# 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/m^{2}), 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 *M*_{e} ("e" for "energetic", to avoid confusion with photometric quantities), is defined as[1]

where

- ∂ is the partial derivative symbol;
- Φ
_{e}is the radiant flux*emitted*; *A*is the 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:

where

- σ is the Stefan–Boltzmann constant;
*T*is the temperature of that surface,

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

where *ε* is the emissivity of that surface.

### Spectral exitance

**Spectral exitance in frequency** of a *surface*, denoted *M*_{e,ν}, is defined as[1]

where *ν* is the frequency.

**Spectral exitance in wavelength** of a *surface*, denoted *M*_{e,λ}, is defined as[1]

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:

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,

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

## SI radiometry units

Quantity | Unit | Dimension | Notes | |||||
---|---|---|---|---|---|---|---|---|

Name | Symbol[nb 1] | Name | Symbol | Symbol | ||||

Radiant energy | Q_{e}[nb 2] |
joule | J | M⋅L^{2}⋅T^{−2} |
Energy of electromagnetic radiation. | |||

Radiant energy density | w_{e} |
joule per cubic metre | J/m^{3} |
M⋅L^{−1}⋅T^{−2} |
Radiant energy per unit volume. | |||

Radiant flux | Φ_{e}[nb 2] |
watt | W = J/s | M⋅L^{2}⋅T^{−3} |
Radiant energy emitted, reflected, transmitted or received, per unit time. This is sometimes also called "radiant power". | |||

Spectral flux | Φ_{e,ν}[nb 3] |
watt per hertz | W/Hz | M⋅L^{2}⋅T^{−2} |
Radiant flux per unit frequency or wavelength. The latter is commonly measured in W⋅nm^{−1}. | |||

Φ_{e,λ}[nb 4] |
watt per metre | W/m | M⋅L⋅T^{−3} | |||||

Radiant intensity | I_{e,Ω}[nb 5] |
watt per steradian | W/sr | M⋅L^{2}⋅T^{−3} |
Radiant flux emitted, reflected, transmitted or received, per unit solid angle. This is a directional quantity. | |||

Spectral intensity | I_{e,Ω,ν}[nb 3] |
watt per steradian per hertz | W⋅sr^{−1}⋅Hz^{−1} |
M⋅L^{2}⋅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. | |||

I_{e,Ω,λ}[nb 4] |
watt per steradian per metre | W⋅sr^{−1}⋅m^{−1} |
M⋅L⋅T^{−3} | |||||

Radiance | L_{e,Ω}[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 | L_{e,Ω,ν}[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". | |||

L_{e,Ω,λ}[nb 4] |
watt per steradian per square metre, per metre | W⋅sr^{−1}⋅m^{−3} |
M⋅L^{−1}⋅T^{−3} | |||||

Irradiance Flux density |
E_{e}[nb 2] |
watt per square metre | W/m^{2} |
M⋅T^{−3} |
Radiant flux received by a surface per unit area. This is sometimes also confusingly called "intensity". | |||

Spectral irradiance Spectral flux density |
E_{e,ν}[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} = 10^{4} Jy). | |||

E_{e,λ}[nb 4] |
watt per square metre, per metre | W/m^{3} |
M⋅L^{−1}⋅T^{−3} | |||||

Radiosity | J_{e}[nb 2] |
watt per square metre | W/m^{2} |
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 | J_{e,ν}[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". | |||

J_{e,λ}[nb 4] |
watt per square metre, per metre | W/m^{3} |
M⋅L^{−1}⋅T^{−3} | |||||

Radiant exitance | M_{e}[nb 2] |
watt per square metre | W/m^{2} |
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 | M_{e,ν}[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". | |||

M_{e,λ}[nb 4] |
watt per square metre, per metre | W/m^{3} |
M⋅L^{−1}⋅T^{−3} | |||||

Radiant exposure | H_{e} |
joule per square metre | J/m^{2} |
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 | H_{e,ν}[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". | |||

H_{e,λ}[nb 4] |
joule per square metre, per metre | J/m^{3} |
M⋅L^{−1}⋅T^{−2} | |||||

Hemispherical emissivity | ε |
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 | ε_{ν}orε_{λ} |
N/A | 1 |
Spectral exitance of a surface, divided by that of a black body at the same temperature as that surface. | ||||

Directional emissivity | ε_{Ω} |
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 | ε_{Ω,ν}orε_{Ω,λ} |
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_{ν}orA_{λ} |
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_{Ω} |
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_{Ω,ν}orA_{Ω,λ} |
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_{ν}orR_{λ} |
N/A | 1 |
Spectral flux reflected by a surface, divided by that received by that surface. | ||||

Directional reflectance | R_{Ω} |
N/A | 1 |
Radiance reflected by a surface, divided by that received by that surface. | ||||

Spectral directional reflectance | R_{Ω,ν}orR_{Ω,λ} |
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_{ν}orT_{λ} |
N/A | 1 |
Spectral flux transmitted by a surface, divided by that received by that surface. | ||||

Directional transmittance | T_{Ω} |
N/A | 1 |
Radiance transmitted by a surface, divided by that received by that surface. | ||||

Spectral directional transmittance | T_{Ω,ν}orT_{Ω,λ} |
N/A | 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 | μ_{ν}orμ_{λ} |
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 | μ_{Ω} |
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 | μ_{Ω,ν}orμ_{Ω,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 |

- Standards organizations recommend that radiometric quantities should be denoted with suffix "e" (for "energetic") to avoid confusion with photometric or photon quantities.
- Alternative symbols sometimes seen:
*W*or*E*for radiant energy,*P*or*F*for radiant flux,*I*for irradiance,*W*for radiant exitance. - 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.
- Spectral quantities given per unit wavelength are denoted with suffix "λ" (Greek).
- Directional quantities are denoted with suffix "Ω" (Greek).

## See also

## References

- "Thermal insulation — Heat transfer by radiation — Physical quantities and definitions".
*ISO 9288:1989*. ISO catalogue. 1989. Retrieved 2015-03-15.