Radiant exitance

Template:Short description 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 (Template:Nobreak), 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 (Template:Nobreak). The CGS unit erg per square centimeter per second (Template:Nobreak) 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 Template:Math ("e" for "energetic", to avoid confusion with photometric quantities), is defined as $$M_{\mathrm{e}}=\frac{\partial {\mathrm{\Phi }}_{\mathrm{e}}}{\partial A},$$ where Template:Math is the partial derivative symbol, Template:Math is the radiant flux emitted, and Template:Mvar 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 Template:Mvar is the Stefan–Boltzmann constant, and Template:Mvar is the temperature of that surface. For a real surface, the radiant exitance is equal to: $$M_{\mathrm{e}}=\epsilon M_{\mathrm{e}}^{\circ }=\epsilon \sigma T^4,$$ where Template:Mvar 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 Template:Mvar 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 Template:Mvar 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:

$\begin{array}{rl}{M}_{\mathrm{e},\nu }^{\circ }& =\pi L_{\mathrm{e},\mathrm{\Omega },\nu }^{\circ }=\frac{2\pi h{\nu }^3}{c^2}\frac{1}{e^{\frac{h\nu }{kT}}-1},\\ M_{\mathrm{e},\lambda }^{\circ }& =\pi L_{\mathrm{e},\mathrm{\Omega },\lambda }^{\circ }=\frac{2\pi h{c}^2}{{\lambda }^5}\frac{1}{e^{\frac{hc}{\lambda kT}}-1},\end{array}$

where Template:Mvar is the Planck constant, Template:Mvar is the frequency, Template:Mvar is the wavelength, Template:Mvar is the Boltzmann constant, Template:Mvar is the speed of light in the medium, Template:Mvar is the temperature of that surface. For a real surface, the spectral exitance is equal to: $$\begin{array}{rl}{M}_{\mathrm{e},\nu }& =\epsilon M_{\mathrm{e},\nu }^{\circ }=\frac{2\pi h\epsilon {\nu }^3}{c^2}\frac{1}{e^{\frac{h\nu }{kT}}-1},\\ M_{\mathrm{e},\lambda }& =\epsilon M_{\mathrm{e},\lambda }^{\circ }=\frac{2\pi h\epsilon c^2}{{\lambda }^5}\frac{1}{e^{\frac{hc}{\lambda kT}}-1}.\end{array}$$

SI radiometry units

Template:Resize

Quantity		Unit		Dimension	Notes
Name	Symbol	Name	Symbol
Radiant energy	Template:Math	joule	J	M⋅L2⋅T−2	Energy of electromagnetic radiation.
Radiant energy density	Template:Math	joule per cubic metre	J/m3	M⋅L−1⋅T−2	Radiant energy per unit volume.
Radiant flux	Template:Math	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", and called luminosity in astronomy.
Spectral flux	Template:Math	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.
	Template:Math	watt per metre	W/m	M⋅L⋅T−3
Radiant intensity	Template:Math	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	Template:Math	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.
	Template:Math	watt per steradian per metre	W⋅sr−1⋅m−1	M⋅L⋅T−3
Radiance	Template:Math	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	Template:Math	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".
	Template:Math	watt per steradian per square metre, per metre	W⋅sr−1⋅m−3	M⋅L−1⋅T−3
Irradiance Flux density	Template:Math	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 irradiance Spectral flux density	Template:Math	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 (Template:Val = Template:Val) and solar flux unit (Template:Val = Template:Val = Template:Val).
	Template:Math	watt per square metre, per metre	W/m3	M⋅L−1⋅T−3
Radiosity	Template:Math	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	Template:Math	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".
	Template:Math	watt per square metre, per metre	W/m3	M⋅L−1⋅T−3
Radiant exitance	Template:Math	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	Template:Math	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".
	Template:Math	watt per square metre, per metre	W/m3	M⋅L−1⋅T−3
Radiant exposure	Template:Math	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	Template:Math	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".
	Template:Math	joule per square metre, per metre	J/m3	M⋅L−1⋅T−2
See also: Template:Hlist

File:Photometry radiometry units.svg

Comparison of photometric and radiometric quantities

See also

• Radiosity