View source for Radiant exitance

{{short description|Radiant flux per unit area}}
In [[radiometry]], '''radiant exitance''' or '''radiant emittance''' is the [[radiant flux]] emitted by a surface per unit area, whereas '''spectral exitance''' or '''spectral emittance''' is the radiant exitance of a surface per unit [[frequency]] or [[wavelength]], depending on whether the [[Spectral radiometric quantity|spectrum]] is taken as a function of frequency or of wavelength. This is the emitted component of [[radiosity (heat transfer)|radiosity]]. The [[International System of Units|SI unit]] of radiant exitance is the [[watt]] per square metre ({{nobreak|W/m<sup>2</sup>}}), while that of spectral exitance in frequency is the watt per square metre per [[hertz]] (W·m<sup>−2</sup>·Hz<sup>−1</sup>) and that of spectral exitance in wavelength is the watt per square metre per metre (W·m<sup>−3</sup>)—commonly the watt per square metre per nanometre ({{nobreak|W·m<sup>−2</sup>·nm<sup>−1</sup>}}). The [[Centimetre–gram–second system of units|CGS unit]] [[erg]] per square centimeter per second ({{nobreak|erg·cm<sup>−2</sup>·s<sup>−1</sup>}}) is often used in [[astronomy]]. Radiant exitance is often called "intensity" in branches of physics other than radiometry, but in radiometry this usage leads to confusion with [[radiant intensity]].

==Mathematical definitions==
===Radiant exitance===
'''Radiant exitance''' of a ''surface'', denoted {{math|''M''<sub>e</sub>}} ("e" for "energetic", to avoid confusion with [[photometry (optics)|photometric]] quantities), is defined as<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>
<math display=block>M_\mathrm{e} = \frac{\partial \Phi_\mathrm{e}}{\partial A},</math>
where
{{math|∂}} is the [[partial derivative]] symbol,
{{math|Φ<sub>e</sub>}} is the [[radiant flux]] ''emitted'', and
{{mvar|A}} is the [[surface area]].

If we want to talk about the radiant flux ''received'' by a surface, we speak of [[irradiance]].

The radiant exitance of a [[Black body|black surface]], according to the [[Stefan–Boltzmann law]], is equal to:
<math display=block>M_\mathrm{e}^\circ = \sigma T^4,</math>
where {{mvar|σ}} is the [[Stefan–Boltzmann constant]], and {{mvar|T}} is the temperature of that surface.
For a real surface, the radiant exitance is equal to:
<math display=block>M_\mathrm{e} = \varepsilon M_\mathrm{e}^\circ = \varepsilon \sigma T^4,</math>
where {{mvar|&epsilon;}} is the [[emissivity]] of that surface.

===Spectral exitance===
'''Spectral exitance in frequency''' of a ''surface'', denoted ''M''<sub>e,ν</sub>, is defined as<ref name="ISO_9288:2022" />
:<math>M_{\mathrm{e},\nu} = \frac{\partial M_\mathrm{e}}{\partial \nu},</math>
where {{mvar|ν}} is the frequency.

'''Spectral exitance in wavelength''' of a ''surface'', denoted ''M''<sub>e,λ</sub>, is defined as<ref name="ISO_9288:2022" />
<math display=block>M_{\mathrm{e},\lambda} = \frac{\partial M_\mathrm{e}}{\partial \lambda},</math>
where {{mvar|λ}} is the wavelength.

The spectral exitance of a [[Black body|black surface]] around a given frequency or wavelength, according to the [[Lambert's cosine law]] and the [[Planck's law]], is equal to:
:<math>
\begin{align}
M_{\mathrm{e},\nu}^\circ & = \pi L_{\mathrm{e},\Omega,\nu}^\circ = \frac{2\pi h\nu^3}{c^2} \frac{1}{e^\frac{h\nu}{kT} - 1}, \\[8pt]
M_{\mathrm{e},\lambda}^\circ & = \pi L_{\mathrm{e},\Omega,\lambda}^\circ = \frac{2\pi hc^2}{\lambda^5} \frac{1}{e^\frac{hc}{\lambda kT} - 1},
\end{align}
</math>
where
{{mvar|h}} is the [[Planck constant]],
{{mvar|ν}} is the frequency,
{{mvar|λ}} is the wavelength,
{{mvar|k}} is the [[Boltzmann constant]],
{{mvar|c}} is the [[speed of light]] in the medium,
{{mvar|T}} is the temperature of that surface.
For a real surface, the spectral exitance is equal to:
<math display=block>
\begin{align}
M_{\mathrm{e},\nu} & = \varepsilon M_{\mathrm{e},\nu}^\circ = \frac{2\pi h\varepsilon \nu^3}{c^2} \frac{1}{e^\frac{h\nu}{kT} - 1}, \\[8pt]
M_{\mathrm{e},\lambda} & = \varepsilon M_{\mathrm{e},\lambda}^\circ = \frac{2\pi h\varepsilon c^2}{\lambda^5} \frac{1}{e^\frac{hc}{\lambda kT} - 1}.
\end{align}
</math>

==SI radiometry units==
{{SI radiometry units}}
[[File:photometry_radiometry_units.svg|thumb|upright=1.5|Comparison of photometric and radiometric quantities]]

==See also==
*[[Radiosity (heat transfer)|Radiosity]]

==References==
{{reflist}}

[[Category:Physical quantities]]
[[Category:Radiometry]]