## Mathematical definitions

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

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 ? is the emissivity of that surface.

### Spectral exitance

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

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

where ? is the frequency.

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

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

where ? 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 \mathrm {h} \nu ^{3}}{c^{2}}}{\frac {1}{e^{\frac {\mathrm {h} \nu }{\mathrm {k} T}}-1}},\\[8pt]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}},\end{aligned}}}

where

so 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 \mathrm {h} \varepsilon \nu ^{3}}{c^{2}}}{\frac {1}{e^{\frac {\mathrm {h} \nu }{\mathrm {k} T}}-1}},\\[8pt]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}}.\end{aligned}}}

Quantity Unit Dimension Notes
Name Symbol[nb 1] Name Symbol Symbol
Radiant energy density we joule per cubic metre J/m3 M?L-1?T-2 Radiant energy per unit volume.
Radiant flux ?e[nb 2] watt W = J/s M?L2?T-3 Radiant energy emitted, reflected, transmitted or received, per unit time. This is sometimes also called "radiant power".
Spectral flux ?e,?[nb 3] watt per hertz W/Hz M?L2?T-2 Radiant flux per unit frequency or wavelength. The latter is commonly measured in W?nm-1.
?e,?[nb 4] watt per metre W/m M?L?T-3
Radiant intensity Ie,?[nb 5] watt per steradian W/sr M?L2?T-3 Radiant flux emitted, reflected, transmitted or received, per unit solid angle. This is a directional quantity.
Spectral intensity Ie,?,?[nb 3] watt per steradian per hertz W?sr-1?Hz-1 M?L2?T-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,?,?[nb 4] watt per steradian per metre W?sr-1?m-1 M?L?T-3
Radiance Le,?[nb 5] watt per steradian per square metre W?sr-1?m-2 M?T-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,?,?[nb 3] watt per steradian per square metre per hertz W?sr-1?m-2?Hz-1 M?T-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,?,?[nb 4] watt per steradian per square metre, per metre W?sr-1?m-3 M?L-1?T-3
Flux density
Ee[nb 2] watt per square metre W/m2 M?T-3 Radiant flux received by a surface per unit area. This is sometimes also confusingly called "intensity".
Spectral flux density
Ee,?[nb 3] watt per square metre per hertz W?m-2?Hz-1 M?T-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,?[nb 4] watt per square metre, per metre W/m3 M?L-1?T-3
Radiosity Je[nb 2] watt per square metre W/m2 M?T-3 Radiant flux leaving (emitted, reflected and transmitted by) a surface per unit area. This is sometimes also confusingly called "intensity".
Spectral radiosity Je,?[nb 3] watt per square metre per hertz W?m-2?Hz-1 M?T-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,?[nb 4] watt per square metre, per metre W/m3 M?L-1?T-3
Radiant exitance Me[nb 2] watt per square metre W/m2 M?T-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,?[nb 3] watt per square metre per hertz W?m-2?Hz-1 M?T-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,?[nb 4] watt per square metre, per metre W/m3 M?L-1?T-3
Radiant exposure He joule per square metre J/m2 M?T-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,?[nb 3] joule per square metre per hertz J?m-2?Hz-1 M?T-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,?[nb 4] joule per square metre, per metre J/m3 M?L-1?T-2
Hemispherical emissivity ? N/A 1 Radiant exitance of a surface, divided by that of a black body at the same temperature as that surface.
Spectral hemispherical emissivity ??
or
??
N/A 1 Spectral exitance of a surface, divided by that of a black body at the same temperature as that surface.
Directional emissivity ?? N/A 1 Radiance emitted by a surface, divided by that emitted by a black body at the same temperature as that surface.
Spectral directional emissivity ??,?
or
??,?
N/A 1 Spectral radiance emitted by a surface, divided by that of a black body at the same temperature as that surface.
Hemispherical absorptance A N/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 A?
or
A?
N/A 1 Spectral flux absorbed by a surface, divided by that received by that surface. This should not be confused with "spectral absorbance".
Directional absorptance A? N/A 1 Radiance absorbed by a surface, divided by the radiance incident onto that surface. This should not be confused with "absorbance".
Spectral directional absorptance A?,?
or
A?,?
N/A 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 N/A 1 Radiant flux reflected by a surface, divided by that received by that surface.
Spectral hemispherical reflectance R?
or
R?
N/A 1 Spectral flux reflected by a surface, divided by that received by that surface.
Directional reflectance R? N/A 1 Radiance reflected by a surface, divided by that received by that surface.
Spectral directional reflectance R?,?
or
R?,?
N/A 1 Spectral radiance reflected by a surface, divided by that received by that surface.
Hemispherical transmittance T N/A 1 Radiant flux transmitted by a surface, divided by that received by that surface.
Spectral hemispherical transmittance T?
or
T?
N/A 1 Spectral flux transmitted by a surface, divided by that received by that surface.
Directional transmittance T? N/A 1 Radiance transmitted by a surface, divided by that received by that surface.
Spectral directional transmittance T?,?
or
T?,?
N/A 1 Spectral radiance transmitted by a surface, divided by that received by that surface.
Hemispherical attenuation coefficient ? 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 ??
or
??
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 ?? 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 ??,?
or
??,?
reciprocal metre m-1 L-1 Spectral radiance absorbed and scattered by a volume per unit length, divided by that received by that volume.
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 "?" (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 "?" (Greek).
5. ^ a b Directional quantities are denoted with suffix "?" (Greek).

## References

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