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Insolation (from Latin insolare, to expose to the sun) is the power per unit area produced by the Sun in the form of electromagnetic radiation. It is also called solar irradiation.

Units

The unit recommended by the World Meteorological Organization is the megajoule per square metre (MJ/m) or joule per square millimetre (J/mm).

An alternate unit of measure is the Langley (1 thermochemical calorie per square centimeter or 41,840 J/m) or irradiance per unit time.

The solar energy business uses watt-hour per square metre (Wh/m). Divided by the recording time, this measure becomes insolation, another unit of irradiance.

Insolation can be measured in space, at the edge of the atmosphere or at a terrestrial object.

Absorption and reflection

Reaching an object, part of the irradiance is absorbed and the remainder reflected. Usually the absorbed radiation is converted to thermal energy, increasing the object's temperature. Manmade or natural systems, however, can convert part of the absorbed radiation into another form such as electricity or chemical bonds, as in the case of photovoltaic cells or plants. The proportion of reflected radiation is the object's reflectivity or albedo.

Projection effect

Insolation onto a surface is largest when the surface directly faces (is normal to) the sun. As the angle between the surface and the Sun moves from normal, the insolation is reduced in proportion to the angle's cosine; see effect of sun angle on climate.

In the figure, the angle shown is between the ground and the sunbeam rather than between the vertical direction and the sunbeam; hence the sine rather than the cosine is appropriate. A sunbeam one mile (1.6 km) wide arrives from directly overhead, and another at a 30° angle to the horizontal. The sine of a 30° angle is 1/2, whereas the sine of a 90° angle is 1. Therefore, the angled sunbeam spreads the light over twice the area. Consequently, half as much light falls on each square mile.

This 'projection effect' is the main reason why Earth's polar regions are much colder than equatorial regions. On an annual average the poles receive less insolation than does the equator, because the poles are always angled more away from the sun than the tropics. At a lower angle the light must travel through more atmosphere. This attenuates it (by absorption and scattering) further reducing insolation.

Earth's insolation

Direct insolation is the solar irradiance measured at a given location on Earth with a surface element perpendicular to the Sun's rays, excluding diffuse insolation (the solar radiation that is scattered or reflected by atmospheric components in the sky). Direct insolation is equal to the solar constant minus the atmospheric losses due to absorption and scattering. While the solar constant varies, losses depend on time of day (length of light's path through the atmosphere depending on the Solar elevation angle), cloud cover, moisture content and other contents. Insolation affects plant metabolism and animal behavior.

Average annual solar radiation arriving at the top of the Earth's atmosphere is roughly 1366 W/m (see solar constant). The radiant power is distributed across the electromagnetic spectrum, although most is visible light. The Sun's rays are attenuated as they pass through the atmosphere, thus reducing maximum normal surface irradiance to approximately 1000 W /m at sealevel on a clear day.

The actual figure varies with the Sun's angle and atmospheric circumstances. Ignoring clouds, the daily average irradiance for the Earth is approximately 250 W/m/hr = 6 kWh/m.

Insolation can also be expressed in Suns, where one Sun equals 1000 W/m at the point of arrival, with kWh/m/day expressed as hours/day. When calculating the output of, for example, a photovoltaic panel, the angle of the sun relative to the panel needs to be considered. One Sun is a unit of power flux, not a standard value for actual insolation. Sometimes this unit is referred to as a Sol, not to be confused with a sol, meaning one solar day.

Solar potential maps

• North America
• South America
• Europe
• Africa and Middle East
• South and South-East Asia
• Australia

Top of the atmosphere

The distribution of solar radiation at the top of the atmosphere is determined by Earth's sphericity and orbital parameters. This applies to any unidirectional beam incident to a rotating sphere. Insolation is essential for numerical weather prediction and understanding seasons and climate change. Application to ice ages is known as Milankovitch cycles.

Distribution is based on a fundamental identity from spherical trigonometry, the spherical law of cosines:

${\displaystyle \cos(c)=\cos(a)\cos(b)+\sin(a)\sin(b)\cos(C)\,}$

where a, b and c are arc lengths, in radians, of the sides of a spherical triangle. C is the angle in the vertex opposite the side which has arc length c. Applied to the calculation of solar zenith angle Θ, the following applies to the spherical law of cosines:

${\displaystyle C=h\,}$

${\displaystyle c=\Theta \,}$

${\displaystyle a={\tfrac {1}{2}}\pi -\phi \,}$

${\displaystyle b={\tfrac {1}{2}}\pi -\delta \,}$

${\displaystyle \cos(\Theta )=\sin(\phi )\sin(\delta )+\cos(\phi )\cos(\delta )\cos(h)\,}$

The separation of Earth from the sun can be denoted R_E and the mean distance can be denoted R₀, approximately 1 AU. The solar constant is denoted S₀. The solar flux density (insolation) onto a plane tangent to the sphere of the Earth, but above the bulk of the atmosphere (elevation 100 km or greater) is:

${\displaystyle Q=S_{o}{\frac {R_{o}^{2}}{R_{E}^{2}}}\cos(\Theta ){\text{ when }}\cos(\Theta )>0}$

and

${\displaystyle Q=0{\text{ when }}\cos(\Theta )\leq 0\,}$

The average of Q over a day is the average of Q over one rotation, or the hour angle progressing from h = π to h = −π:

${\displaystyle {\overline {Q}}^{\text{day}}=-{\frac {1}{2\pi }}{\int _{\pi }^{-\pi }Q\,dh}}$

Let h₀ be the hour angle when Q becomes positive. This could occur at sunrise when ${\displaystyle \Theta ={\tfrac {1}{2}}\pi }$, or for h₀ as a solution of

${\displaystyle \sin(\phi )\sin(\delta )+\cos(\phi )\cos(\delta )\cos(h_{o})=0\,}$

or

${\displaystyle \cos(h_{o})=-\tan(\phi )\tan(\delta )}$

If tan(φ)tan(δ) > 1, then the sun does not set and the sun is already risen at h = π, so h_o = π. If tan(φ)tan(δ) < −1, the sun does not rise and ${\displaystyle {\overline {Q}}^{\mathrm {day} }=0}$.

${\displaystyle {\frac {R_{o}^{2}}{R_{E}^{2}}}}$ is nearly constant over the course of a day, and can be taken outside the integral

${\displaystyle \int _{\pi }^{-\pi }Q\,dh=\int _{h_{o}}^{-h_{o}}Q\,dh=S_{o}{\frac {R_{o}^{2}}{R_{E}^{2}}}\int _{h_{o}}^{-h_{o}}\cos(\Theta )\,dh}$

${\displaystyle \int _{\pi }^{-\pi }Q\,dh=S_{o}{\frac {R_{o}^{2}}{R_{E}^{2}}}\left_{h=h_{o}}^{h=-h_{o}}}$

${\displaystyle \int _{\pi }^{-\pi }Q\,dh=-2S_{o}{\frac {R_{o}^{2}}{R_{E}^{2}}}\left}$

${\displaystyle {\overline {Q}}^{\text{day}}={\frac {S_{o}}{\pi }}{\frac {R_{o}^{2}}{R_{E}^{2}}}\left}$

Let θ be the conventional polar angle describing a planetary orbit. Let θ = 0 at the vernal equinox. The declination δ as a function of orbital position is

${\displaystyle \sin \delta =\sin \varepsilon ~\sin(\theta -\varpi )\,}$

where ε is the obliquity. The conventional longitude of perihelion ϖ is defined relative to the vernal equinox, so for the elliptical orbit:

${\displaystyle R_{E}={\frac {R_{o}}{1+e\cos(\theta -\varpi )}}}$

or

${\displaystyle {\frac {R_{o}}{R_{E}}}={1+e\cos(\theta -\varpi )}}$

With knowledge of ϖ, ε and e from astrodynamical calculations and S_o from a consensus of observations or theory, ${\displaystyle {\overline {Q}}^{\mathrm {day} }}$can be calculated for any latitude φ and θ. Because of the elliptical orbit, and as a consequence of Kepler's second law, θ does not progress uniformly with time. Nevertheless, θ = 0° is exactly the time of the vernal equinox, θ = 90° is exactly the time of the summer solstice, θ = 180° is exactly the time of the autumnal equinox and θ = 270° is exactly the time of the winter solstice.

Application to Milankovitch cycles

Obtaining a time series for a ${\displaystyle {\overline {Q}}^{\mathrm {day} }}$ for a particular time of year, and particular latitude, is a useful application in the theory of Milankovitch cycles. For example, at the summer solstice, the declination δ is equal to the obliquity ε. The distance from the sun is

${\displaystyle {\frac {R_{o}}{R_{E}}}=1+e\cos(\theta -\varpi )=1+e\cos({\tfrac {\pi }{2}}-\varpi )=1+e\sin(\varpi )}$

For this summer solstice calculation, the role of the elliptical orbit is entirely contained within the important product ${\displaystyle e\sin(\varpi )}$, the precession index, whose variation dominates the variations in insolation at 65° N when eccentricity is large. For the next 100,000 years, with variations in eccentricity being relatively small, variations in obliquity dominate.

Applications

Buildings

In construction, insolation is an important consideration when designing a building for a particular site.

The projection effect can be used to design buildings that are cool in summer and warm in winter, by providing vertical windows on the equator-facing side of the building (the south face in the northern hemisphere, or the north face in the southern hemisphere): this maximizes insolation in the winter months when the Sun is low in the sky and minimizes it in the summer when the Sun is high. (The Sun's north/south path through the sky spans 47 degrees through the year).

Solar power

Insolation figures are used as an input to worksheets to size solar power systems. Because (except for asphalt solar collectors) panels are almost always mounted at an angle towards the sun, insolation must be adjusted to prevent estimates that are inaccurately low for winter and inaccurately high for summer. In many countries the figures can be obtained from an insolation map or from insolation tables that reflect data over the prior 30–50 years. Photovoltaic panels are rated under standard conditions to determine the Wp rating (watts peak), which can then be used with insolation to determine the expected output, adjusted by factors such as tilt, tracking and shading (which can be included to create the installed Wp rating). Insolation values range from 800 to 950 kWh/(kWp·y) in Norway to up to 2,900 in Australia.

Other

Insolation is the primary variable affecting equilibrium temperature in spacecraft design and planetology.

In civil engineering and hydrology, numerical models of snowmelt runoff use observations of insolation. This permits estimation of the rate at which water is released from a melting snowpack. Field measurement is accomplished using a pyranometer.

See also

• Albedo
• Earth's energy budget
• Flux
• Power density
• Sky footage
• Sun chart
• Sunlight

References

1. "Insolation - Definition of insolation by Merriam-Webster". merriam-webster.com.
2. "Online Etymology Dictionary". etymonline.com.
3. WMO Webteam. "World Meteorological Organization - WMO" (PDF). wmo.int.
4. C.Michael Hogan. 2010. Abiotic factor. Encyclopedia of Earth. eds Emily Monosson and C. Cleveland. National Council for Science and the Environment. Washington DC
5. Satellite observations of total solar irradiance
6. "Construction of a Composite Total Solar Irradiance (TSI) Time Series from 1978 to present". Figure 4 & figure 5. Retrieved February 2, 2009.
7. U.S. Solar Radiation Resource Maps retrieved 29 October 2012
8. Michael Allison; Robert Schmunk (5 August 2008). "Technical Notes on Mars Solar Time". NASA. Retrieved 16 January 2012. {{cite web}}: Unknown parameter |last-author-amp= ignored (|name-list-style= suggested) (help)
10. Nall, D. H. "Looking across the water: Climate-adaptive buildings in the United States & Europe" (PDF). The Construction Specifier. 57 (2004–11): 50–56.
11. "Determining your solar power requirements and planning the number of components".
12. "Asphalt Solar Collector Renewable Heat for IHT - Solar Collectors - Solar Recharge for GSHP - Pavement Solar Collectors - Road Solar Thermal Collector". icax.co.uk.
13. "Optimum solar panel angle". macslab.com.
14. "Heliostat Concepts". redrok.com.
16. How Do Solar Panels Work?

External links

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• San Francisco Solar Map
• European Commission- Interactive Maps
• Yesterday‘s Australian Solar Radiation Map
• Net surface solar radiation
• Maps of Solar Radiation
• Solar Radiation using Google Maps
• Sample Calculations based on US Insolation Map
• Solar Radiation on a Tilted Collector (U.S.A. only) choose "Theoretically Perfect Collector" to receive results for the insolation on a tilted surface
• Annual Optimal Orientation of Fixed Tilt Solar Collectors (U.S.A. only)
• SMARTS, software to compute solar insolation of each date/location of earth
• Solar Radiation and Clouds - A Discussion
• NASA Surface meteorology and Solar Energy
• insol: R package for insolation on complex terrain
• Online insolation calculator

• Atmospheric radiation
• Photovoltaics