# Radiant exitance

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

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

so 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:

${\displaystyle M_{\mathrm {e} ,\nu }^{\circ }=\pi L_{\mathrm {e} ,\Omega ,\nu }^{\circ }={\frac {2\pi \mathrm {h} \nu ^{3}}{c^{2}}}{\frac {1}{e^{\frac {\mathrm {h} \nu }{\mathrm {k} T}}-1}},}$
${\displaystyle M_{\mathrm {e} ,\lambda }^{\circ }=\pi L_{\mathrm {e} ,\Omega ,\lambda }^{\circ }={\frac {2\pi \mathrm {h} c^{2}}{\lambda ^{5}}}{\frac {1}{e^{\frac {\mathrm {h} c}{\lambda \mathrm {k} T}}-1}},}$

where

so for a real surface, the spectral exitance is equal to:

${\displaystyle M_{\mathrm {e} ,\nu }=\varepsilon M_{\mathrm {e} ,\nu }^{\circ }={\frac {2\pi \mathrm {h} \varepsilon \nu ^{3}}{c^{2}}}{\frac {1}{e^{\frac {\mathrm {h} \nu }{\mathrm {k} T}}-1}},}$
${\displaystyle M_{\mathrm {e} ,\lambda }=\varepsilon M_{\mathrm {e} ,\lambda }^{\circ }={\frac {2\pi \mathrm {h} \varepsilon c^{2}}{\lambda ^{5}}}{\frac {1}{e^{\frac {\mathrm {h} c}{\lambda \mathrm {k} T}}-1}}.}$

## SI radiometry units

SI radiometry units

Quantity Unit Dimension Notes
Name Symbol[nb 1] Name Symbol 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".
Spectral flux Φe,ν[nb 3]
or
Φe,λ[nb 4]
watt per hertz
or
watt per metre
W/Hz
or
W/m
ML2T−2
or
MLT−3
Radiant flux per unit frequency or wavelength. The latter is commonly measured in W⋅nm−1.
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]
or
Ie,Ω,λ[nb 4]
watt per steradian per hertz
or
watt per steradian per metre
W⋅sr−1⋅Hz−1
or
W⋅sr−1⋅m−1
ML2T−2
or
MLT−3
Radiant intensity per unit frequency or wavelength. The latter is commonly measured in W⋅sr−1⋅nm−1. This is a directional quantity.
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 Le,Ω,ν[nb 3]
or
Le,Ω,λ[nb 4]
watt per steradian per square metre per hertz
or
watt per steradian per square metre, per metre
W⋅sr−1⋅m−2⋅Hz−1
or
W⋅sr−1⋅m−3
MT−2
or
ML−1T−3
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".
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]
or
Ee,λ[nb 4]
watt per square metre per hertz
or
watt per square metre, per metre
W⋅m−2⋅Hz−1
or
W/m3
MT−2
or
ML−1T−3
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).
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]
or
Je,λ[nb 4]
watt per square metre per hertz
or
watt per square metre, per metre
W⋅m−2⋅Hz−1
or
W/m3
MT−2
or
ML−1T−3
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".
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]
or
Me,λ[nb 4]
watt per square metre per hertz
or
watt per square metre, per metre
W⋅m−2⋅Hz−1
or
W/m3
MT−2
or
ML−1T−3
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".
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]
or
He,λ[nb 4]
joule per square metre per hertz
or
joule per square metre, per metre
J⋅m−2⋅Hz−1
or
J/m3
MT−1
or
ML−1T−2
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".
Hemispherical emissivity ε 1 Radiant exitance of a surface, divided by that of a black body at the same temperature as that surface.
Spectral hemispherical emissivity εν
or
ελ
1 Spectral exitance of a surface, divided by that of a black body at the same temperature as that surface.
Directional emissivity εΩ 1 Radiance emitted by a surface, divided by that emitted by a black body at the same temperature as that surface.
Spectral directional emissivity εΩ,ν
or
εΩ,λ
1 Spectral radiance emitted by a surface, divided by that of a black body at the same temperature as that surface.
Hemispherical absorptance A 1 Radiant flux absorbed by a surface, divided by that received by that surface. This should not be confused with "absorbance".
Spectral hemispherical absorptance Aν
or
Aλ
1 Spectral flux absorbed by a surface, divided by that received by that surface. This should not be confused with "spectral absorbance".
Directional absorptance AΩ 1 Radiance absorbed by a surface, divided by the radiance incident onto that surface. This should not be confused with "absorbance".
Spectral directional absorptance AΩ,ν
or
AΩ,λ
1 Spectral radiance absorbed by a surface, divided by the spectral radiance incident onto that surface. This should not be confused with "spectral absorbance".
Hemispherical reflectance R 1 Radiant flux reflected by a surface, divided by that received by that surface.
Spectral hemispherical reflectance Rν
or
Rλ
1 Spectral flux reflected by a surface, divided by that received by that surface.
Directional reflectance RΩ 1 Radiance reflected by a surface, divided by that received by that surface.
Spectral directional reflectance RΩ,ν
or
RΩ,λ
1 Spectral radiance reflected by a surface, divided by that received by that surface.
Hemispherical transmittance T 1 Radiant flux transmitted by a surface, divided by that received by that surface.
Spectral hemispherical transmittance Tν
or
Tλ
1 Spectral flux transmitted by a surface, divided by that received by that surface.
Directional transmittance TΩ 1 Radiance transmitted by a surface, divided by that received by that surface.
Spectral directional transmittance TΩ,ν
or
TΩ,λ
1 Spectral radiance transmitted by a surface, divided by that received by that surface.
Hemispherical attenuation coefficient μ reciprocal metre m−1 L−1 Radiant flux absorbed and scattered by a volume per unit length, divided by that received by that volume.
Spectral hemispherical attenuation coefficient μν
or
μλ
reciprocal metre m−1 L−1 Spectral radiant flux absorbed and scattered by a volume per unit length, divided by that received by that volume.
Directional attenuation coefficient μΩ reciprocal metre m−1 L−1 Radiance absorbed and scattered by a volume per unit length, divided by that received by that volume.
Spectral directional attenuation coefficient μΩ,ν
or
μΩ,λ
reciprocal metre m−1 L−1 Spectral radiance absorbed and scattered by a volume per unit length, divided by that received by that volume.
See also: SI · Radiometry · Photometry
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)—not to be confused with suffix "v" (for "visual") indicating a photometric quantity.
4. Spectral quantities given per unit wavelength are denoted with suffix "λ" (Greek).
5. ^ a b Directional quantities are denoted with suffix "Ω" (Greek).

## References

1. ^ a b c "Thermal insulation — Heat transfer by radiation — Physical quantities and definitions". ISO 9288:1989. ISO catalogue. 1989. Retrieved 2015-03-15.