Black body

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For the radiation from a black body in thermal equilibrium, see Black-body radiation.
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 white body is one with a "rough surface [that] reflects all incident rays completely and uniformly in all directions."[1]

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.

A black body in thermal equilibrium has two notable properties:[2]

  1. It is an ideal emitter: at every frequency, it emits as much energy as – or more energy than – 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 surface is a hole in the wall of a large enclosure (see 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.[3][4]

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.[5][6] Construction of black bodies with emissivity as close to one as possible remains a topic of current interest.[7]

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.

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.[8]

A more modern definition drops the reference to "infinitely small thicknesses":[9]

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. [10]

Idealizations[edit]

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

An approximate realization of a black body as a tiny hole in an insulated enclosure.

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.[10] 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.[11]

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.[11] 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,[12] although the time it takes to do so may be very long.[13] Typically, equilibrium is reached by continual absorption and re-emission of radiation by material in the cavity or its walls.[3][4][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 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.[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]

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 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%.[7][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]

Another example of a nearly perfect black material is super black, made by chemically etching a nickelphosphorus alloy.[34]

Stars and planets[edit]

For more about the UBV color index, see Photometric system.
An idealized view of the cross-section of a star. The photosphere contains photons 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.[35]

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.[36]

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,[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 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.[40] For example, the Sun has an effective temperature of 5780 K,[41] 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 km 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:

T=\frac {\hbar c^3}{8\pi Gk_BM} \ ,

where c is the speed of light, ℏ is the reduced Planck constant, kB 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 2.7K. 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[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:

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" emissivity ε ≤ 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 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 km, rather than black holes as originally conjectured.[55] It should be noted that 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

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. ISBN 3-540-22292-8. 
  3. ^ a b 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. Thermodynamics and statistical mechanics (Reprint of Oxford University Press 1978 ed.). Courier Dover Publications. p. 209. ISBN 0-486-66493-7. 
  4. ^ a b Planck 1914, p. 44, §52
  5. ^ 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. ISBN 0-471-43463-9.  and Ronald G. Driggers (2003). Encyclopedia of optical engineering, Volume 3. CRC Press. p. 2303. ISBN 0-8247-4252-4. 
  6. ^ 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. 
  7. ^ a b Ai Lin Chun (25 Jan 2008). "Carbon nanotubes: Blacker than black". Nature Nanotechnology. doi:10.1038/nnano.2008.29. 
  8. ^ 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 (Taylor & Francis) 20 (130). 
  9. ^ The notion of an infinitely thin layer was dropped by Planck. See Planck 1914, p. 10, footnote 2, .
  10. ^ a b Siegel, Robert; Howell, John R. (2002). Thermal Radiation Heat Transfer; Volume 1 (4th ed.). Taylor & Francis. p. 7. ISBN 1-56032-839-8. 
  11. ^ a b 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 Roger Dale Van Zee, J. Patrick Looney (2002). Cavity-enhanced spectroscopies. Academic Press. p. 202. ISBN 0-12-475987-4. 
  12. ^ Clement John Adkins (1983). "§4.1 The function of the second law". Equilibrium thermodynamics (3rd ed.). Cambridge University Press. p. 50. ISBN 0-521-27456-7. 
  13. ^ 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. ISBN 0521821436. 
  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. ISBN 3-11-014318-6. "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. ISBN 0-521-58990-8. 
  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. ISBN 0-486-63896-0. "...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. ISBN 0-7167-3821-X. 
  21. ^ Massoud Kaviany (2002). "Figure 4.3(b) Radiation properties of an opaque surface". Principles of heat transfer. Wiley-IEEE. p. 381. ISBN 0-471-43463-9. 
  22. ^ BA Venkanna (2010). "§10.3.4 Absorptivity, reflectivity, and transmissivity". Fundamentals of heat and mass transfer. PHI Learning Pvt. Ltd. pp. 385–386. ISBN 81-203-4031-0. 
  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. ISBN 0-387-95174-1. 
  28. ^ Kangro 1976, p. 159
  29. ^ Lummer & Kurlbaum 1901
  30. ^ CF Lewis (June 1988). "Materials keep a low profile". Mech. Eng.: 37–41. 
  31. ^ Bradley Quinn (2010). Textile Futures. Berg. p. 68. ISBN 1-84520-807-2. 
  32. ^ a b K. Mizuno et al. (2009). "A black body absorber from vertically aligned single-walled carbon nanotubes" (free download). Proceedings of the National Academy of Sciences 106 (15): 6044–6077. Bibcode:2009PNAS..106.6044M. doi:10.1073/pnas.0900155106. PMC 2669394. PMID 19339498. 
  33. ^ Zu-Po Yang et al. (2008). "Experimental observation of an extremely dark material made by a low-density nanotube array". Nano Letters (American Chemical Society) 8: 446–451. Bibcode:2008NanoL...8..446Y. doi:10.1021/nl072369t. PMID 18181658. 
  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. ISBN 0-521-54622-2. "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. ISBN 0-521-34869-2. 
  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. ISBN 1-4419-7623-X. 
  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. Retrieved 2012-01-26. 
  39. ^ M Golay (1974). "Table IX: U-B Indices". Introduction to astronomical photometry. Springer. p. 82. ISBN 90-277-0428-7. 
  40. ^ Lawrence Hugh Aller (1991). Atoms, stars, and nebulae (3rd ed.). Cambridge University Press. p. 61. ISBN 0-521-31040-7. 
  41. ^ Kenneth R. Lang (2006). Astrophysical formulae, Volume 1 (3rd ed.). Birkhäuser. p. 23. ISBN 3-540-29692-1. 
  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. ISBN 1-4020-1428-7. 
  43. ^ Schutz, Bernard (2004). Gravity From the Group Up: An Introductory Guide to Gravity and General Relativity (1st ed.). Cambridge University Press. p. 304. ISBN 0-521-45506-5. 
  44. ^ PCW Davies (1978). "Thermodynamics of black holes". Rep Prog Phys 41 (8): 1313 ff. Bibcode:1978RPPh...41.1313D. doi:10.1088/0034-4885/41/8/004. 
  45. ^ Robert M Wald (2005). "The thermodynamics of black holes". In Andrés Gomberoff, Donald Marolf, eds. Lectures on quantum gravity. Springer. pp. 1 ff. ISBN 0-387-23995-2. 
  46. ^ Bernard J Carr and Steven B Giddings (2008). "Chapter 6: Quantum black holes". Beyond Extreme Physics: Cutting-edge science. Rosen Publishing Group, Scientific American (COR). p. 30. ISBN 1-4042-1402-X. 
  47. ^ Valeri P. Frolov, Andrei Zelnikov (2011). "Equation 9.7.1". Introduction to Black Hole Physics. Oxford University Press. p. 321. ISBN 0-19-969229-7. 
  48. ^ Robert M Wald. "The thermodynamics of black holes". cited work. p. 28. ISBN 0-387-23995-2. "... no results on black hole thermodynamics have been subject to any experimental or observational tests, ..." 
  49. ^ White, M. (1999). "Anisotropies in the CMB". 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. ISBN 81-317-5513-4. 
  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. ISBN 3-527-63087-2. 
  53. ^ Robert Osiander, M. Ann Garrison Darrin, John Champion (2006). MEMS and Microstructures in aerospace applications. CRC Press. p. 187. ISBN 0-8247-2637-5. 
  54. ^ Neutrino emission is a mechanism of cooling in neutron stars, for example; see Mikhail A. Shifman (2001). "Cooling by neutrino emission". In B. L. Ioffe, Mikhail A. Shifman, eds. World Scientific. p. 2135. ISBN 981-02-4969-1. 
  55. ^ Walter Lewin with Warren Goldstein (2011). "X-ray bursters!". For the love of physics. Simon and Schuster. pp. 251 ff. ISBN 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. ISBN 3-540-30312-X. 

Bibliography[edit]

External links[edit]