatsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin: 662–665.
• Kirchhoff, G.; [11 December 1859] (1860b). 'Über den Zusammenhang zwischen Emission und Absorption von Licht und Wärme' [On the relation between emission and absorption of light and heat]. Monatsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin: 783–787.
• Kirchhoff, G. (1860c). 'Ueber das Verhältniss zwischen dem Emissionsvermögen und dem Absorptionsvermögen der Körper für Wärme and Licht' [On the relation between bodies' emission capacity and absorption capacity for heat and light]. Annalen der Physik und Chemie. 109 (2): 275–301. Bibcode:1860AnP...185..275K. doi:10.1002/andp.18601850205. Translated by Guthrie, F. as Kirchhoff, G. (1860). 'On the relation between the radiating and absorbing powers of different bodies for light and heat'. Philosophical Magazine. Series 4, volume 20: 1–21.
• Kirchhoff, G. (1882) [1862], 'Ueber das Verhältniss zwischen dem Emissionsvermögen und dem Absorptionsvermögen der Körper für Wärme und Licht', Gessamelte Abhandlungen, Leipzig: Johann Ambrosius Barth, pp. 571–598
• Kondepudi, D.; Prigogine, I. (1998). Modern Thermodynamics. From Heat Engines to Dissipative Structures. John Wiley & Sons. ISBN0-471-97393-9.
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• Kuhn, T. S. (1978). Black–Body Theory and the Quantum Discontinuity. Oxford University Press. ISBN0-19-502383-8.
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• Lavenda, Bernard Howard (1991). Statistical Physics: A Probabilistic Approach. John Wiley & Sons. pp. 41–42. ISBN978-0-471-54607-8.
• Loudon, R. (2000) [1973]. The Quantum Theory of Light (third ed.). Cambridge University Press. ISBN0-19-850177-3.
• Mandel, L.; Wolf, E. (1995). Optical Coherence and Quantum Optics. Cambridge University Press. ISBN0-521-41711-2.
• Mehra, J.; Rechenberg, H. (1982). The Historical Development of Quantum Theory. volume 1, part 1. Springer-Verlag. ISBN0-387-90642-8.
• Mihalas, D.; Weibel-Mihalas, B. (1984). Foundations of Radiation Hydrodynamics. Oxford University Press. ISBN0-19-503437-6.
• Milne, E.A. (1930). 'Thermodynamics of the Stars'. Handbuch der Astrophysik. 3, part 1: 63–255.
• Partington, J.R. (1949). An Advanced Treatise on Physical Chemistry. Volume 1. Fundamental Principles. The Properties of Gases. Longmans, Green and Co.
• Planck, M. (1914) [1912]. The Theory of Heat Radiation. translated by Masius, M. P. Blakiston's Sons & Co.
• Rybicki, G. B.; Lightman, A. P. (1979). Radiative Processes in Astrophysics. John Wiley & Sons. ISBN0-471-82759-2.
• Schirrmacher, A. (2001). Experimenting theory: the proofs of Kirchhoff's radiation law before and after Planck. Münchner Zentrum für Wissenschafts und Technikgeschichte.
• Siegel, D.M. (1976). 'Balfour Stewart and Gustav Robert Kirchhoff: two independent approaches to 'Kirchhoff's radiation law''. Isis. 67 (4): 565–600. doi:10.1086/351669.
• Stewart, B. (1858). 'An account of some experiments on radiant heat'. Transactions of the Royal Society of Edinburgh. 22: 1–20.
• Wien, W. (1894). 'Temperatur und Entropie der Strahlung' [Temperature and entropy of radiation]. Annalen der Physik. 288 (5): 132–165. Bibcode:1894AnP...288..132W. doi:10.1002/andp.18942880511.

Further reading[edit]

• Kroemer, Herbert; Kittel, Charles (1980). Thermal Physics (2nd ed.). W. H. Freeman Company. ISBN0-7167-1088-9.
• Tipler, Paul; Llewellyn, Ralph (2002). Modern Physics (4th ed.). W. H. Freeman. ISBN0-7167-4345-0.

External links[edit]

Black Body Radiation Physics Pdf

• Black-body radiation JavaScript Interactives Black-body radiation by Fu-Kwun Hwang and Loo Kang Wee
• Calculating Black-body Radiation Interactive calculator with Doppler Effect. Includes most systems of units.
• Color-to-Temperature demonstration at Academo.org
• Cooling Mechanisms for Human Body – From Hyperphysics
• 'Blackbody Spectrum' by Jeff Bryant, Wolfram Demonstrations Project, 2007.

Blackbody Radiation Temperature