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Radiant exitance

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Radiant flux per unit area

Inradiometry,radiant exitance orradiant emittance is theradiant flux emitted by a surface per unit area, whereasspectral exitance orspectral emittance is the radiant exitance of a surface per unitfrequency orwavelength, depending on whether thespectrum is taken as a function of frequency or of wavelength. This is the emitted component ofradiosity. TheSI unit of radiant exitance is thewatt per square metre (W/m²), while that of spectral exitance in frequency is the watt per square metre perhertz (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). TheCGS unit erg per square centimeter per second (erg·cm⁻²·s⁻¹) is often used inastronomy. Radiant exitance is often called "intensity" in branches of physics other than radiometry, but in radiometry this usage leads to confusion withradiant intensity.

Mathematical definitions

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Radiant exitance

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Radiant exitance of asurface, denotedM_e ("e" for "energetic", to avoid confusion withphotometric quantities), is defined as^[1] $M_{\mathrm {e} }={\frac {\partial \Phi _{\mathrm {e} }}{\partial A}},$ where∂ is thepartial derivative symbol,Φ_e is theradiant fluxemitted, andA is thesurface area.

The radiant fluxreceived by a surface is calledirradiance.

The radiant exitance of ablack surface, according to theStefan–Boltzmann law, is equal to: $M_{\mathrm {e} }^{\circ }=\sigma T^{4},$ whereσ is theStefan–Boltzmann constant, andT is the temperature of that surface.For a real surface, the radiant exitance is equal to: $M_{\mathrm {e} }=\varepsilon M_{\mathrm {e} }^{\circ }=\varepsilon \sigma T^{4},$ whereε is theemissivity of that surface.

Spectral exitance

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Spectral exitance in frequency of asurface, denotedM_e,ν, is defined as^[1]

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

whereν is the frequency.

Spectral exitance in wavelength of asurface, denotedM_e,λ, is defined as^[1] $M_{\mathrm {e} ,\lambda }={\frac {\partial M_{\mathrm {e} }}{\partial \lambda }},$ whereλ is the wavelength.

The spectral exitance of ablack surface around a given frequency or wavelength, according toLambert's cosine law andPlanck'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}}

whereh is thePlanck constant,ν is the frequency,λ is the wavelength,k is theBoltzmann constant,c is thespeed of light in vacuum,T is the temperature of that surface.For a real surface, the spectral exitance is equal to: ${\begin{aligned}M_{\mathrm {e} ,\nu }&=\varepsilon M_{\mathrm {e} ,\nu }^{\circ }={\frac {2\pi h\varepsilon \nu ^{3}}{c^{2}}}{\frac {1}{e^{\frac {h\nu }{kT}}-1}},\\[8pt]M_{\mathrm {e} ,\lambda }&=\varepsilon M_{\mathrm {e} ,\lambda }^{\circ }={\frac {2\pi h\varepsilon c^{2}}{\lambda ^{5}}}{\frac {1}{e^{\frac {hc}{\lambda kT}}-1}}.\end{aligned}}$ where $\varepsilon$ is the emittance of the surface.

SI radiometry units

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SI radiometry units
v
t
e
Quantity		Unit		Dimension	Notes
Name	Symbol^{[nb 1]}	Name	Symbol	Dimension	Notes
Radiant energy	Q_e^{[nb 2]}	joule	J	M⋅L²⋅T⁻²	Energy of electromagnetic radiation.
Radiant energy density	w_e	joule per cubic metre	J/m³	M⋅L⁻¹⋅T⁻²	Radiant energy per unit volume.
Radiant flux	Φ_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", and calledluminosity in astronomy.
Spectral flux	Φ_e,ν^{[nb 3]}	watt perhertz	W/Hz	M⋅L²⋅T⁻²	Radiant flux per unit frequency or wavelength. The latter is commonly measured in W⋅nm⁻¹.
Spectral flux	Φ_e,λ^{[nb 4]}	watt per metre	W/m	M⋅L⋅T⁻³
Radiant intensity	I_e,Ω^{[nb 5]}	watt persteradian	W/sr	M⋅L²⋅T⁻³	Radiant flux emitted, reflected, transmitted or received, per unit solid angle. This is adirectional quantity.
Spectral intensity	I_e,Ω,ν^{[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 adirectional quantity.
Spectral intensity	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 asurface, per unit solid angle per unit projected area. This is adirectional quantity. This is sometimes also confusingly called "intensity".
Spectral radiance Specific intensity	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 asurface per unit frequency or wavelength. The latter is commonly measured in W⋅sr⁻¹⋅m⁻²⋅nm⁻¹. This is adirectional quantity. This is sometimes also confusingly called "spectral intensity".
Spectral radiance Specific 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/m²	M⋅T⁻³	Radiant fluxreceived by asurface 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 asurface per unit frequency or wavelength. This is sometimes also confusingly called "spectral intensity". Non-SI units of spectral flux density includejansky (1 Jy =10⁻²⁶ W⋅m⁻²⋅Hz⁻¹) andsolar flux unit (1 sfu =10⁻²² W⋅m⁻²⋅Hz⁻¹ =10⁴ Jy).
Spectral irradiance Spectral flux density	E_e,λ^{[nb 4]}	watt per square metre, per metre	W/m³	M⋅L⁻¹⋅T⁻³
Radiosity	J_e^{[nb 2]}	watt per square metre	W/m²	M⋅T⁻³	Radiant fluxleaving (emitted, reflected and transmitted by) asurface 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 asurface 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^{[nb 4]}	watt per square metre, per metre	W/m³	M⋅L⁻¹⋅T⁻³
Radiant exitance	M_e^{[nb 2]}	watt per square metre	W/m²	M⋅T⁻³	Radiant fluxemitted by asurface 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 asurface 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^{[nb 4]}	watt per square metre, per metre	W/m³	M⋅L⁻¹⋅T⁻³
Radiant exposure	H_e	joule per square metre	J/m²	M⋅T⁻²	Radiant energy received by asurface per unit area, or equivalently irradiance of asurface 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 asurface 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^{[nb 4]}	joule per square metre, per metre	J/m³	M⋅L⁻¹⋅T⁻²
See also: SI Radiometry Photometry

^Standards organizations recommend that radiometricquantities should be denoted with suffix "e" (for "energetic") to avoid confusion with photometric orphoton quantities.
^^a ^b ^c ^d ^eAlternative symbols sometimes seen:W orE for radiant energy,P orF for radiant flux,I for irradiance,W for radiant exitance.
^^a ^b ^c ^d ^e ^f ^gSpectral quantities given per unitfrequency are denoted with suffix "ν" (Greek letternu, not to be confused with a letter "v", indicating a photometric quantity.)
^^a ^b ^c ^d ^e ^f ^gSpectral quantities given per unitwavelength are denoted with suffix "λ".
^^a ^bDirectional quantities are denoted with suffix "Ω".