# Radiant exitance

(Redirected from Radiant emittance)

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 spectrum is taken as a function of frequency or of wavelength. This is the emitted component of radiosity. The SI unit of radiant exitance is the watt per square metre (W/m2), while that of spectral exitance in frequency is the watt per square metre per hertz (W·m−2·Hz−1) and that of spectral exitance in wavelength is the watt per square metre per metre (W·m−3)—commonly the watt per square metre per nanometre (W·m−2·nm−1). The CGS unit erg per square centimeter per second (erg·cm−2·s−1) 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 Me ("e" for "energetic", to avoid confusion with photometric quantities), is defined as[1]

${\displaystyle M_{\mathrm {e} }={\frac {\partial \Phi _{\mathrm {e} }}{\partial A}},}$
where is the partial derivative symbol, Φe is the radiant flux emitted, and 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 surface, according to the Stefan–Boltzmann law, is equal to:

${\displaystyle M_{\mathrm {e} }^{\circ }=\sigma T^{4},}$
where σ is the Stefan–Boltzmann constant, and T is the temperature of that surface. For a real surface, the radiant exitance is equal to:
${\displaystyle M_{\mathrm {e} }=\varepsilon M_{\mathrm {e} }^{\circ }=\varepsilon \sigma T^{4},}$
where ε is the emissivity of that surface.

### Spectral exitance

Spectral exitance in frequency of a surface, denoted Me,ν, is defined as[1]

${\displaystyle M_{\mathrm {e} ,\nu }={\frac {\partial M_{\mathrm {e} }}{\partial \nu }},}$

where ν is the frequency.

Spectral exitance in wavelength of a surface, denoted Me,λ, is defined as[1]

${\displaystyle M_{\mathrm {e} ,\lambda }={\frac {\partial M_{\mathrm {e} }}{\partial \lambda }},}$
where λ is the wavelength.

The spectral exitance of a black surface around a given frequency or wavelength, according to the Lambert's cosine law and the Planck's law, is equal to:

{\displaystyle {\begin{aligned}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{aligned}}}

where h is the Planck constant, ν is the frequency, λ is the wavelength, k is the Boltzmann constant, c is the speed of light in the medium, T is the temperature of that surface. For a real surface, the spectral exitance is equal to:

{\displaystyle {\begin{aligned}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{aligned}}}

## SI radiometry units

Quantity Unit Dimension Notes
Name Symbol[nb 1] Name Symbol
Radiant energy Qe[nb 2] joule J ML2T−2 Energy of electromagnetic radiation.
Radiant energy density we joule per cubic metre J/m3 ML−1T−2 Radiant energy per unit volume.
Radiant flux Φe[nb 2] watt W = J/s ML2T−3 Radiant energy emitted, reflected, transmitted or received, per unit time. This is sometimes also called "radiant power", and called luminosity in Astronomy.
Spectral flux Φe,ν[nb 3] watt per hertz W/Hz ML2T −2 Radiant flux per unit frequency or wavelength. The latter is commonly measured in W⋅nm−1.
Φe,λ[nb 4] watt per metre W/m MLT−3
Radiant intensity Ie,Ω[nb 5] watt per steradian W/sr ML2T−3 Radiant flux emitted, reflected, transmitted or received, per unit solid angle. This is a directional quantity.
Spectral intensity Ie,Ω,ν[nb 3] watt per steradian per hertz W⋅sr−1⋅Hz−1 ML2T−2 Radiant intensity per unit frequency or wavelength. The latter is commonly measured in W⋅sr−1⋅nm−1. This is a directional quantity.
Ie,Ω,λ[nb 4] watt per steradian per metre W⋅sr−1⋅m−1 MLT−3
Radiance Le,Ω[nb 5] watt per steradian per square metre W⋅sr−1⋅m−2 MT−3 Radiant flux emitted, reflected, transmitted or received by a surface, per unit solid angle per unit projected area. This is a directional quantity. This is sometimes also confusingly called "intensity".
Spectral radiance
Specific intensity
Le,Ω,ν[nb 3] watt per steradian per square metre per hertz W⋅sr−1⋅m−2⋅Hz−1 MT−2 Radiance of a surface per unit frequency or wavelength. The latter is commonly measured in W⋅sr−1⋅m−2⋅nm−1. This is a directional quantity. This is sometimes also confusingly called "spectral intensity".
Le,Ω,λ[nb 4] watt per steradian per square metre, per metre W⋅sr−1⋅m−3 ML−1T−3
Irradiance
Flux density
Ee[nb 2] watt per square metre W/m2 MT−3 Radiant flux received by a surface per unit area. This is sometimes also confusingly called "intensity".
Spectral irradiance
Spectral flux density
Ee,ν[nb 3] watt per square metre per hertz W⋅m−2⋅Hz−1 MT−2 Irradiance of a surface per unit frequency or wavelength. This is sometimes also confusingly called "spectral intensity". Non-SI units of spectral flux density include jansky (1 Jy = 10−26 W⋅m−2⋅Hz−1) and solar flux unit (1 sfu = 10−22 W⋅m−2⋅Hz−1 = 104 Jy).
Ee,λ[nb 4] watt per square metre, per metre W/m3 ML−1T−3
Radiosity Je[nb 2] watt per square metre W/m2 MT−3 Radiant flux leaving (emitted, reflected and transmitted by) a surface per unit area. This is sometimes also confusingly called "intensity".
Spectral radiosity Je,ν[nb 3] watt per square metre per hertz W⋅m−2⋅Hz−1 MT−2 Radiosity of a surface per unit frequency or wavelength. The latter is commonly measured in W⋅m−2⋅nm−1. This is sometimes also confusingly called "spectral intensity".
Je,λ[nb 4] watt per square metre, per metre W/m3 ML−1T−3
Radiant exitance Me[nb 2] watt per square metre W/m2 MT−3 Radiant flux emitted by a surface per unit area. This is the emitted component of radiosity. "Radiant emittance" is an old term for this quantity. This is sometimes also confusingly called "intensity".
Spectral exitance Me,ν[nb 3] watt per square metre per hertz W⋅m−2⋅Hz−1 MT−2 Radiant exitance of a surface per unit frequency or wavelength. The latter is commonly measured in W⋅m−2⋅nm−1. "Spectral emittance" is an old term for this quantity. This is sometimes also confusingly called "spectral intensity".
Me,λ[nb 4] watt per square metre, per metre W/m3 ML−1T−3
Radiant exposure He joule per square metre J/m2 MT−2 Radiant energy received by a surface per unit area, or equivalently irradiance of a surface integrated over time of irradiation. This is sometimes also called "radiant fluence".
Spectral exposure He,ν[nb 3] joule per square metre per hertz J⋅m−2⋅Hz−1 MT−1 Radiant exposure of a surface per unit frequency or wavelength. The latter is commonly measured in J⋅m−2⋅nm−1. This is sometimes also called "spectral fluence".
He,λ[nb 4] joule per square metre, per metre J/m3 ML−1T−2
See also:
1. ^ Standards organizations recommend that radiometric quantities should be denoted with suffix "e" (for "energetic") to avoid confusion with photometric or photon quantities.
2. Alternative symbols sometimes seen: W or E for radiant energy, P or F for radiant flux, I for irradiance, W for radiant exitance.
3. Spectral quantities given per unit frequency are denoted with suffix "ν" (Greek letter nu, not to be confused with a letter "v", indicating a photometric quantity.)
4. Spectral quantities given per unit wavelength are denoted with suffix "λ".
5. ^ a b Directional quantities are denoted with suffix "Ω".

## References

1. ^ a b c "Thermal insulation — Heat transfer by radiation — Vocabulary". ISO_9288:2022. International Organization for Standardization. 2022. Retrieved 2023-06-17.