Editing Stefan–Boltzmann law

{{Short description|Physical law on the emissive power of black body}}
{{redirect-distinguish|Stefan's law|Stefan's equation|Stefan's formula}}
[[File:Stefan Boltzmann 001.svg|thumb|upright=1.4|Total emitted energy, <math>j \equiv M^{\circ}</math>, of a black body as a function of its temperature, <math>T</math>. The upper (black) curve depicts the Stefan–Boltzmann law, <math>M^{\circ} = \sigma\,T^4</math>. The lower (blue) curve is total energy according to the [[Wien approximation]], <math> M^{\circ}_{W} = M^{\circ} / \zeta(4) \approx 0.924 \, \sigma T^{4} \!\, </math>]]

The '''Stefan–Boltzmann law''', also known as ''Stefan's law'', describes the intensity of the [[thermal radiation]] emitted by matter in terms of that matter's [[temperature]].  It is named for [[Josef Stefan]], who empirically derived the relationship, and [[Ludwig Boltzmann]] who derived the law theoretically.

For an ideal absorber/emitter or [[black body]], the Stefan–Boltzmann law states that the total [[energy]] radiated per unit [[area|surface area]] per unit [[time]] (also known as the ''[[radiant exitance]]'') is directly [[Proportionality (mathematics)|proportional]] to the fourth power of the black body's temperature, {{math|''T''}}:
<math display="block"> M^{\circ} = \sigma\, T^{4}.</math>

The [[constant of proportionality]], <math>\sigma</math>, is called the '''Stefan–Boltzmann constant'''. It has the value
{{block indent | em = 1.5 | text = {{physconst|sigma|symbol=yes|after=.}}}}

In the general case, the Stefan–Boltzmann law for radiant exitance takes the form:
<math display="block"> M = \varepsilon\, M^{\circ} = \varepsilon\,\sigma\, T^4 ,</math>
where <math>\varepsilon</math> is the [[emissivity]] of the surface emitting the radiation.  The emissivity is generally between zero and one.  An emissivity of one corresponds to a black body.

== Detailed explanation ==
The ''[[radiant exitance]]'' (previously called ''radiant emittance''), <math>M</math>, has [[Dimensional analysis#Formulation|dimensions]] of [[energy flux]] (energy per unit time per unit area), and the [[SI units]] of measure are [[joule]]s per second per square metre (J⋅s{{sup|−1}}⋅m{{sup|−2}}), or equivalently, [[watt]]s per square metre (W⋅m{{sup|−2}}).<ref name="ISO_9288:2022"/> The SI unit for [[temperature|absolute temperature]], {{math|''T''}}, is the [[kelvin]] (K).

To find the total [[Power (physics)|power]], <math>P</math>, radiated from an object, multiply the radiant exitance by the object's surface area, <math>A</math>:
<math display="block"> P = A \cdot M = A \, \varepsilon\,\sigma\, T^{4}.</math>

Matter that does not absorb all incident radiation emits less total energy than a black body. Emissions are reduced by a factor <math>\varepsilon</math>, where the [[emissivity]], <math>\varepsilon</math>, is a material property which, for most matter, satisfies <math>0 \leq \varepsilon \leq 1</math>. Emissivity can in general depend on [[wavelength]], direction, and [[Polarization (physics)|polarization]]. However, the emissivity which appears in the non-directional form of the Stefan–Boltzmann law is the [[emissivity|hemispherical total emissivity]], which reflects emissions as totaled over all wavelengths, directions, and polarizations.<ref name="SH92"/>{{rp|p=60}}

The form of the Stefan–Boltzmann law that includes emissivity is applicable to all matter, provided that matter is in a state of [[Radiative transfer#Local thermodynamic equilibrium|local thermodynamic equilibrium (LTE)]] so that its temperature is well-defined.<ref name="SH92"/>{{rp|pp=66n,541}} (This is a trivial conclusion, since the emissivity, <math>\varepsilon</math>, is defined to be the quantity that makes this equation valid. What is non-trivial is the proposition that <math>\varepsilon \leq 1</math>, which is a consequence of [[Kirchhoff's law of thermal radiation]].<ref name="Reif">{{cite book |last1=Reif |first1=F. |title=Fundamentals of Statistical and Thermal Physics |date=1965 |publisher=Waveland Press |isbn=978-1-57766-612-7}}</ref>{{rp|p=385}})

A so-called ''grey body'' is a body for which the [[emissivity|spectral emissivity]] is independent of wavelength, so that the total emissivity, <math>\varepsilon</math>, is a constant.<ref name="SH92">{{cite book |last1=Siegel |first1=Robert |last2=Howell |first2=John R. |title=Thermal Radiation Heat Transfer |date=1992 |publisher=Taylor & Francis |isbn=0-89116-271-2 |edition=3}}</ref>{{rp|p=71}} In the more general (and realistic) case, the spectral emissivity depends on wavelength. The total emissivity, as applicable to the Stefan–Boltzmann law, may be calculated as a [[weighted average]] of the spectral emissivity, with the [[Planck's law|blackbody emission spectrum]] serving as the [[weighting function]]. It follows that if the spectral emissivity depends on wavelength then the total emissivity depends on the temperature, i.e., <math>\varepsilon = \varepsilon(T)</math>.<ref name="SH92"/>{{rp|p=60}} However, if the dependence on wavelength is small, then the dependence on temperature will be small as well.

Wavelength- and subwavelength-scale particles,<ref name="Bohren">
{{cite book
 |last1=Bohren |first1=Craig F.
 |last2=Huffman |first2=Donald R.
 |year=1998
 |title=Absorption and scattering of light by small particles
 |url=https://books.google.com/books?id=ib3EMXXIRXUC
 |publisher=Wiley
 |isbn=978-0-471-29340-8
 |pages=123–126
}}</ref> [[metamaterial]]s,<ref>
{{Cite conference
 |last1=Narimanov |first1=Evgenii E.
 |last2=Smolyaninov |first2=Igor I.
 |year=2012
 |chapter=Beyond Stefan–Boltzmann Law: Thermal Hyper-Conductivity
 |arxiv=1109.5444
 |title=Conference on Lasers and Electro-Optics 2012
 |series=OSA Technical Digest
 |publisher=Optical Society of America
 |pages=QM2E.1
 |doi=10.1364/QELS.2012.QM2E.1
 |isbn=978-1-55752-943-5
 |citeseerx=10.1.1.764.846
 |s2cid=36550833
}}</ref> and other nanostructures<ref name="Golyk2012">{{cite journal |last1=Golyk |first1=V. A. |last2=Krüger |first2=M. |last3=Kardar |first3=M. |title=Heat radiation from long cylindrical objects |journal=Phys. Rev. E |date=2012 |volume=85 |issue=4 |page=046603 |doi=10.1103/PhysRevE.85.046603 |pmid=22680594 |arxiv=1109.1769 |bibcode=2012PhRvE..85d6603G |s2cid=27489038 |url=https://link.aps.org/accepted/10.1103/PhysRevE.85.046603|hdl=1721.1/71630 |hdl-access=free }}</ref> are not subject to ray-optical limits and may be designed to have an emissivity greater than 1.

In national and [[international standard]]s documents, the symbol <math>M</math> is recommended to denote ''radiant exitance''; a superscript circle (°) indicates a term relate to a black body.<ref name="ISO_9288:2022">{{cite web | url = https://www.iso.org/standard/82088.html | title=Thermal insulation — Heat transfer by radiation — Vocabulary | work=ISO_9288:2022 |publisher=[[International Organization for Standardization]] | year=2022 | accessdate=2023-06-17}}</ref> (A subscript "e" is added when it is important to distinguish the energetic ([[radiometric]]) quantity ''radiant exitance'', <math>M_\mathrm{e}</math>, from the analogous human vision ([[photometry (optics)|photometric]]) quantity, ''[[luminous exitance]]'', denoted <math>M_\mathrm{v}</math>.<ref name="IECrex">{{cite web |title=radiant exitance |url=https://www.electropedia.org/iev/iev.nsf/display?openform&ievref=845-21-080 |website=Electropedia: The World's Online Electrotechnical Vocabulary |publisher=International Electrotechnical Commission |access-date=20 June 2023}}</ref>) In common usage, the symbol used for radiant exitance (often called ''radiant emittance'') varies among different texts and in different fields.

The ''Stefan–Boltzmann law'' may be expressed as a formula for ''[[radiance]]'' as a function of temperature. Radiance is measured in watts per square metre per [[steradian]] (W⋅m{{sup|−2}}⋅sr{{sup|−1}}). The Stefan–Boltzmann law for the radiance of a black body is:<ref name="Goody89">{{cite book |last1=Goody |first1=R. M. |last2=Yung |first2=Y. L. |title=Atmospheric Radiation: Theoretical Basis |date=1989 |publisher=Oxford University Press |isbn=0-19-505134-3}}</ref>{{rp|p=26}}<ref name="GraingerCh3">{{cite web |last1=Grainger |first1=R. G. |title=A Primer on Atmospheric Radiative Transfer: Chapter 3. Radiometric Basics |url=http://eodg.atm.ox.ac.uk/user/grainger/research/book/protected/Chapter3.pdf |publisher=Earth Observation Data Group, Department of Physics, University of Oxford |access-date=15 June 2023 |date=2020}}</ref>
<math display="block"> L^\circ_\Omega = \frac{M^{\circ}}\pi = \frac\sigma\pi\, T^{4}.</math>

The ''Stefan–Boltzmann law'' expressed as a formula for ''[[Radiant energy density|radiation energy density]]'' is:<ref name="hyperphysics">{{cite web |title=Radiation Energy Density |url=http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/raddens.html |website=HyperPhysics |access-date=20 June 2023}}</ref>
<math display="block"> w^\circ_\mathrm{e} = \frac{4}{c} \, M^\circ = \frac{4}{c} \, \sigma\, T^{4} ,</math>
where <math>c</math> is the speed of light.

== History ==
In 1864, [[John Tyndall]] presented measurements of the infrared emission by a platinum filament and the corresponding color of the filament.<ref>{{cite journal |last1=Tyndall|first1=John|title=On luminous [i.e., visible] and obscure [i.e., infrared] radiation|journal=Philosophical Magazine |date=1864 |volume=28 |pages=329–341 | url=https://babel.hathitrust.org/cgi/pt?id=umn.31951000614117o;view=1up;seq=357 |series=4th series}} ; see p. 333.</ref><ref>In his physics textbook of 1875, [[Adolf Wüllner|Adolph Wüllner]] quoted Tyndall's results and then added estimates of the temperature that corresponded to the platinum filament's color:
{{cite book| last1=Wüllner|first1=Adolph |title=Lehrbuch der Experimentalphysik |trans-title=Textbook of experimental physics | volume=3|date=1875|publisher=B.G. Teubner | location=Leipzig, Germany| page=215| url=https://babel.hathitrust.org/cgi/pt?id=uc1.b4062759;view=1up;seq=231 | language=de}}</ref><ref>From {{harvnb|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.)</ref><ref name=wisniak2002>{{cite journal| last1=Wisniak|first1=Jaime| title=Heat radiation law – from Newton to Stefan| journal=Indian Journal of Chemical Technology|date=November 2002| volume=9| url=http://nopr.niscair.res.in/bitstream/123456789/18926/1/IJCT%209%286%29%20545-555.pdf | pages=551–552 |access-date=2023-06-15 }}</ref>
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.<ref>Stefan stated {{harv|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.)</ref>

A derivation of the law  from theoretical considerations was presented by [[Ludwig Boltzmann]] (1844–1906) in 1884, drawing upon the work of [[Adolfo Bartoli]].<ref>{{cite journal| last1=Boltzmann| first1=Ludwig| title=Ableitung des Stefan'schen Gesetzes, betreffend die Abhängigkeit der Wärmestrahlung von der Temperatur aus der electromagnetischen Lichttheorie| journal=Annalen der Physik und Chemie| date=1884| volume=258| issue=6| pages=291–294| doi=10.1002/andp.18842580616| url=https://babel.hathitrust.org/cgi/pt?id=uc1.a0002763670;view=1up;seq=327| trans-title=Derivation of Stefan's law, concerning the dependency of heat radiation on temperature, from the electromagnetic theory of light| language=de| bibcode=1884AnP...258..291B| doi-access=free}}</ref>
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 Friedrich Weber|Heinrich Weber]] in 1888 pointed out deviations at higher temperatures, but perfect accuracy within measurement uncertainties was confirmed up to temperatures of 1535&nbsp;K by 1897.<ref>{{cite book | last=Badino | first=M. | title=The Bumpy Road: Max Planck from Radiation Theory to the Quantum (1896–1906) | publisher=Springer International Publishing | series=SpringerBriefs in History of Science and Technology | year=2015 | isbn=978-3-319-20031-6 | url=https://books.google.com/books?id=JcvWCQAAQBAJ | access-date=2023-06-15 | page=31}}</ref>
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 [[#Derivation from Planck's law|direct consequence]] of [[Planck's law]] as formulated in 1900.

== Stefan–Boltzmann constant ==

The Stefan–Boltzmann constant, {{math|''σ''}}, is derived from other known [[physical constant]]s:
<math display="block">\sigma = \frac{2 \pi^5 k^4}{15 c^2 h^3} </math>
where {{math|''k''}} is the [[Boltzmann constant]], the {{math|''h''}} is the [[Planck constant]], and {{math|''c''}} is the [[speed of light|speed of light in vacuum]].<ref name="TECS">{{cite web |title=Thermodynamic derivation of the Stefan–Boltzmann Law |url=https://www.tec-science.com/thermodynamics/temperature/thermodynamic-derivation-of-the-stefan-boltzmann-law/ |website=TECS |date=21 February 2020 |access-date=20 June 2023}}</ref><ref name="Reif"/>{{rp|p=388}}

As of the [[2019 revision of the SI]], which establishes exact fixed values for {{math|''k''}}, {{math|''h''}}, and {{math|''c''}}, the Stefan–Boltzmann constant is exactly:
<math display="block">
\sigma = \left[\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}\right]\,\frac{\mathrm{W}}{\mathrm{m}^2{\cdot}\mathrm{K}^4}
</math>
Thus,
{{ block indent | em = 1.5 | text = ''σ'' = {{val|5.670374419|end=...|e=-8|u=W⋅m<sup>−2</sup>⋅K<sup>−4</sup>}} {{OEIS|A081820}}.}}

Prior to this, the value of <math>\sigma</math> was calculated from the measured value of the [[gas constant]].<ref>{{cite journal|title=Measurement of the Universal Gas Constant ''R'' Using a Spherical Acoustic Resonator |first1=M. R. |last1=Moldover |first2=J. P. M. |last2=Trusler |first3=T. J. |last3=Edwards |first4=J. B. |last4=Mehl |first5=R. S. |last5=Davis | journal = Physical Review Letters |volume=60 |pages=249–252 | date=1988-01-25 |issue=4 | doi=10.1103/PhysRevLett.60.249 | pmid=10038493 |bibcode=1988PhRvL..60..249M |url=http://researchrepository.murdoch.edu.au/id/eprint/34689/ }}</ref>

The numerical value of the Stefan–Boltzmann constant is different in other systems of units, as shown in the table below.

{| class="wikitable"
|+ Stefan–Boltzmann constant, ''σ'' <ref>{{cite book |title=Heat and Mass Transfer: a Practical Approach |edition=3rd |first=Yunus A. |last=Çengel |publisher=McGraw Hill |year=2007}}</ref>
|-
! Context !! Value !! Units
|-
| [[SI]] || {{val|5.670374419|end=...|e=-8}} || W⋅m{{sup|−2}}⋅K{{sup|−4}}
|-
| [[CGS]] || {{val|5.670374419|end=...|e=-5}} || erg⋅cm{{sup|−2}}⋅s{{sup|−1}}⋅K{{sup|−4}}
|-
|[[United States customary units|US customary units]]|| {{val| 1.713441|end=...|e=-9}} || BTU⋅hr{{sup|−1}}⋅ft{{sup|−2}}⋅°R{{sup|−4}}
|-
|[[Thermochemistry]] ||{{val|1.170937|end=...|e=-7}}|| [[calorie|cal]]⋅[[centimeter|cm]]<sup>−2</sup>⋅[[day]]<sup>−1</sup>⋅[[Kelvin|K]]<sup>−4</sup>
|}

== Examples ==
=== Temperature of the Sun ===
[[File:Blackbody peak wavelength exitance vs temperature.svg|thumb|Log–log graphs of [[Wien's displacement law|peak emission wavelength]] and [[radiant exitance]] vs. [[black-body]] temperature. Red arrows show that [[photosphere|5780 K]] black bodies have 501 nm peak and 63.3 MW/m<sup>2</sup> radiant exitance.]]
With his law, Stefan also determined the temperature of the [[Sun]]'s surface.<ref>{{harvnb|Stefan|1879|pp=426–427}}</ref> He inferred from the data of [[Jacques-Louis Soret]] (1827&ndash;1890)<ref>{{Cite book |last=Soret |first=J.L. |url=https://babel.hathitrust.org/cgi/pt?id=wu.89048214449;view=1up;seq=684 |title=Comparaison des intensités calorifiques du rayonnement solaire et du rayonnement d'un corps chauffé à la lampe oxyhydrique |date=1872 |publisher=Archives des sciences physiques et naturelles |series=2nd series |volume=44 |location=Geneva, Switzerland |pages=220–229 |language=fr |trans-title=Comparison of the heat intensities of solar radiation and of radiation from a body heated with an oxy-hydrogen torch}}</ref> that the energy flux density from the Sun is 29 times greater than the energy flux density of a certain warmed metal [[Lamella (materials)|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 [[Celsius|°C]] to 2000&nbsp;°C. Stefan surmised that 1/3 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 (electromagnetic radiation)|absorption]] were not made until 1888 and 1904. The temperature Stefan obtained was a median value of previous ones, 1950&nbsp;°C and the absolute thermodynamic one 2200&nbsp;K. As 2.57<sup>4</sup> = 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&nbsp;°C or 5700&nbsp;K. This was the first sensible value for the temperature of the Sun. Before this, values ranging from as low as 1800&nbsp;°C to as high as {{val|13,000,000|u=°C}}<ref>{{cite journal| last1=Waterston|first1=John James|title=An account of observations on solar radiation | journal=Philosophical Magazine |date=1862 |volume=23|pages=497–511|url=https://books.google.com/books?id=v1YEAAAAYAAJ&pg=PA497 |series=4th series |issue=2 |doi=10.1093/mnras/22.2.60 |bibcode=1861MNRAS..22...60W | doi-access=free}} On p. 505, the Scottish physicist [[John James Waterston]] estimated that the temperature of the sun's surface could be 12,880,000°.</ref> were claimed. The lower value of 1800&nbsp;°C was determined by [[Claude Pouillet]] (1790–1868) in 1838 using the [[Dulong–Petit law]].<ref>{{cite journal| last1=Pouillet |title=Mémoire sur la chaleur solaire, sur les pouvoirs rayonnants et absorbants de l'air atmosphérique, et sur la température de l'espace|journal=Comptes Rendus| date=1838| volume=7| issue=2| pages=24–65| url=https://www.biodiversitylibrary.org/item/81350#page/34/mode/1up | trans-title=Memoir on solar heat, on the radiating and absorbing powers of the atmospheric air, and on the temperature of space | language=fr}}  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° … )</ref><ref>English translation:  {{cite book | last1=Taylor | first1=R. | last2=Woolf | first2=H. | title=Scientific Memoirs, Selected from the Transactions of Foreign Academies of Science and Learned Societies, and from Foreign Journals | publisher=Johnson Reprint Corporation | issue=v. 4 | year=1966 | url=https://books.google.com/books?id=X5Y1AAAAIAAJ | access-date=2023-06-15 }}</ref> Pouillet also took just half the value of the Sun's correct energy flux.

=== Temperature of stars ===
The temperature of [[star]]s other than the Sun can be approximated using a similar means by treating the emitted energy as a [[black body]] radiation.<ref name="luminosity">{{cite web |url=http://outreach.atnf.csiro.au/education/senior/astrophysics/photometry_luminosity.html |title=Luminosity of Stars |publisher=Australian Telescope Outreach and Education |access-date=2006-08-13 }}</ref> So:
<math display="block">L = 4 \pi R^2 \sigma T^4</math>
where {{math|''L''}} is the [[luminosity]], {{math|''σ''}} is the Stefan–Boltzmann constant, {{math|''R''}} is the stellar radius and {{math|''T''}} is the [[effective temperature]]. This formula can then be rearranged to calculate the temperature:
<math display="block">T = \sqrt[4]{\frac{L}{4 \pi R^2 \sigma}}</math>
or alternatively the radius:
<math display="block">R = \sqrt{\frac{L}{4 \pi \sigma T^4}}</math>

The same formulae can also be simplified to compute the parameters relative to the Sun:
<math display="block">\begin{align}
\frac{L}{L_\odot} &= \left(\frac{R}{R_\odot}\right)^2 \left(\frac{T}{T_\odot}\right)^4 \\[1ex]
\frac{T}{T_\odot} &= \left(\frac{L}{L_\odot}\right)^{1/4} \left(\frac{R_\odot}{R}\right)^{1/2} \\[1ex]
\frac{R}{R_\odot} &= \left(\frac{T_\odot}{T}\right)^2 \left(\frac{L}{L_\odot}\right)^{1/2}
\end{align}</math>
where <math>R_\odot</math> is the [[solar radius]], and so forth. They can also be rewritten in terms of the surface area ''A'' and radiant exitance <math> M^{\circ}</math>:
<math display="block">\begin{align}
L &= A M^{\circ} \\[1ex]
M^{\circ} &= \frac{L}{A} \\[1ex]
A &= \frac{L}{M^{\circ}}
\end{align}</math>
where <math>A = 4 \pi R^2</math> and <math>M^{\circ} = \sigma T^{4}.</math>

With the Stefan–Boltzmann law, [[astronomer]]s can easily infer the radii of stars. The law is also met in the [[Black hole thermodynamics|thermodynamics]] of [[black hole]]s in so-called [[Hawking radiation]].

=== Effective temperature of the Earth ===
Similarly we can calculate the [[effective temperature]] of the Earth ''T''<sub>⊕</sub> 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''<sub>⊙</sub>, is given by:
<math display="block">
L_\odot = 4\pi R_\odot^2 \sigma T_\odot^4
</math>

At Earth, this energy is passing through a sphere with a radius of ''a''<sub>0</sub>, the distance between the Earth and the Sun, and the [[irradiance]] (received power per unit area) is given by
<math display="block">
E_\oplus = \frac{L_\odot}{4\pi a_0^2}
</math>

The Earth has a radius of ''R''<sub>⊕</sub>, and therefore has a cross-section of <math>\pi R_\oplus^2</math>.  The [[radiant flux]] (i.e. solar power) absorbed by the Earth is thus given by:
<math display="block">
\Phi_\text{abs} = \pi R_\oplus^2 \times E_\oplus
</math>

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:
<math display="block">
\begin{align}
4\pi R_\oplus^2 \sigma T_\oplus^4 &= \pi R_\oplus^2 \times E_\oplus \\
 &= \pi R_\oplus^2 \times \frac{4\pi R_\odot^2\sigma T_\odot^4}{4\pi a_0^2} \\
\end{align}
</math>

''T''<sub>⊕</sub> can then be found:
<math display="block">
\begin{align}
T_\oplus^4 &= \frac{R_\odot^2 T_\odot^4}{4 a_0^2} \\
T_\oplus &= T_\odot \times \sqrt\frac{R_\odot}{2 a_0} \\
& = 5780 \; {\rm K} \times \sqrt{6.957 \times 10^{8} \; {\rm m} \over 2 \times 1.495\ 978\ 707 \times 10^{11} \; {\rm m} } \\
& \approx 279 \; {\rm K}
\end{align}
</math>
where ''T''<sub>⊙</sub> is the temperature of the Sun, ''R''<sub>⊙</sub> the radius of the Sun, and ''a''<sub>0</sub> is the distance between the Earth and the Sun. This gives an effective temperature of 6&nbsp;°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 0.7<sup>1/4</sup>, giving {{convert|255|K}}.<ref name= "IPCC4_ch01">{{cite report |url=http://www.ipcc.ch/pdf/assessment-report/ar4/wg1/ar4-wg1-chapter1.pdf |title=Intergovernmental Panel on Climate Change Fourth Assessment Report. Chapter 1: Historical overview of climate change science |archive-url=https://web.archive.org/web/20181126204443/http://www.ipcc.ch/pdf/assessment-report/ar4/wg1/ar4-wg1-chapter1.pdf |archive-date=2018-11-26 |page=97 }}</ref><ref>{{cite web |url=http://eesc.columbia.edu/courses/ees/climate/lectures/radiation/ |title=Solar Radiation and the Earth's Energy Balance |access-date=2010-08-16 |archive-date=2012-07-17 |archive-url=https://archive.today/20120717025320/http://eesc.columbia.edu/courses/ees/climate/lectures/radiation/ |url-status=dead }}</ref>

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 {{convert|288|K}}, which is higher than the {{convert|255|K}} effective temperature, and even higher than the {{convert|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&nbsp;W/m<sup>2</sup>.<ref name="Solar constant at ground level">{{cite web|title=Introduction to Solar Radiation|url= http://www.newport.com/Introduction-to-Solar-Radiation/411919/1033/content.aspx|publisher= Newport Corporation|url-status= live|archive-url= https://web.archive.org/web/20131029234117/http://www.newport.com/Introduction-to-Solar-Radiation/411919/1033/content.aspx |archive-date=October 29, 2013}}</ref> The Stefan–Boltzmann law then gives a temperature of
<math display="block">T=\left(\frac{1120\text{ W/m}^2}\sigma\right)^{1/4}\approx 375\text{ K}</math>
or {{convert|102|C}}. (Above the atmosphere, the result is even higher: {{convert|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 <math>T^{4}</math> can be derived using thermodynamics.<ref>{{Cite web|url=http://www.pha.jhu.edu/~kknizhni/StatMech/Derivation_of_Stefan_Boltzmann_Law.pdf |title=Derivation of the Stefan–Boltzmann Law |last=Knizhnik |first=Kalman |website=Johns Hopkins University – Department of Physics & Astronomy |archive-url=https://web.archive.org/web/20160304133636/http://www.pha.jhu.edu/~kknizhni/StatMech/Derivation_of_Stefan_Boltzmann_Law.pdf |archive-date=2016-03-04 |url-status=dead|access-date=2018-09-03}}</ref><ref name=wisniak2002/> This derivation uses the relation between the [[radiation pressure]] ''p'' and the [[internal energy]] density <math>u</math>, a relation that [[Radiation pressure#Compression in a uniform radiation field|can be shown]] using the form of the [[electromagnetic stress–energy tensor]]. This relation is:
<math display="block"> p = \frac{u}{3}.</math>

Now, from the [[fundamental thermodynamic relation]]
<math display="block"> dU = T \, dS - p \, dV, </math>
we obtain the following expression, after dividing by <math> dV </math> and fixing <math> T </math>:
<math display="block"> \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. </math>

The last equality comes from the following [[Maxwell relations|Maxwell relation]]:
<math display="block"> \left(\frac{\partial S}{\partial V}\right)_T = \left(\frac{\partial p}{\partial T}\right)_V. </math>

From the definition of energy density it follows that
<math display="block"> U = u V </math>
where the energy density of radiation only depends on the temperature, therefore
<math display="block"> \left(\frac{\partial U}{\partial V}\right)_T = u \left(\frac{\partial V}{\partial V}\right)_T = u. </math>

Now, the equality is
<math display="block"> u = T \left(\frac{\partial p}{\partial T}\right)_V - p, </math> after substitution of <math> \left(\frac{\partial U}{\partial V}\right)_{T}.</math>

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,
<math display="block"> u = \frac{T}{3} \left(\frac{\partial u}{\partial T}\right)_V - \frac{u}{3}, </math>
where the factor 1/3 comes from the projection of the momentum transfer onto the normal to the wall of the container.

Since the partial derivative <math> \left(\frac{\partial u}{\partial T}\right)_V </math> can be expressed as a relationship between only <math> u </math> and <math> T </math> (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
<math display="block"> \frac{du}{4u} = \frac{dT}{T}, </math>
which leads immediately to <math> u = A T^4 </math>, with <math> A </math> as some constant of integration.

=== Derivation from Planck's law ===
[[File:Stefan-Boltzmann_Law.png|thumb|Deriving the Stefan–Boltzmann Law using [[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'' = <sup>{{pi}}</sup>/<sub>2</sub>.

The intensity of the light emitted from the blackbody surface is given by [[Planck's law]],
<math display="block">I(\nu,T) =\frac{2 h\nu^3}{c^2}\frac{1}{ e^{h\nu/(kT)}-1},</math>
where
* <math>I(\nu,T)</math> is the amount of [[Power (physics)|power]] per unit [[surface area]] per unit [[solid angle]] per unit [[frequency]] emitted at a frequency <math>\nu </math> by a black body at temperature ''T''.
* <math>h </math> is the [[Planck constant]]
* <math>c </math> is the [[speed of light]], and
* <math>k </math> is the [[Boltzmann constant]].

The quantity <math>I(\nu,T) ~A \cos \theta ~d\nu ~d\Omega</math> is the [[Power (physics)|power]] radiated by a surface of area A through a [[solid angle]] {{math|''d''Ω}} in the frequency range between {{mvar|ν}} and {{math|''ν'' + ''dν''}}.

The Stefan–Boltzmann law gives the power emitted per unit area of the emitting body,
<math display="block">\frac{P}{A} = \int_0^\infty I(\nu,T) \, d\nu \int \cos \theta \, d\Omega </math>

{{anchor|Integration of intensity derivation}}<!-- Linked from [[Radiance#Description]] -->
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 <math display="inline">d\Omega = \sin \theta\, d\theta \, d\varphi</math> over the half-sphere and integrate <math>\nu</math> from 0 to ∞.

<math display="block">
\begin{align}
\frac{P}{A} & = \int_0^\infty I(\nu,T) \, d\nu \int_0^{2\pi} \, d\varphi \int_0^{\pi/2} \cos \theta \sin \theta \, d\theta  \\
& = \pi \int_0^\infty I(\nu,T) \, d\nu
\end{align}
</math>

Then we plug in for ''I'':
<math display="block">\frac{P}{A} = \frac{2 \pi h}{c^2} \int_0^\infty \frac{\nu^3}{ e^{\frac{h\nu}{kT}} - 1} \, d\nu </math>

To evaluate this integral, do a substitution,
<math display="block">
\begin{align}
u & = \frac{h \nu}{k T} \\[6pt]
du & = \frac{h}{k T} \, d\nu
\end{align}
</math>
which gives:
<math display="block">\frac{P}{A} = \frac{2 \pi h }{c^2} \left(\frac{k T}{h} \right)^4 \int_0^\infty \frac{u^3}{ e^u - 1} \, du .</math>

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]] <math> \zeta(s) </math>. The value of the integral is <math> \Gamma(4)\zeta(4) = \frac{\pi^4}{15} </math> (where <math>\Gamma(s)</math> is the [[Gamma function]]), giving the result that, for a perfect blackbody surface:
<math display="block">M^\circ
= \sigma T^4 ~, ~~
\sigma = \frac{2 \pi^5 k^4 }{15 c^2 h^3}
= \frac{\pi^2 k^4}{60 \hbar^3 c^2}. </math>

Finally, this proof started out only considering a small flat surface. However, any [[Differentiable manifold|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 set|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'':
<math display="block">U = \frac{1}{c} \int_0^\infty I(\nu,T) \, d\nu \int \, d\Omega </math>
Thus <math display="inline">\int_0^{\pi/2} \cos \theta \sin \theta \, d\theta </math> is replaced by <math display="inline"> \int_0^{\pi} \sin \theta \, d\theta </math>, giving an extra factor of 4.

Thus, in total:
<math display="block">U = \frac{4}{c} \, \sigma \, T^4 </math>
The product <math>\frac{4}{c} \sigma</math> is sometimes known as the '''radiation constant''' or '''radiation density constant'''.<ref>{{Cite book |last1=Lemons |first1=Don S. |url=https://books.google.com/books?id=i0pTEAAAQBAJ |title=On the Trail of Blackbody Radiation: Max Planck and the Physics of his Era |last2=Shanahan |first2=William R. |last3=Buchholtz |first3=Louis J. |date=2022-09-13 |publisher=MIT Press |isbn=978-0-262-37038-7 |pages=38 |language=en}}</ref><ref>{{Cite journal |last1=Campana |first1=S. |last2=Mangano |first2=V. |last3=Blustin |first3=A. J. |last4=Brown |first4=P. |last5=Burrows |first5=D. N. |last6=Chincarini |first6=G. |last7=Cummings |first7=J. R. |last8=Cusumano |first8=G. |last9=Valle |first9=M. Della |last10=Malesani |first10=D. |last11=Mészáros |first11=P. |last12=Nousek |first12=J. A. |last13=Page |first13=M. |last14=Sakamoto |first14=T. |last15=Waxman |first15=E. |date=August 2006 |title=The association of GRB 060218 with a supernova and the evolution of the shock wave |journal=Nature |language=en |volume=442 |issue=7106 |pages=1008–1010 |doi=10.1038/nature04892 |pmid=16943830 |arxiv=astro-ph/0603279 |bibcode=2006Natur.442.1008C |s2cid=119357877 |issn=0028-0836}}</ref>

== Decomposition in terms of photons ==
The Stefan–Boltzmann law can be expressed as<ref name="Marr2012">{{cite journal |last1=Marr |first1=Jonathan M. |last2=Wilkin |first2=Francis P. |title=A Better Presentation of Planck's Radiation Law |journal=Am. J. Phys. |date=2012 |volume=80 |issue=5 |page=399 |doi=10.1119/1.3696974 |arxiv=1109.3822 |bibcode=2012AmJPh..80..399M |s2cid=10556556 |url=https://scholarship.haverford.edu/cgi/viewcontent.cgi?article=1579&context=physics_facpubs}}</ref>
<math display="block">M^{\circ} = \sigma\, T^4 = N_\mathrm{phot} \, \langle E_\mathrm{phot} \rangle</math>
where the flux of photons, <math>N_\mathrm{phot}</math>, is given by
<math display="block">N_\mathrm{phot} = \pi \int_0^\infty \frac{B_\nu}{h\nu}\,\mathrm{d}\nu</math>
<math display="block">N_\mathrm{phot} = \left({ 1.5205\times10^{15}} \; \textrm{photons}{\cdot}\textrm{s}^{-1}{\cdot}\textrm{m}^{-2}{\cdot}\mathrm{K}^{-3}\right)\cdot T^3</math>
and the average energy per photon,<math>\langle E_\textrm{phot}\rangle</math>, is given by
<math display="block">\langle E_\textrm{phot}\rangle = \frac{\pi^4}{30\,\zeta(3)} k\,T= \left({3.7294 \times 10^{-23}} \mathrm{J}{\cdot}\mathrm{K}^{-1}\right) \cdot T\,.</math>

Marr and Wilkin (2012) recommend that students be taught about <math>\langle E_\textrm{phot}\rangle</math> instead of being taught [[Wien's displacement law]], and that the above decomposition be taught when the Stefan–Boltzmann law is taught.<ref name="Marr2012"/>

== See also ==
* [[Black-body radiation]]
* [[Rayleigh–Jeans law]]
* [[Sakuma–Hattori equation]]

== Notes ==
{{reflist}}

== References ==
{{refbegin}}
* {{Cite journal |last=Stefan |first=J. |year=1879 |title=Über die Beziehung zwischen der Wärmestrahlung und der Temperatur |trans-title=On the relationship between heat radiation and temperature |url=http://www.ing-buero-ebel.de/strahlung/Original/Stefan1879.pdf |journal=Sitzungsberichte der Mathematisch-naturwissenschaftlichen Classe der Kaiserlichen Akademie der Wissenschaften |language=de |volume=79 |pages=391–428}}
* {{Cite journal |last=Boltzmann |first=Ludwig |year=1884 |title=Ableitung des Stefan'schen Gesetzes, betreffend die Abhängigkeit der Wärmestrahlung von der Temperatur aus der electromagnetischen Lichttheorie |trans-title=Derivation of Stefan's little law concerning the dependence of thermal radiation on the temperature of the electro-magnetic theory of light |journal=Annalen der Physik und Chemie |language=en |volume=258 |issue=6 |pages=291–294 |bibcode=1884AnP...258..291B |doi=10.1002/andp.18842580616 |issn=0003-3804 |doi-access=free}}
{{refend}}

{{blackbody radiation laws}}

{{DEFAULTSORT:Stefan-Boltzmann law}}
[[Category:Laws of thermodynamics]]
[[Category:Power laws]]
[[Category:Heat transfer]]
[[Category:Ludwig Boltzmann]]