# 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 $M_{\mathrm {e} }={\frac {\partial \Phi _{\mathrm {e} }}{\partial A}},$ where ∂ is the partial derivative symbol, Φ$_e$ is the radiant flux emitted, and A is the surface area.
The radiant flux received by a surface is called irradiance.
The radiant exitance of a black surface, according to the Stefan–Boltzmann law, is equal to: $M_{\mathrm {e} }^{\circ }=\sigma T^{4},$ 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: $M_{\mathrm {e} }=\varepsilon M_{\mathrm {e} }^{\circ }=\varepsilon \sigma T^{4},$ where ε is the emissivity of that surface.

### Spectral exitance

Spectral exitance in frequency of a surface, denoted M$_{e,ν}$, is defined as
    $M_{\mathrm {e} ,\nu }={\frac {\partial M_{\mathrm {e} }}{\partial \nu }},$
where ν is the frequency.
Spectral exitance in wavelength of a surface, denoted M$_{e,λ}$, is defined as $M_{\mathrm {e} ,\lambda }={\frac {\partial M_{\mathrm {e} }}{\partial \lambda }},$ where λ is the wavelength.
The spectral exitance of a black surface around a given frequency or wavelength, according to Lambert's cosine law and Planck's law, is equal to:
    ${\begin{aligned}M_{\mathrm {e} ,\nu }^{\circ }&=\pi L_{\mathrm {e} ,\Omega ,\nu }^{\circ }={\frac {2\pi h\nu ^{3}}{c^{2}}}{\frac {1}{e^{\frac {h\nu }{kT}}-1}},\\[8pt]M_{\mathrm {e} ,\lambda }^{\circ }&=\pi L_{\mathrm {e} ,\Omega ,\lambda }^{\circ }={\frac {2\pi hc^{2}}{\lambda ^{5}}}{\frac {1}{e^{\frac {hc}{\lambda kT}}-1}},\end{aligned}}$
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: ${\begin{aligned}M_{\mathrm {e} ,\nu }&=\varepsilon M_{\mathrm {e} ,\nu }^{\circ }={\frac {2\pi h\varepsilon \nu ^{3}}{c^{2}}}{\frac {1}{e^{\frac {h\nu }{kT}}-1}},\\[8pt]M_{\mathrm {e} ,\lambda }&=\varepsilon M_{\mathrm {e} ,\lambda }^{\circ }={\frac {2\pi h\varepsilon c^{2}}{\lambda ^{5}}}{\frac {1}{e^{\frac {hc}{\lambda kT}}-1}}.\end{aligned}}$

## SI radiometry units

SI radiometry unitsvte :
| Quantity | Quantity | Unit | Unit | Dimension | Notes |
| Name | Symbol | Name | Symbol | Dimension | Notes |
| Radiant energy | Q$_e$ | 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$_e$ | 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", and called luminosity in astronomy. |
| Spectral flux | Φ$_{e,ν}$ | 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$^{−1}$. |
| Spectral flux | Φ$_{e,λ}$ | watt per metre | W/m | M⋅L⋅T$^{−3}$ | Radiant flux per unit frequency or wavelength. The latter is commonly measured in W⋅nm$^{−1}$. |
| Radiant intensity | I$_{e,Ω}$ | 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$_{e,Ω,ν}$ | 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. |
| Spectral intensity | I$_{e,Ω,λ}$ | watt per steradian per metre | W⋅sr$^{−1}$⋅m$^{−1}$ | M⋅L⋅T$^{−3}$ | Radiant intensity per unit frequency or wavelength. The latter is commonly measured in W⋅sr$^{−1}$⋅nm$^{−1}$. This is a directional quantity. |
| Radiance | L$_{e,Ω}$ | 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 Specific intensity | L$_{e,Ω,ν}$ | 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". |
| Spectral radiance Specific intensity | L$_{e,Ω,λ}$ | watt per steradian per square metre, per metre | W⋅sr$^{−1}$⋅m$^{−3}$ | M⋅L$^{−1}$⋅T$^{−3}$ | 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 | E$_e$ | 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$_{e,ν}$ | 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). |
| Spectral irradiance Spectral flux density | E$_{e,λ}$ | watt per square metre, per metre | W/m$^3$ | M⋅L$^{−1}$⋅T$^{−3}$ | 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$^{−1}$ = 10$^4$ Jy). |
| Radiosity | J$_e$ | 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$_{e,ν}$ | 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". |
| Spectral radiosity | J$_{e,λ}$ | watt per square metre, per metre | W/m$^3$ | M⋅L$^{−1}$⋅T$^{−3}$ | 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 | M$_e$ | 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$_{e,ν}$ | 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". |
| Spectral exitance | M$_{e,λ}$ | watt per square metre, per metre | W/m$^3$ | M⋅L$^{−1}$⋅T$^{−3}$ | 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 | 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$_{e,ν}$ | 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". |
| Spectral exposure | H$_{e,λ}$ | joule per square metre, per metre | J/m$^3$ | M⋅L$^{−1}$⋅T$^{−2}$ | 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". |
| See also: SIRadiometryPhotometry ||||||
* ↑ Standards organizations recommend that radiometric quantities should be denoted with suffix "e" (for "energetic") to avoid confusion with photometric or photon quantities.
* 1 2 3 4 5 Alternative symbols sometimes seen: W or E for radiant energy, P or F for radiant flux, I for irradiance, W for radiant exitance.
* 1 2 3 4 5 6 7 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.)
* 1 2 3 4 5 6 7 Spectral quantities given per unit wavelength are denoted with suffix "λ".
* 1 2 Directional quantities are denoted with suffix "Ω".