Irradiance

File:photometry_radiometry_units.svg

Irradiance of a surface, denoted E_e ("e" for "energetic", to avoid confusion with photometric quantities), is defined as{{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=ISO catalogue|year=1989|access-date=2015-03-15}}

:$E\_\backslash mathrm\{e\}\; =\; \backslash frac\{\backslash partial\; \backslash Phi\_\backslash mathrm\{e\}\}\{\backslash partial\; A\},$

where

• ∂ is the partial derivative symbol;
• Φ_e is the radiant flux received;
• A is the area.

The radiant flux emitted by a surface is called radiant exitance.

Spectral irradiance in frequency of a surface, denoted E_e,ν, is defined as

:$E\_\{\backslash mathrm\{e\},\backslash nu\}\; =\; \backslash frac\{\backslash partial\; E\_\backslash mathrm\{e\}\}\{\backslash partial\; \backslash nu\},$

where ν is the frequency.

Spectral irradiance in wavelength of a surface, denoted E_e,λ, is defined as

:$E\_\{\backslash mathrm\{e\},\backslash lambda\}\; =\; \backslash frac\{\backslash partial\; E\_\backslash mathrm\{e\}\}\{\backslash partial\; \backslash lambda\},$

where λ is the wavelength.

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:

:$E\_\backslash mathrm\{e\}\; =\; \backslash langle|\backslash mathbf\{S\}|\backslash rangle\; \backslash cos\; \backslash alpha,$

where

• {{math|⟨ • ⟩}} is the time-average;
• S is the Poynting vector;
• α is the angle between a unit vector normal to the surface and S.

For a propagating sinusoidal 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{{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}}

:$E\_\backslash mathrm\{e\}\; =\; \backslash frac\{n\}\{2\; \backslash mu\_0\; \backslash mathrm\{c\}\}\; E\_\backslash mathrm\{m\}^2\; \backslash cos\; \backslash alpha$

= \frac{n \varepsilon_0 \mathrm{c}}{2} E_\mathrm{m}^2 \cos \alpha \,or

\frac{n }{2Z_0} E_\mathrm{m}^2 \cos \alpha,

where

• E_m 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;

• μ₀ is the vacuum permeability;

• ε₀ is the vacuum permittivity;
• $c=\{\backslash frac\; \{1\}\{\backslash sqrt\; \{\backslash varepsilon\_0\; \backslash mu\_0\; \}\}\}$
• $Z\_0=\backslash mu\_0c$ is the impedance of free space.

This formula assumes that the magnetic susceptibility is negligible; i.e. that μ_r ≈ 1 (μ ≈ μ₀) where μ_r is the relative magnetic permeability of the propagation medium. This assumption is typically valid in transparent media in the optical frequency range.

A point source of light produces spherical wavefronts. The irradiance in this case varies inversely with the square of the distance from the source.

:$$

E = \frac P A = \frac P {4 \pi r^2},

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 71%.

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 L_☉, distance: 4.34 ly) is about 2.7 × 10⁻⁸ W/m² on Earth.

{{main|Solar irradiance}}

The global irradiance on a horizontal surface on Earth consists of the direct irradiance E_e,dir and diffuse irradiance E_e,diff. On a tilted plane, there is another irradiance component, E_e,refl, 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_e on a tilted plane consists of three components:{{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}}

:$E\_\backslash mathrm\{e\}\; =\; E\_\{\backslash mathrm\{e\},\backslash mathrm\{dir\}\}\; +\; E\_\{\backslash mathrm\{e\},\backslash mathrm\{diff\}\}\; +\; E\_\{\backslash mathrm\{e\},\backslash mathrm\{refl\}\}.$

The integral of solar irradiance over a time period is called "solar exposure" or "insolation".{{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 }}

Average solar irradiance at the top of the Earth's atmosphere is roughly 1361 W/m², but at surface irradiance is approximately 1000 W/m² on a clear day.

{{SI radiometry units}}

File:photometry_radiometry_units.svg

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• Albedo
• Fluence
• Illuminance
• Insolation
• Light diffusion
• PI curve (photosynthesis-irradiance curve)
• Solar azimuth angle
• Solar irradiance
• Solar noon
• Spectral flux density
• Stefan–Boltzmann law

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Category:Physical quantities

Category:Radiometry