View source for Irradiance

{{Short description|Measure of radiant energy over surface area}}
In [[radiometry]], '''irradiance''' is the [[radiant flux]] ''received'' by a ''surface'' per unit area.  The [[International System of Units|SI unit]] of irradiance is the [[watt]] per square metre (W⋅m<sup>−2</sup>). The [[Centimetre–gram–second system of units|CGS unit]] [[erg]] per square centimetre per second (erg⋅cm<sup>−2</sup>⋅s<sup>−1</sup>) is often used in [[astronomy]]. Irradiance is often called [[Intensity (physics)|intensity]], but this term is avoided in radiometry where such usage leads to confusion with [[radiant intensity]]. In astrophysics, irradiance is called ''radiant flux''.<ref>{{Cite book |title=An introduction to modern astrophysics |last=Carroll |first=Bradley W. |isbn=978-1-108-42216-1 |oclc=991641816 |page=60|date = 2017-09-07}}</ref>

'''Spectral irradiance''' is the irradiance of a surface per unit [[frequency]] or [[wavelength]], depending on whether the [[Spectral radiometric quantity|spectrum]] is taken as a function of frequency or of wavelength. The two forms have different [[dimensional analysis|dimensions]] and units: spectral irradiance of a frequency spectrum is measured in watts per square metre per [[hertz]] (W⋅m<sup>−2</sup>⋅Hz<sup>−1</sup>), while spectral irradiance of a wavelength spectrum is measured in watts per square metre per metre (W⋅m<sup>−3</sup>), or more commonly watts per square metre per nanometre (W⋅m<sup>−2</sup>⋅nm<sup>−1</sup>).

==Mathematical definitions==
===Irradiance===
Irradiance of a surface, denoted ''E''<sub>e</sub> ("e" for "energetic", to avoid confusion with [[photometry (optics)|photometric]] quantities), is defined as<ref name="ISO_9288-1989">{{cite web|url=http://www.iso.org/iso/home/store/catalogue_tc/catalogue_detail.htm?csnumber=16943|title=Thermal insulation — Heat transfer by radiation — Physical quantities and definitions|work=ISO 9288:1989|publisher=[[International Organization for Standardization|ISO]] catalogue|year=1989|access-date=2015-03-15}}</ref>
:<math>E_\mathrm{e} = \frac{\partial \Phi_\mathrm{e}}{\partial A},</math>
where
*∂ is the [[partial derivative]] symbol;
*Φ<sub>e</sub> is the radiant flux received;
*''A'' is the area.

If we want to talk about the radiant flux ''emitted'' by a surface, we speak of [[radiant exitance]].

===Spectral irradiance===
Spectral irradiance in frequency of a surface, denoted ''E''<sub>e,ν</sub>, is defined as<ref name="ISO_9288-1989" />
:<math>E_{\mathrm{e},\nu} = \frac{\partial E_\mathrm{e}}{\partial \nu},</math>
where ''ν'' is the frequency.

Spectral irradiance in wavelength of a surface, denoted ''E''<sub>e,λ</sub>, is defined as<ref name="ISO_9288-1989" />
:<math>E_{\mathrm{e},\lambda} = \frac{\partial E_\mathrm{e}}{\partial \lambda},</math>
where ''λ'' is the wavelength.

==Property==
Irradiance of a surface is also, according to the definition of [[radiant flux]], equal to the time-average of the component of the [[Poynting vector]] perpendicular to the surface:
:<math>E_\mathrm{e} = \langle|\mathbf{S}|\rangle \cos \alpha,</math>
where
*{{math|⟨ • ⟩}} is the time-average;
*'''S''' is the Poynting vector;
*''α'' is the angle between a unit vector [[Normal (geometry)|normal]] to the surface and '''S'''.

For a propagating ''sinusoidal'' [[Linear polarisation|linearly polarized]] electromagnetic [[plane wave]], the Poynting vector always points to the direction of propagation while oscillating in magnitude. The irradiance of a surface is then given by<ref name=griffiths>{{cite book|last=Griffiths|first=David J.|title=Introduction to electrodynamics|date=1999|publisher=[[Prentice-Hall]]|location=Upper Saddle River, NJ [u.a.]|isbn=0-13-805326-X|url=https://archive.org/details/introductiontoel00grif_0|edition=3. ed., reprint. with corr.|url-access=registration}}</ref>
:<math>E_\mathrm{e} = \frac{n}{2 \mu_0 \mathrm{c}} E_\mathrm{m}^2 \cos \alpha 
= \frac{n \varepsilon_0 \mathrm{c}}{2} E_\mathrm{m}^2 \cos \alpha
= \frac{n }{2Z_0} E_\mathrm{m}^2 \cos \alpha,</math>
where
*''E''<sub>m</sub> is the amplitude of the wave's electric field;
*''n'' is the [[refractive index]] of the medium of propagation;
*''c'' is the [[speed of light]] in [[vacuum]];
*μ<sub>0</sub> is the [[vacuum permeability]];
*ε<sub>0</sub> is the [[vacuum permittivity]];
*Z<sub>0</sub> is the [[impedance of free space]].

This formula assumes that the [[magnetic susceptibility]] is negligible; i.e. that ''μ''<sub>r</sub> ≈ 1 where ''μ''<sub>r</sub> is the [[magnetic permeability]] of the propagation medium. This assumption is typically valid in transparent media in the [[visible spectrum|optical frequency range]].

==Point source==
A [[point source]] of light produces spherical wavefronts. The irradiance in this case varies inversely with the square of the distance from the source.
:<math>
E = \frac P A = \frac P {4 \pi r^2}. \,
</math>
where
*{{mvar|r}} is the distance;
*{{mvar|P}} is the [[radiant flux]];
*{{mvar|A}} is the surface area of a sphere of radius {{mvar|r}}.

For quick approximations, this equation indicates that doubling the distance reduces irradiation to one quarter; or similarly, to double irradiation, reduce the distance to 0.7.

In astronomy, stars are routinely treated as point sources even though they are much larger than the Earth. This is a good approximation because the distance from even a nearby star to the Earth is much larger than the star's diameter. For instance, the irradiance of [[Alpha Centauri A]] (radiant flux: 1.5 [[Solar luminosity|L<sub>☉</sub>]], distance: 4.34 [[light-year|ly]]) is about 2.7 × 10<sup>−8</sup> W/m<sup>2</sup> on Earth.

==Solar irradiance==
{{main|Solar irradiance}}
The global irradiance on a horizontal surface on Earth consists of the direct irradiance ''E''<sub>e,dir</sub> and diffuse irradiance ''E''<sub>e,diff</sub>. On a tilted plane, there is another irradiance component, ''E''<sub>e,refl</sub>, which is the component that is reflected from the ground. The average ground reflection is about 20% of the global irradiance. Hence, the irradiance ''E''<sub>e</sub> on a tilted plane consists of three components:<ref name=Quaschning>{{cite journal |last=Quaschning |first=Volker |author-link=Volker Quaschning |title=Technology fundamentals—The sun as an energy resource |journal=Renewable Energy World |volume=6 |date=2003 |issue=5 |pages=90–93 |url=http://www.volker-quaschning.de/articles/fundamentals1/index_e.html}}</ref>
:<math>E_\mathrm{e} = E_{\mathrm{e},\mathrm{dir}} + E_{\mathrm{e},\mathrm{diff}} + E_{\mathrm{e},\mathrm{refl}}.</math>

The [[integral]] of solar irradiance over a time period is called "[[Radiant exposure|solar exposure]]" or "[[insolation]]".<ref name=Quaschning/><ref>{{Cite journal | last1 = Liu | first1 = B. Y. H. | last2 = Jordan | first2 = R. C. | doi = 10.1016/0038-092X(60)90062-1 | title = The interrelationship and characteristic distribution of direct, diffuse and total solar radiation | journal = Solar Energy | volume = 4 | issue = 3 | pages = 1 | year = 1960 |bibcode = 1960SoEn....4....1L }}</ref>

==SI radiometry units==
{{SI radiometry units}}
[[File:photometry_radiometry_units.svg|thumb|upright=1.5|Comparison of photometric and radiometric quantities]]

==See also==
{{Div col|small=yes}}
*[[Albedo]]
*[[Fluence]]
*[[Illuminance]]
*[[Insolation]]
*[[Light diffusion]]
*[[PI curve]] (photosynthesis-irradiance curve)
*[[Solar azimuth angle]]
*[[Solar irradiance]]
*[[Solar noon]]
*[[Spectral flux density]]
*[[Stefan–Boltzmann law]]
{{Div col end}}

==References==
{{Reflist}}

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[[Category:Physical quantities]]
[[Category:Radiometry]]