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Stefan–Boltzmann law

The Stefan–Boltzmann law describes the power radiated from a black body in terms of its temperature. Specifically, the Stefan–Boltzmann law states that the total energy radiated per unit surface area of a black body across all wavelengths per unit time (also known as the black-body radiant emittance) is directly proportional to the fourth power of the black body's thermodynamic temperature T:

Graph of a function of total emitted energy of a black body proportional to its thermodynamic temperature . In blue is a total energy according to the Wien approximation,

The constant of proportionality σ, called the Stefan–Boltzmann constant, is derived from other known physical constants. Since 2019, the value of the constant is

where k is the Boltzmann constant, h is the Planck constant, and c is the speed of light in vacuum. The radiance from a specified angle of view (watts per square metre per steradian) is given by

A body that does not absorb all incident radiation (sometimes known as a grey body) emits less total energy than a black body and is characterized by an emissivity, :

The radiant emittance has dimensions of energy flux (energy per unit time per unit area), and the SI units of measure are joules per second per square metre, or equivalently, watts per square metre. The SI unit for absolute temperature T is the kelvin. is the emissivity of the grey body; if it is a perfect blackbody, . In the still more general (and realistic) case, the emissivity depends on the wavelength, .

To find the total power radiated from an object, multiply by its surface area, :

Wavelength- and subwavelength-scale particles,[1] metamaterials,[2] and other nanostructures are not subject to ray-optical limits and may be designed to exceed the Stefan–Boltzmann law.

History

In 1864, John Tyndall presented measurements of the infrared emission by a platinum filament and the corresponding color of the filament.[3] The proportionality to the fourth power of the absolute temperature was deduced by Josef Stefan (1835–1893) in 1877 on the basis of Tyndall's experimental measurements, in the article Über die Beziehung zwischen der Wärmestrahlung und der Temperatur (On the relationship between thermal radiation and temperature) in the Bulletins from the sessions of the Vienna Academy of Sciences.[4][5]

A derivation of the law from theoretical considerations was presented by Ludwig Boltzmann (1844–1906) in 1884, drawing upon the work of Adolfo Bartoli.[6] Bartoli in 1876 had derived the existence of radiation pressure from the principles of thermodynamics. Following Bartoli, Boltzmann considered an ideal heat engine using electromagnetic radiation instead of an ideal gas as working matter.

The law was almost immediately experimentally verified. Heinrich Weber in 1888 pointed out deviations at higher temperatures, but perfect accuracy within measurement uncertainties was confirmed up to temperatures of 1535 K by 1897.[7] The law, including the theoretical prediction of the Stefan–Boltzmann constant as a function of the speed of light, the Boltzmann constant and the Planck constant, is a direct consequence of Planck's law as formulated in 1900.

As of the 2019 redefinition of SI base units, which fixes the values of the Boltzmann constant k, the Planck constant h, and the speed of light c, the Stefan–Boltzmann constant is exactly

 
σ = 5.670374419...×10−8 W/(m2K4).[8]

Prior to this, the value of   was calculated from the measured value of the gas constant.[9]

The numerical value of the Stefan–Boltzmann constant is different in other systems of units. For example, in thermochemistry it is often expressed in calcm−2day−1K−4:

σ = 1.170937...×10−7 cal cm−2⋅day−1⋅K−4.

Likewise, in US customary units the Stefan–Boltzmann constant is[10]

σ = 1.713441...×10−9 BTU⋅hr−1⋅ft−2⋅°R−4.

Examples

Temperature of the Sun

 
Log–log graphs of peak emission wavelength and radiant exitance vs. black-body temperature – red arrows show that 5780 K black bodies have 501 nm peak and 63.3 MW/m2 radiant exitance

With his law, Stefan also determined the temperature of the Sun's surface.[11] He inferred from the data of Jacques-Louis Soret (1827–1890)[12] that the energy flux density from the Sun is 29 times greater than the energy flux density of a certain warmed metal lamella (a thin plate). A round lamella was placed at such a distance from the measuring device that it would be seen at the same angular diameter as the Sun. Soret estimated the temperature of the lamella to be approximately 1900 °C to 2000 °C. Stefan surmised that ⅓ of the energy flux from the Sun is absorbed by the Earth's atmosphere, so he took for the correct Sun's energy flux a value 3/2 times greater than Soret's value, namely 29 × 3/2 = 43.5.

Precise measurements of atmospheric absorption were not made until 1888 and 1904. The temperature Stefan obtained was a median value of previous ones, 1950 °C and the absolute thermodynamic one 2200 K. As 2.574 = 43.5, it follows from the law that the temperature of the Sun is 2.57 times greater than the temperature of the lamella, so Stefan got a value of 5430 °C or 5700 K. This was the first sensible value for the temperature of the Sun. Before this, values ranging from as low as 1800 °C to as high as 13,000,000 °C[13] were claimed. The lower value of 1800 °C was determined by Claude Pouillet (1790–1868) in 1838 using the Dulong–Petit law.[14] Pouillet also took just half the value of the Sun's correct energy flux.

Temperature of stars

The temperature of stars other than the Sun can be approximated using a similar means by treating the emitted energy as a black body radiation.[15] So:

 

where L is the luminosity, σ is the Stefan–Boltzmann constant, R is the stellar radius and T is the effective temperature. This formula can then be rearranged to calculate the temperature:

 

or alternatively the radius:

 

The same formulae can also be simplified to compute the parameters relative to the Sun:

 
 
 

where   is the solar radius, and so forth. They can also be rewritten in terms of the surface area A and radiant emittance  :

 
 
 

where   and  

With the Stefan–Boltzmann law, astronomers can easily infer the radii of stars. The law is also met in the thermodynamics of black holes in so-called Hawking radiation.

Effective temperature of the Earth

Similarly we can calculate the effective temperature of the Earth T by equating the energy received from the Sun and the energy radiated by the Earth, under the black-body approximation (Earth's own production of energy being small enough to be negligible). The luminosity of the Sun, L, is given by:

 

At Earth, this energy is passing through a sphere with a radius of a0, the distance between the Earth and the Sun, and the irradiance (received power per unit area) is given by

 

The Earth has a radius of R, and therefore has a cross-section of  . The radiant flux (i.e. solar power) absorbed by the Earth is thus given by:

 

Because the Stefan–Boltzmann law uses a fourth power, it has a stabilizing effect on the exchange and the flux emitted by Earth tends to be equal to the flux absorbed, close to the steady state where:

 

T can then be found:

 

where T is the temperature of the Sun, R the radius of the Sun, and a0 is the distance between the Earth and the Sun. This gives an effective temperature of 6 °C on the surface of the Earth, assuming that it perfectly absorbs all emission falling on it and has no atmosphere.

The Earth has an albedo of 0.3, meaning that 30% of the solar radiation that hits the planet gets scattered back into space without absorption. The effect of albedo on temperature can be approximated by assuming that the energy absorbed is multiplied by 0.7, but that the planet still radiates as a black body (the latter by definition of effective temperature, which is what we are calculating). This approximation reduces the temperature by a factor of 0.71/4, giving 255 K (−18 °C).[16][17]

The above temperature is Earth's as seen from space, not ground temperature but an average over all emitting bodies of Earth from surface to high altitude. Because of the greenhouse effect, the Earth's actual average surface temperature is about 288 K (15 °C), which is higher than the 255 K effective temperature, and even higher than the 279 K temperature that a black body would have.

In the above discussion, we have assumed that the whole surface of the earth is at one temperature. Another interesting question is to ask what the temperature of a blackbody surface on the earth would be assuming that it reaches equilibrium with the sunlight falling on it. This of course depends on the angle of the sun on the surface and on how much air the sunlight has gone through. When the sun is at the zenith and the surface is horizontal, the irradiance can be as high as 1120 W/m2.[18] The Stefan–Boltzmann law then gives a temperature of

 

or 102 °C. (Above the atmosphere, the result is even higher: 394 K.) We can think of the earth's surface as "trying" to reach equilibrium temperature during the day, but being cooled by the atmosphere, and "trying" to reach equilibrium with starlight and possibly moonlight at night, but being warmed by the atmosphere.

Origination

Thermodynamic derivation of the energy density

The fact that the energy density of the box containing radiation is proportional to   can be derived using thermodynamics.[19][20] This derivation uses the relation between the radiation pressure p and the internal energy density  , a relation that can be shown using the form of the electromagnetic stress–energy tensor. This relation is:

 

Now, from the fundamental thermodynamic relation

 

we obtain the following expression, after dividing by   and fixing   :

 

The last equality comes from the following Maxwell relation:

 

From the definition of energy density it follows that

 

where the energy density of radiation only depends on the temperature, therefore

 

Now, the equality

 

after substitution of   Meanwhile, the pressure is the rate of momentum change per unit area. Since the momentum of a photon is the same as the energy divided by the speed of light,

 

where the 1/3 factor comes from the projection of the momentum transfer onto the normal to the wall of the container.

Since the partial derivative   can be expressed as a relationship between only   and   (if one isolates it on one side of the equality), the partial derivative can be replaced by the ordinary derivative. After separating the differentials the equality becomes

 

which leads immediately to  , with   as some constant of integration.

Derivation from Planck's law

 
Deriving the Stefan–Boltzmann Law using the Planck's law.

The law can be derived by considering a small flat black body surface radiating out into a half-sphere. This derivation uses spherical coordinates, with θ as the zenith angle and φ as the azimuthal angle; and the small flat blackbody surface lies on the xy-plane, where θ = π/2.

The intensity of the light emitted from the blackbody surface is given by Planck's law,

 
where

The quantity   is the power radiated by a surface of area A through a solid angle dΩ in the frequency range between ν and ν + .

The Stefan–Boltzmann law gives the power emitted per unit area of the emitting body,

 

Note that the cosine appears because black bodies are Lambertian (i.e. they obey Lambert's cosine law), meaning that the intensity observed along the sphere will be the actual intensity times the cosine of the zenith angle. To derive the Stefan–Boltzmann law, we must integrate   over the half-sphere and integrate   from 0 to ∞.

 

Then we plug in for I:

 

To evaluate this integral, do a substitution,

 

which gives:

 

The integral on the right is standard and goes by many names: it is a particular case of a Bose–Einstein integral, the polylogarithm, or the Riemann zeta function  . The value of the integral is   (where   is the Gamma function), giving the result that, for a perfect blackbody surface:

 

Finally, this proof started out only considering a small flat surface. However, any differentiable surface can be approximated by a collection of small flat surfaces. So long as the geometry of the surface does not cause the blackbody to reabsorb its own radiation, the total energy radiated is just the sum of the energies radiated by each surface; and the total surface area is just the sum of the areas of each surface—so this law holds for all convex blackbodies, too, so long as the surface has the same temperature throughout. The law extends to radiation from non-convex bodies by using the fact that the convex hull of a black body radiates as though it were itself a black body.

Energy density

The total energy density U can be similarly calculated, except the integration is over the whole sphere and there is no cosine, and the energy flux (U c) should be divided by the velocity c to give the energy density U:

 
Thus   is replaced by  , giving an extra factor of 4.

Thus, in total:

 
The product   is sometimes known as the radiation constant or radiation density constant.[21][22]

See also

Notes

  1. ^ Bohren, Craig F.; Huffman, Donald R. (1998). Absorption and scattering of light by small particles. Wiley. pp. 123–126. ISBN 978-0-471-29340-8.
  2. ^ Narimanov, Evgenii E.; Smolyaninov, Igor I. (2012). "Beyond Stefan–Boltzmann Law: Thermal Hyper-Conductivity". Conference on Lasers and Electro-Optics 2012. OSA Technical Digest. Optical Society of America. pp. QM2E.1. CiteSeerX 10.1.1.764.846. doi:10.1364/QELS.2012.QM2E.1. ISBN 978-1-55752-943-5. S2CID 36550833.
  3. ^
    • Tyndall, John (1864). "On luminous [i.e., visible] and obscure [i.e., infrared] radiation". Philosophical Magazine. 4th series. 28: 329–341. ; see p. 333.
    In his physics textbook of 1875, Adolph Wüllner quoted Tyndall's results and then added estimates of the temperature that corresponded to the platinum filament's color:
    • Wüllner, Adolph (1875). Lehrbuch der Experimentalphysik [Textbook of experimental physics] (in German). Vol. 3. Leipzig, Germany: B.G. Teubner. p. 215.
    From (Wüllner, 1875), p. 215: "Wie aus gleich zu besprechenden Versuchen von Draper hervorgeht, … also fast um das 12fache zu." (As follows from the experiments of Draper, which will be discussed shortly, a temperature of about 525°[C] corresponds to the weak red glow; a [temperature] of about 1200°[C], to the full white glow. Thus, while the temperature climbed only somewhat more than double, the intensity of the radiation increased from 10.4 to 122; thus, almost 12-fold.)
    See also:
    • Wisniak, Jaime (November 2002). "Heat radiation law – from Newton to Stefan". Indian Journal of Chemical Technology. 9: 545–555. ; see pp. 551–552. Available at: National Institute of Science Communication and Information Resources (New Delhi, India)
  4. ^ Stefan, J. (1879). "Über die Beziehung zwischen der Wärmestrahlung und der Temperatur" [On the relation between heat radiation and temperature]. Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften: Mathematisch-Naturwissenschaftliche Classe (Proceedings of the Imperial Philosophical Academy [of Vienna]: Mathematical and Scientific Class) (in German). 79: 391–428.
  5. ^ Stefan stated (Stefan, 1879), p. 421: "Zuerst will ich hier die Bemerkung anführen, … die Wärmestrahlung der vierten Potenz der absoluten Temperatur proportional anzunehmen." (First of all, I want to point out here the observation which Wüllner, in his textbook, added to the report of Tyndall's experiments on the radiation of a platinum wire that was brought to glowing by an electric current, because this observation first caused me to suppose that thermal radiation is proportional to the fourth power of the absolute temperature.)
  6. ^ Boltzmann, Ludwig (1884). "Ableitung des Stefan'schen Gesetzes, betreffend die Abhängigkeit der Wärmestrahlung von der Temperatur aus der electromagnetischen Lichttheorie" [Derivation of Stefan's law, concerning the dependency of heat radiation on temperature, from the electromagnetic theory of light]. Annalen der Physik und Chemie (in German). 258 (6): 291–294. Bibcode:1884AnP...258..291B. doi:10.1002/andp.18842580616.
  7. ^ Massimiliano Badino, The Bumpy Road: Max Planck from Radiation Theory to the Quantum (1896–1906) (2015), p. 31.
  8. ^ Sloane, N. J. A. (ed.). "Sequence A081820". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  9. ^ Moldover, M. R.; Trusler, J. P. M.; Edwards, T. J.; Mehl, J. B.; Davis, R. S. (1988-01-25). "Measurement of the Universal Gas Constant R Using a Spherical Acoustic Resonator". Physical Review Letters. 60: 249. doi:10.1103/PhysRevLett.60.249.
  10. ^ Çengel, Yunus A. (2007). Heat and Mass Transfer: a Practical Approach (3rd ed.). McGraw Hill.
  11. ^ (Stefan, 1879), pp. 426–427.
  12. ^ Soret, J.L. (1872) "Comparaison des intensités calorifiques du rayonnement solaire et du rayonnement d'un corps chauffé à la lampe oxyhydrique" [Comparison of the heat intensities of solar radiation and of radiation from a body heated with an oxy-hydrogen torch], Archives des sciences physiques et naturelles (Geneva, Switzerland), 2nd series, 44: 220–229; 45: 252–256.
  13. ^ Waterston, John James (1862). "An account of observations on solar radiation". Philosophical Magazine. 4th series. 23 (2): 497–511. Bibcode:1861MNRAS..22...60W. doi:10.1093/mnras/22.2.60. On p. 505, the Scottish physicist John James Waterston estimated that the temperature of the sun's surface could be 12,880,000°.
  14. ^ See:
    • Pouillet (1838). "Mémoire sur la chaleur solaire, sur les pouvoirs rayonnants et absorbants de l'air atmosphérique, et sur la température de l'espace" [Memoir on solar heat, on the radiating and absorbing powers of the atmospheric air, and on the temperature of space]. Comptes Rendus (in French). 7 (2): 24–65. On p. 36, Pouillet estimates the sun's temperature: " … cette température pourrait être de 1761° … " ( … this temperature [i.e., of the Sun] could be 1761° … )
    • English translation: Pouillet (1838) "Memoir on the solar heat, on the radiating and absorbing powers of atmospheric air, and on the temperature of space" in: Taylor, Richard, ed. (1846) Scientific Memoirs, Selected from the Transactions of Foreign Academies of Science and Learned Societies, and from Foreign Journals. vol. 4. London, England: Richard and John E. Taylor. pp. 44–90; see pp. 55–56.
  15. ^ "Luminosity of Stars". Australian Telescope Outreach and Education. Retrieved 2006-08-13.
  16. ^ Intergovernmental Panel on Climate Change Fourth Assessment Report. Chapter 1: Historical overview of climate change science 2018-11-26 at the Wayback Machine page 97
  17. ^ Solar Radiation and the Earth's Energy Balance
  18. ^ "Introduction to Solar Radiation". Newport Corporation. from the original on October 29, 2013.
  19. ^ Knizhnik, Kalman. (PDF). Johns Hopkins University – Department of Physics & Astronomy. Archived from the original (PDF) on 2016-03-04. Retrieved 2018-09-03.
  20. ^ (Wisniak, 2002), p. 554.
  21. ^ Lemons, Don S.; Shanahan, William R.; Buchholtz, Louis J. (2022-09-13). On the Trail of Blackbody Radiation: Max Planck and the Physics of his Era. MIT Press. p. 38. ISBN 978-0-262-37038-7.
  22. ^ Campana, S.; Mangano, V.; Blustin, A. J.; Brown, P.; Burrows, D. N.; Chincarini, G.; Cummings, J. R.; Cusumano, G.; Valle, M. Della; Malesani, D.; Mészáros, P.; Nousek, J. A.; Page, M.; Sakamoto, T.; Waxman, E. (August 2006). "The association of GRB 060218 with a supernova and the evolution of the shock wave". Nature. 442 (7106): 1008–1010. doi:10.1038/nature04892. ISSN 0028-0836.

References

  • Stefan, J. (1879), "Über die Beziehung zwischen der Wärmestrahlung und der Temperatur" [On the relationship between heat radiation and temperature] (PDF), Sitzungsberichte der Mathematisch-naturwissenschaftlichen Classe der Kaiserlichen Akademie der Wissenschaften (in German), 79: 391–428
  • Boltzmann, L. (1884), "Ableitung des Stefan'schen Gesetzes, betreffend die Abhängigkeit der Wärmestrahlung von der Temperatur aus der electromagnetischen Lichttheorie" [Derivation of Stefan's little law concerning the dependence of thermal radiation on the temperature of the electro-magnetic theory of light], Annalen der Physik und Chemie (in German), 258 (6): 291–294, Bibcode:1884AnP...258..291B, doi:10.1002/andp.18842580616

stefan, boltzmann, this, article, includes, list, general, references, lacks, sufficient, corresponding, inline, citations, please, help, improve, this, article, introducing, more, precise, citations, january, 2023, learn, when, remove, this, template, message. This article includes a list of general references but it lacks sufficient corresponding inline citations Please help to improve this article by introducing more precise citations January 2023 Learn how and when to remove this template message Stefan s law redirects here Not to be confused with Stefan s equation or Stefan s formula The Stefan Boltzmann law describes the power radiated from a black body in terms of its temperature Specifically the Stefan Boltzmann law states that the total energy radiated per unit surface area of a black body across all wavelengths per unit time j displaystyle j star also known as the black body radiant emittance is directly proportional to the fourth power of the black body s thermodynamic temperature T Graph of a function of total emitted energy of a black body j displaystyle j star proportional to its thermodynamic temperature T displaystyle T In blue is a total energy according to the Wien approximation j W j z 4 0 924 s T 4 displaystyle j W star j star zeta 4 approx 0 924 sigma T 4 j s T 4 displaystyle j star sigma T 4 The constant of proportionality s called the Stefan Boltzmann constant is derived from other known physical constants Since 2019 the value of the constant is s 2 p 5 k 4 15 c 2 h 3 5 670 374 419 10 8 W m 2 K 4 displaystyle sigma frac 2 pi 5 k 4 15c 2 h 3 5 670 374 419 times 10 8 mathrm W m 2 K 4 where k is the Boltzmann constant h is the Planck constant and c is the speed of light in vacuum The radiance from a specified angle of view watts per square metre per steradian is given by L j p s p T 4 displaystyle L frac j star pi frac sigma pi T 4 A body that does not absorb all incident radiation sometimes known as a grey body emits less total energy than a black body and is characterized by an emissivity 0 lt e lt 1 displaystyle 0 lt varepsilon lt 1 j e s T 4 displaystyle j star varepsilon sigma T 4 The radiant emittance j displaystyle j star has dimensions of energy flux energy per unit time per unit area and the SI units of measure are joules per second per square metre or equivalently watts per square metre The SI unit for absolute temperature T is the kelvin e displaystyle varepsilon is the emissivity of the grey body if it is a perfect blackbody e 1 displaystyle varepsilon 1 In the still more general and realistic case the emissivity depends on the wavelength e e l displaystyle varepsilon varepsilon lambda To find the total power radiated from an object multiply by its surface area A displaystyle A P A j A e s T 4 displaystyle P Aj star A varepsilon sigma T 4 Wavelength and subwavelength scale particles 1 metamaterials 2 and other nanostructures are not subject to ray optical limits and may be designed to exceed the Stefan Boltzmann law Contents 1 History 2 Examples 2 1 Temperature of the Sun 2 2 Temperature of stars 2 3 Effective temperature of the Earth 3 Origination 3 1 Thermodynamic derivation of the energy density 3 2 Derivation from Planck s law 3 3 Energy density 4 See also 5 Notes 6 ReferencesHistory EditIn 1864 John Tyndall presented measurements of the infrared emission by a platinum filament and the corresponding color of the filament 3 The proportionality to the fourth power of the absolute temperature was deduced by Josef Stefan 1835 1893 in 1877 on the basis of Tyndall s experimental measurements in the article Uber die Beziehung zwischen der Warmestrahlung und der Temperatur On the relationship between thermal radiation and temperature in the Bulletins from the sessions of the Vienna Academy of Sciences 4 5 A derivation of the law from theoretical considerations was presented by Ludwig Boltzmann 1844 1906 in 1884 drawing upon the work of Adolfo Bartoli 6 Bartoli in 1876 had derived the existence of radiation pressure from the principles of thermodynamics Following Bartoli Boltzmann considered an ideal heat engine using electromagnetic radiation instead of an ideal gas as working matter The law was almost immediately experimentally verified Heinrich Weber in 1888 pointed out deviations at higher temperatures but perfect accuracy within measurement uncertainties was confirmed up to temperatures of 1535 K by 1897 7 The law including the theoretical prediction of the Stefan Boltzmann constant as a function of the speed of light the Boltzmann constant and the Planck constant is a direct consequence of Planck s law as formulated in 1900 As of the 2019 redefinition of SI base units which fixes the values of the Boltzmann constant k the Planck constant h and the speed of light c the Stefan Boltzmann constant is exactly s 2 p 5 k 4 15 c 2 h 3 2 p 5 1 380 649 10 23 4 15 2 997 924 58 10 8 2 6 626 070 15 10 34 3 W m 2 K 4 displaystyle sigma frac 2 pi 5 k 4 15c 2 h 3 frac 2 pi 5 left 1 380 649 times 10 23 right 4 15 left 2 997 924 58 times 10 8 right 2 left 6 626 070 15 times 10 34 right 3 mathrm frac W m 2 cdot K 4 s 5 670374 419 10 8 W m2K4 8 Prior to this the value of s displaystyle sigma was calculated from the measured value of the gas constant 9 The numerical value of the Stefan Boltzmann constant is different in other systems of units For example in thermochemistry it is often expressed in cal cm 2 day 1 K 4 s 1 170937 10 7 cal cm 2 day 1 K 4 Likewise in US customary units the Stefan Boltzmann constant is 10 s 1 713441 10 9 BTU hr 1 ft 2 R 4 Examples EditTemperature of the Sun Edit Log log graphs of peak emission wavelength and radiant exitance vs black body temperature red arrows show that 5780 K black bodies have 501 nm peak and 63 3 MW m2 radiant exitance With his law Stefan also determined the temperature of the Sun s surface 11 He inferred from the data of Jacques Louis Soret 1827 1890 12 that the energy flux density from the Sun is 29 times greater than the energy flux density of a certain warmed metal lamella a thin plate A round lamella was placed at such a distance from the measuring device that it would be seen at the same angular diameter as the Sun Soret estimated the temperature of the lamella to be approximately 1900 C to 2000 C Stefan surmised that of the energy flux from the Sun is absorbed by the Earth s atmosphere so he took for the correct Sun s energy flux a value 3 2 times greater than Soret s value namely 29 3 2 43 5 Precise measurements of atmospheric absorption were not made until 1888 and 1904 The temperature Stefan obtained was a median value of previous ones 1950 C and the absolute thermodynamic one 2200 K As 2 574 43 5 it follows from the law that the temperature of the Sun is 2 57 times greater than the temperature of the lamella so Stefan got a value of 5430 C or 5700 K This was the first sensible value for the temperature of the Sun Before this values ranging from as low as 1800 C to as high as 13 000 000 C 13 were claimed The lower value of 1800 C was determined by Claude Pouillet 1790 1868 in 1838 using the Dulong Petit law 14 Pouillet also took just half the value of the Sun s correct energy flux Temperature of stars Edit The temperature of stars other than the Sun can be approximated using a similar means by treating the emitted energy as a black body radiation 15 So L 4 p R 2 s T 4 displaystyle L 4 pi R 2 sigma T 4 where L is the luminosity s is the Stefan Boltzmann constant R is the stellar radius and T is the effective temperature This formula can then be rearranged to calculate the temperature T L 4 p R 2 s 4 displaystyle T sqrt 4 frac L 4 pi R 2 sigma or alternatively the radius R L 4 p s T 4 displaystyle R sqrt frac L 4 pi sigma T 4 The same formulae can also be simplified to compute the parameters relative to the Sun L L R R 2 T T 4 displaystyle frac L L odot left frac R R odot right 2 left frac T T odot right 4 T T L L 1 4 R R displaystyle frac T T odot left frac L L odot right 1 4 sqrt frac R odot R R R T T 2 L L displaystyle frac R R odot left frac T odot T right 2 sqrt frac L L odot where R displaystyle R odot is the solar radius and so forth They can also be rewritten in terms of the surface area A and radiant emittance j displaystyle j star L A j displaystyle L Aj star j L A displaystyle j star frac L A A L j displaystyle A frac L j star where A 4 p R 2 displaystyle A 4 pi R 2 and j s T 4 displaystyle j star sigma T 4 With the Stefan Boltzmann law astronomers can easily infer the radii of stars The law is also met in the thermodynamics of black holes in so called Hawking radiation Effective temperature of the Earth Edit Similarly we can calculate the effective temperature of the Earth T by equating the energy received from the Sun and the energy radiated by the Earth under the black body approximation Earth s own production of energy being small enough to be negligible The luminosity of the Sun L is given by L 4 p R 2 s T 4 displaystyle L odot 4 pi R odot 2 sigma T odot 4 At Earth this energy is passing through a sphere with a radius of a0 the distance between the Earth and the Sun and the irradiance received power per unit area is given by E L 4 p a 0 2 displaystyle E oplus frac L odot 4 pi a 0 2 The Earth has a radius of R and therefore has a cross section of p R 2 displaystyle pi R oplus 2 The radiant flux i e solar power absorbed by the Earth is thus given by F abs p R 2 E displaystyle Phi text abs pi R oplus 2 times E oplus Because the Stefan Boltzmann law uses a fourth power it has a stabilizing effect on the exchange and the flux emitted by Earth tends to be equal to the flux absorbed close to the steady state where 4 p R 2 s T 4 p R 2 E p R 2 4 p R 2 s T 4 4 p a 0 2 displaystyle begin aligned 4 pi R oplus 2 sigma T oplus 4 amp pi R oplus 2 times E oplus amp pi R oplus 2 times frac 4 pi R odot 2 sigma T odot 4 4 pi a 0 2 end aligned T can then be found T 4 R 2 T 4 4 a 0 2 T T R 2 a 0 5780 K 6 957 10 8 m 2 1 495 978 707 10 11 m 279 K displaystyle begin aligned T oplus 4 amp frac R odot 2 T odot 4 4a 0 2 T oplus amp T odot times sqrt frac R odot 2a 0 amp 5780 rm K times sqrt 6 957 times 10 8 rm m over 2 times 1 495 978 707 times 10 11 rm m amp approx 279 rm K end aligned where T is the temperature of the Sun R the radius of the Sun and a0 is the distance between the Earth and the Sun This gives an effective temperature of 6 C on the surface of the Earth assuming that it perfectly absorbs all emission falling on it and has no atmosphere The Earth has an albedo of 0 3 meaning that 30 of the solar radiation that hits the planet gets scattered back into space without absorption The effect of albedo on temperature can be approximated by assuming that the energy absorbed is multiplied by 0 7 but that the planet still radiates as a black body the latter by definition of effective temperature which is what we are calculating This approximation reduces the temperature by a factor of 0 71 4 giving 255 K 18 C 16 17 The above temperature is Earth s as seen from space not ground temperature but an average over all emitting bodies of Earth from surface to high altitude Because of the greenhouse effect the Earth s actual average surface temperature is about 288 K 15 C which is higher than the 255 K effective temperature and even higher than the 279 K temperature that a black body would have In the above discussion we have assumed that the whole surface of the earth is at one temperature Another interesting question is to ask what the temperature of a blackbody surface on the earth would be assuming that it reaches equilibrium with the sunlight falling on it This of course depends on the angle of the sun on the surface and on how much air the sunlight has gone through When the sun is at the zenith and the surface is horizontal the irradiance can be as high as 1120 W m2 18 The Stefan Boltzmann law then gives a temperature of T 1120 W m 2 s 1 4 375 K displaystyle T left frac 1120 text W m 2 sigma right 1 4 approx 375 text K or 102 C Above the atmosphere the result is even higher 394 K We can think of the earth s surface as trying to reach equilibrium temperature during the day but being cooled by the atmosphere and trying to reach equilibrium with starlight and possibly moonlight at night but being warmed by the atmosphere Origination EditThermodynamic derivation of the energy density Edit The fact that the energy density of the box containing radiation is proportional to T 4 displaystyle T 4 can be derived using thermodynamics 19 20 This derivation uses the relation between the radiation pressure p and the internal energy density u displaystyle u a relation that can be shown using the form of the electromagnetic stress energy tensor This relation is p u 3 displaystyle p frac u 3 Now from the fundamental thermodynamic relation d U T d S p d V displaystyle dU T dS p dV we obtain the following expression after dividing by d V displaystyle dV and fixing T displaystyle T U V T T S V T p T p T V p displaystyle left frac partial U partial V right T T left frac partial S partial V right T p T left frac partial p partial T right V p The last equality comes from the following Maxwell relation S V T p T V displaystyle left frac partial S partial V right T left frac partial p partial T right V From the definition of energy density it follows that U u V displaystyle U uV where the energy density of radiation only depends on the temperature therefore U V T u V V T u displaystyle left frac partial U partial V right T u left frac partial V partial V right T u Now the equality U V T T p T V p displaystyle left frac partial U partial V right T T left frac partial p partial T right V p after substitution of U V T displaystyle left frac partial U partial V right T Meanwhile the pressure is the rate of momentum change per unit area Since the momentum of a photon is the same as the energy divided by the speed of light u T 3 u T V u 3 displaystyle u frac T 3 left frac partial u partial T right V frac u 3 where the 1 3 factor comes from the projection of the momentum transfer onto the normal to the wall of the container Since the partial derivative u T V displaystyle left frac partial u partial T right V can be expressed as a relationship between only u displaystyle u and T displaystyle T if one isolates it on one side of the equality the partial derivative can be replaced by the ordinary derivative After separating the differentials the equality becomes d u 4 u d T T displaystyle frac du 4u frac dT T which leads immediately to u A T 4 displaystyle u AT 4 with A displaystyle A as some constant of integration Derivation from Planck s law Edit Deriving the Stefan Boltzmann Law using the Planck s law The law can be derived by considering a small flat black body surface radiating out into a half sphere This derivation uses spherical coordinates with 8 as the zenith angle and f as the azimuthal angle and the small flat blackbody surface lies on the xy plane where 8 p 2 The intensity of the light emitted from the blackbody surface is given by Planck s law I n T 2 h n 3 c 2 1 e h n k T 1 displaystyle I nu T frac 2h nu 3 c 2 frac 1 e h nu kT 1 where I n T displaystyle I nu T is the amount of power per unit surface area per unit solid angle per unit frequency emitted at a frequency n displaystyle nu by a black body at temperature T h displaystyle h is the Planck constant c displaystyle c is the speed of light and k displaystyle k is the Boltzmann constant The quantity I n T A cos 8 d n d W displaystyle I nu T A cos theta d nu d Omega is the power radiated by a surface of area A through a solid angle dW in the frequency range between n and n dn The Stefan Boltzmann law gives the power emitted per unit area of the emitting body P A 0 I n T d n cos 8 d W displaystyle frac P A int 0 infty I nu T d nu int cos theta d Omega Note that the cosine appears because black bodies are Lambertian i e they obey Lambert s cosine law meaning that the intensity observed along the sphere will be the actual intensity times the cosine of the zenith angle To derive the Stefan Boltzmann law we must integrate d W sin 8 d 8 d f textstyle d Omega sin theta d theta d varphi over the half sphere and integrate n displaystyle nu from 0 to P A 0 I n T d n 0 2 p d f 0 p 2 cos 8 sin 8 d 8 p 0 I n T d n displaystyle begin aligned frac P A amp int 0 infty I nu T d nu int 0 2 pi d varphi int 0 pi 2 cos theta sin theta d theta amp pi int 0 infty I nu T d nu end aligned Then we plug in for I P A 2 p h c 2 0 n 3 e h n k T 1 d n displaystyle frac P A frac 2 pi h c 2 int 0 infty frac nu 3 e frac h nu kT 1 d nu To evaluate this integral do a substitution u h n k T d u h k T d n displaystyle begin aligned u amp frac h nu kT 6pt du amp frac h kT d nu end aligned which gives P A 2 p h c 2 k T h 4 0 u 3 e u 1 d u displaystyle frac P A frac 2 pi h c 2 left frac kT h right 4 int 0 infty frac u 3 e u 1 du The integral on the right is standard and goes by many names it is a particular case of a Bose Einstein integral the polylogarithm or the Riemann zeta function z s displaystyle zeta s The value of the integral is G 4 z 4 p 4 15 displaystyle Gamma 4 zeta 4 frac pi 4 15 where G s displaystyle Gamma s is the Gamma function giving the result that for a perfect blackbody surface j s T 4 s 2 p 5 k 4 15 c 2 h 3 p 2 k 4 60 ℏ 3 c 2 displaystyle j star sigma T 4 sigma frac 2 pi 5 k 4 15c 2 h 3 frac pi 2 k 4 60 hbar 3 c 2 Finally this proof started out only considering a small flat surface However any differentiable surface can be approximated by a collection of small flat surfaces So long as the geometry of the surface does not cause the blackbody to reabsorb its own radiation the total energy radiated is just the sum of the energies radiated by each surface and the total surface area is just the sum of the areas of each surface so this law holds for all convex blackbodies too so long as the surface has the same temperature throughout The law extends to radiation from non convex bodies by using the fact that the convex hull of a black body radiates as though it were itself a black body Energy density Edit The total energy density U can be similarly calculated except the integration is over the whole sphere and there is no cosine and the energy flux U c should be divided by the velocity c to give the energy density U U 1 c 0 I n T d n d W displaystyle U frac 1 c int 0 infty I nu T d nu int d Omega Thus 0 p 2 cos 8 sin 8 d 8 textstyle int 0 pi 2 cos theta sin theta d theta is replaced by 0 p sin 8 d 8 textstyle int 0 pi sin theta d theta giving an extra factor of 4 Thus in total U 4 c s T 4 displaystyle U frac 4 c sigma T 4 The product 4 c s displaystyle frac 4 c sigma is sometimes known as the radiation constant or radiation density constant 21 22 See also EditBlack body radiation Rayleigh Jeans law Planck s law Sakuma Hattori equation Rado von KovesligethyNotes Edit Bohren Craig F Huffman Donald R 1998 Absorption and scattering of light by small particles Wiley pp 123 126 ISBN 978 0 471 29340 8 Narimanov Evgenii E Smolyaninov Igor I 2012 Beyond Stefan Boltzmann Law Thermal Hyper Conductivity Conference on Lasers and Electro Optics 2012 OSA Technical Digest Optical Society of America pp QM2E 1 CiteSeerX 10 1 1 764 846 doi 10 1364 QELS 2012 QM2E 1 ISBN 978 1 55752 943 5 S2CID 36550833 Tyndall John 1864 On luminous i e visible and obscure i e infrared radiation Philosophical Magazine 4th series 28 329 341 see p 333 In his physics textbook of 1875 Adolph Wullner quoted Tyndall s results and then added estimates of the temperature that corresponded to the platinum filament s color Wullner Adolph 1875 Lehrbuch der Experimentalphysik Textbook of experimental physics in German Vol 3 Leipzig Germany B G Teubner p 215 From Wullner 1875 p 215 Wie aus gleich zu besprechenden Versuchen von Draper hervorgeht also fast um das 12fache zu As follows from the experiments of Draper which will be discussed shortly a temperature of about 525 C corresponds to the weak red glow a temperature of about 1200 C to the full white glow Thus while the temperature climbed only somewhat more than double the intensity of the radiation increased from 10 4 to 122 thus almost 12 fold See also Wisniak Jaime November 2002 Heat radiation law from Newton to Stefan Indian Journal of Chemical Technology 9 545 555 see pp 551 552 Available at National Institute of Science Communication and Information Resources New Delhi India Stefan J 1879 Uber die Beziehung zwischen der Warmestrahlung und der Temperatur On the relation between heat radiation and temperature Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften Mathematisch Naturwissenschaftliche Classe Proceedings of the Imperial Philosophical Academy of Vienna Mathematical and Scientific Class in German 79 391 428 Stefan stated Stefan 1879 p 421 Zuerst will ich hier die Bemerkung anfuhren die Warmestrahlung der vierten Potenz der absoluten Temperatur proportional anzunehmen First of all I want to point out here the observation which Wullner in his textbook added to the report of Tyndall s experiments on the radiation of a platinum wire that was brought to glowing by an electric current because this observation first caused me to suppose that thermal radiation is proportional to the fourth power of the absolute temperature Boltzmann Ludwig 1884 Ableitung des Stefan schen Gesetzes betreffend die Abhangigkeit der Warmestrahlung von der Temperatur aus der electromagnetischen Lichttheorie Derivation of Stefan s law concerning the dependency of heat radiation on temperature from the electromagnetic theory of light Annalen der Physik und Chemie in German 258 6 291 294 Bibcode 1884AnP 258 291B doi 10 1002 andp 18842580616 Massimiliano Badino The Bumpy Road Max Planck from Radiation Theory to the Quantum 1896 1906 2015 p 31 Sloane N J A ed Sequence A081820 The On Line Encyclopedia of Integer Sequences OEIS Foundation Moldover M R Trusler J P M Edwards T J Mehl J B Davis R S 1988 01 25 Measurement of the Universal Gas Constant R Using a Spherical Acoustic Resonator Physical Review Letters 60 249 doi 10 1103 PhysRevLett 60 249 Cengel Yunus A 2007 Heat and Mass Transfer a Practical Approach 3rd ed McGraw Hill Stefan 1879 pp 426 427 Soret J L 1872 Comparaison des intensites calorifiques du rayonnement solaire et du rayonnement d un corps chauffe a la lampe oxyhydrique Comparison of the heat intensities of solar radiation and of radiation from a body heated with an oxy hydrogen torch Archives des sciences physiques et naturelles Geneva Switzerland 2nd series 44 220 229 45 252 256 Waterston John James 1862 An account of observations on solar radiation Philosophical Magazine 4th series 23 2 497 511 Bibcode 1861MNRAS 22 60W doi 10 1093 mnras 22 2 60 On p 505 the Scottish physicist John James Waterston estimated that the temperature of the sun s surface could be 12 880 000 See Pouillet 1838 Memoire sur la chaleur solaire sur les pouvoirs rayonnants et absorbants de l air atmospherique et sur la temperature de l espace Memoir on solar heat on the radiating and absorbing powers of the atmospheric air and on the temperature of space Comptes Rendus in French 7 2 24 65 On p 36 Pouillet estimates the sun s temperature cette temperature pourrait etre de 1761 this temperature i e of the Sun could be 1761 English translation Pouillet 1838 Memoir on the solar heat on the radiating and absorbing powers of atmospheric air and on the temperature of space in Taylor Richard ed 1846 Scientific Memoirs Selected from the Transactions of Foreign Academies of Science and Learned Societies and from Foreign Journals vol 4 London England Richard and John E Taylor pp 44 90 see pp 55 56 Luminosity of Stars Australian Telescope Outreach and Education Retrieved 2006 08 13 Intergovernmental Panel on Climate Change Fourth Assessment Report Chapter 1 Historical overview of climate change science Archived 2018 11 26 at the Wayback Machine page 97 Solar Radiation and the Earth s Energy Balance Introduction to Solar Radiation Newport Corporation Archived from the original on October 29 2013 Knizhnik Kalman Derivation of the Stefan Boltzmann Law PDF Johns Hopkins University Department of Physics amp Astronomy Archived from the original PDF on 2016 03 04 Retrieved 2018 09 03 Wisniak 2002 p 554 Lemons Don S Shanahan William R Buchholtz Louis J 2022 09 13 On the Trail of Blackbody Radiation Max Planck and the Physics of his Era MIT Press p 38 ISBN 978 0 262 37038 7 Campana S Mangano V Blustin A J Brown P Burrows D N Chincarini G Cummings J R Cusumano G Valle M Della Malesani D Meszaros P Nousek J A Page M Sakamoto T Waxman E August 2006 The association of GRB 060218 with a supernova and the evolution of the shock wave Nature 442 7106 1008 1010 doi 10 1038 nature04892 ISSN 0028 0836 References EditStefan J 1879 Uber die Beziehung zwischen der Warmestrahlung und der Temperatur On the relationship between heat radiation and temperature PDF Sitzungsberichte der Mathematisch naturwissenschaftlichen Classe der Kaiserlichen Akademie der Wissenschaften in German 79 391 428 Boltzmann L 1884 Ableitung des Stefan schen Gesetzes betreffend die Abhangigkeit der Warmestrahlung von der Temperatur aus der electromagnetischen Lichttheorie Derivation of Stefan s little law concerning the dependence of thermal radiation on the temperature of the electro magnetic theory of light Annalen der Physik und Chemie in German 258 6 291 294 Bibcode 1884AnP 258 291B doi 10 1002 andp 18842580616 Retrieved from https en wikipedia org w index php title Stefan Boltzmann law amp oldid 1147761437, wikipedia, wiki, book, books, library,

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