Editing Stefan–Boltzmann law (section)

== 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.