# Radiant flux

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Measure of radiant energy over time

Not to be confused with [Radiation flux](/source/Radiation_flux).

"Spectral power" redirects here; not to be confused with [Spectral power density](/source/Spectral_power_density).

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A flow chart describing the relationship of various physical quantities, including radiant flux and exitance

In [radiometry](/source/Radiometry), **radiant flux** or **radiant power** is the [radiant energy](/source/Radiant_energy) emitted, reflected, transmitted, or received per unit time, and **spectral flux** or **spectral power** is the radiant flux per unit [frequency](/source/Frequency) or [wavelength](/source/Wavelength), depending on whether the [spectrum](/source/Spectral_radiometric_quantity) is taken as a function of frequency or of wavelength. The [SI unit](/source/International_System_of_Units) of radiant flux is the [watt](/source/Watt) (W), one [joule](/source/Joule) per second (J/s), while that of spectral flux in frequency is the watt per [hertz](/source/Hertz) (W/Hz) and that of spectral flux in wavelength is the watt per metre (W/m)—commonly the watt per nanometre (W/nm). Radiant flux is sometimes called **[luminosity](/source/Luminosity)**, especially in astronomy contexts.

## Mathematical definitions

### Radiant flux

Radiant flux, denoted [Φe](/source/Phi) ('e' for "energetic", to avoid confusion with [photometric](/source/Photometry_(optics)) quantities), is defined as[1] Φ e = d Q e d t Q e = ∫ T ∫ Σ S ⋅ n ^ d A d t {\displaystyle {\begin{aligned}\Phi _{\mathrm {e} }&={\frac {dQ_{\mathrm {e} }}{dt}}\\[2pt]Q_{\mathrm {e} }&=\int _{T}\int _{\Sigma }\mathbf {S} \cdot {\hat {\mathbf {n} }}\,dAdt\end{aligned}}} where

- *Q*e is the [radiant energy](/source/Radiant_energy) passing out of a [closed surface](/source/Closed_surface) Σ in time interval T;

- t is time;

- A is the area of the surface Σ;

- **S** is the [Poynting vector](/source/Poynting_vector), representing the *directional* flow of energy per unit time, per unit area;

- **n** is the unit [normal vector](/source/Normal_vector) to the differential area element *dA*.

The rate of energy flow through the surface fluctuates at the frequency of the radiation, but radiation detectors only respond to the average rate of flow. This is represented by replacing the Poynting vector with the time average of its norm, giving Φ e ≈ ∫ Σ ⟨ | S | ⟩ cos ⁡ α d A , {\displaystyle \Phi _{\mathrm {e} }\approx \int _{\Sigma }\langle |\mathbf {S} |\rangle \cos \alpha \ dA,} where ⟨-⟩ is the time average, and α is the angle between **n** and **S**.

### Spectral flux

**Spectral flux in frequency**, denoted Φe,*ν*, is defined as[1] Φ e , ν = ∂ Φ e ∂ ν , {\displaystyle \Phi _{\mathrm {e} ,\nu }={\frac {\partial \Phi _{\mathrm {e} }}{\partial \nu }},} where ν is the frequency.

**Spectral flux in wavelength**, denoted Φe,*λ*, is defined as[1] Φ e , λ = ∂ Φ e ∂ λ , {\displaystyle \Phi _{\mathrm {e} ,\lambda }={\frac {\partial \Phi _{\mathrm {e} }}{\partial \lambda }},} where λ is the wavelength.

## SI radiometry units

Comparison of photometric and radiometric quantities

SI radiometry units v t e Quantity Unit Dimension Notes Name Symbol[nb 1] Name Symbol Radiant energy Qe[nb 2] joule J M⋅L2⋅T−2 Energy of electromagnetic radiation. Radiant energy density we joule per cubic metre J/m3 M⋅L−1⋅T−2 Radiant energy per unit volume. Radiant flux Φe[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 called luminosity in astronomy. Spectral flux Φe,ν[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. Φe,λ[nb 4] watt per metre W/m M⋅L⋅T−3 Radiant intensity Ie,Ω[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,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 a directional quantity. Ie,Ω,λ[nb 4] watt per steradian per metre W⋅sr−1⋅m−1 M⋅L⋅T−3 Radiance Le,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 called "intensity". Spectral radiance Specific intensity Le,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 called "spectral intensity". Le,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 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 called "intensity". Spectral irradiance Spectral flux density Ee,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 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,λ[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 called "intensity". Spectral radiosity Je,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 called "spectral intensity". Je,λ[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 called "intensity". Spectral exitance Me,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 called "spectral intensity". Me,λ[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,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". He,λ[nb 4] joule per square metre, per metre J/m3 M⋅L−1⋅T−2 See also: SI Radiometry Photometry

1. **[^](#cite_ref-note-suffix-e_2-0)** [Standards organizations](/source/Standards_organization) recommend that radiometric [quantities](/source/Physical_quantity) should be denoted with suffix "e" (for "energetic") to avoid confusion with photometric or [photon](/source/Photon) quantities.

1. ^ [***a***](#cite_ref-note-alternative-symbol-radiometric_3-0) [***b***](#cite_ref-note-alternative-symbol-radiometric_3-1) [***c***](#cite_ref-note-alternative-symbol-radiometric_3-2) [***d***](#cite_ref-note-alternative-symbol-radiometric_3-3) [***e***](#cite_ref-note-alternative-symbol-radiometric_3-4) Alternative symbols sometimes seen: W or E for radiant energy, P or F for radiant flux, I for irradiance, W for radiant exitance.

1. ^ [***a***](#cite_ref-note-suffix-nu_4-0) [***b***](#cite_ref-note-suffix-nu_4-1) [***c***](#cite_ref-note-suffix-nu_4-2) [***d***](#cite_ref-note-suffix-nu_4-3) [***e***](#cite_ref-note-suffix-nu_4-4) [***f***](#cite_ref-note-suffix-nu_4-5) [***g***](#cite_ref-note-suffix-nu_4-6) Spectral quantities given per unit [frequency](/source/Frequency) are denoted with suffix "*[ν](/source/%CE%9D)*" (Greek letter [nu](/source/Nu_(Greek)), not to be confused with a letter "v", indicating a photometric quantity.)

1. ^ [***a***](#cite_ref-note-suffix-lambda_5-0) [***b***](#cite_ref-note-suffix-lambda_5-1) [***c***](#cite_ref-note-suffix-lambda_5-2) [***d***](#cite_ref-note-suffix-lambda_5-3) [***e***](#cite_ref-note-suffix-lambda_5-4) [***f***](#cite_ref-note-suffix-lambda_5-5) [***g***](#cite_ref-note-suffix-lambda_5-6) Spectral quantities given per unit [wavelength](/source/Wavelength) are denoted with suffix "*[λ](/source/%CE%9B)*".

1. ^ [***a***](#cite_ref-note-suffix-omega_6-0) [***b***](#cite_ref-note-suffix-omega_6-1) Directional quantities are denoted with suffix "[Ω](/source/%CE%A9)".

## See also

- [Luminous flux](/source/Luminous_flux)

- [Heat flux](/source/Heat_flux)

- [Power (physics)](/source/Power_(physics))

- [Radiosity (heat transfer)](/source/Radiosity_(heat_transfer))

## References

1. ^ [***a***](#cite_ref-ISO_9288-1989_1-0) [***b***](#cite_ref-ISO_9288-1989_1-1) [***c***](#cite_ref-ISO_9288-1989_1-2) ["Thermal insulation — Heat transfer by radiation — Physical quantities and definitions"](http://www.iso.org/iso/home/store/catalogue_tc/catalogue_detail.htm?csnumber=16943). *ISO 9288:1989*. [ISO](/source/International_Organization_for_Standardization) catalogue. 1989. Retrieved 2015-03-15.

## Further reading

- [Boyd, Robert](/source/Robert_W._Boyd_(physicist)) (1983). *Radiometry and the Detection of Optical Radiation (Pure & Applied Optics Series)*. Wiley-Interscience. [ISBN](/source/ISBN_(identifier)) [978-0-471-86188-1](https://en.wikipedia.org/wiki/Special:BookSources/978-0-471-86188-1).

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Adapted from the Wikipedia article [Radiant flux](https://en.wikipedia.org/wiki/Radiant_flux) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Radiant_flux?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.