# Insolation

Insolation (from Latin insolare, to expose to the sun) is the total amount of solar radiation energy received on a given surface area during a given time. It is also called solar irradiation and expressed as "hourly irradiation" if recorded during an hour or "daily irradiation" if recorded during a day. The unit recommended by the World Meteorological Organization is the megajoule per square metre (MJ/m2) or joule per square millimetre (J/mm2). An alternate unit of measure is the Langley (1 thermochemical calorie per square centimeter or 41,840 J/m2). Practitioners in the business of solar energy may use the unit watt-hour per square metre (Wh/m2). If this energy is divided by the recording time in hours, it is then a density of power called irradiance, expressed in watts per square metre (W/m2).

## Absorption and reflection

The object or surface that solar radiation strikes may be a planet, a terrestrial object inside the atmosphere of a planet, or an object exposed to solar rays outside of an atmosphere, such as spacecraft. Some of the radiation will be absorbed and the remainder reflected. Usually the absorbed solar radiation is converted to thermal energy, causing an increase in the object's temperature. Manmade or natural systems, however, may convert a portion of the absorbed radiation into another form, as in the case of photovoltaic cells or plants. The proportion of radiation reflected or absorbed depends on the object's reflectivity or albedo.

## Projection effect One sunbeam one mile wide shines on the ground at a 90° angle, and another at a 30° angle. The oblique sunbeam distributes its light energy over twice as much area.

The insolation into a surface is largest when the surface directly faces the Sun. As the angle increases between the direction at a right angle to the surface and the direction of the rays of sunlight, the insolation is reduced in proportion to the cosine of the angle; see effect of sun angle on climate.

In this illustration, 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 falls on the ground from directly overhead, and another hits the ground at a 30° angle to the horizontal. Trigonometry tells us that the sine of a 30° angle is 1/2, whereas the sine of a 90° angle is 1. Therefore, the sunbeam hitting the ground at a 30° angle spreads the same amount of light over twice as much area (if we imagine the sun shining from the south at noon, the north-south width doubles; the east-west width does not). Consequently, the amount of light falling on each square mile is only half as much.

This 'projection effect' is the main reason why the polar regions are much colder than equatorial regions on Earth. On an annual average the poles receive less insolation than does the equator, because at the poles the Earth's surface are angled away from the Sun.

## 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 with the Earth-Sun distance and solar cycles, the losses depend on the time of day (length of light's path through the atmosphere depending on the Solar elevation angle), cloud cover, moisture content, and other impurities. Insolation is a fundamental abiotic factor affecting the metabolism of plants and the behavior of animals.

Over the course of a year the average solar radiation arriving at the top of the Earth's atmosphere at any point in time is roughly 1366 W/m2  (see solar constant). The radiant power is distributed across the entire electromagnetic spectrum, although most of the power is in the visible light portion of the spectrum. The Sun's rays are attenuated as they pass through the atmosphere, thus reducing the irradiance at the Earth's surface to approximately 1000 W /m2 for a surface perpendicular to the Sun's rays at sea level on a clear day.

The actual figure varies with the Sun angle at different times of year, according to the distance the sunlight travels through the air, and depending on the extent of atmospheric haze and cloud cover. Ignoring clouds, the daily average irradiance for the Earth is approximately 250 W/m2 (i.e., a daily irradiation of 6 kWh/m2), taking into account the lower radiation intensity in early morning and evening, and its near-absence at night.

The insolation of the sun can also be expressed in Suns, where one Sun equals 1000 W/m2 at the point of arrival, with kWh/m2/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 taken into account as well as the insolation. (The insolation, taking into account the attenuation of the atmosphere, should be multiplied by the cosine of the angle between the normal to the panel and the direction of the sun from it). 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 on a different planet, such as Mars.

## Distribution of insolation at the top of the atmosphere Spherical triangle for application of the spherical law of cosines for the calculation the solar zenith angle Θ for observer at latitude φ and longitude λ from knowledge of the hour angle h and solar declination δ. (δ is latitude of subsolar point, and h is relative longitude of subsolar point). ${\overline {Q}}^{\mathrm {day} }$ , the theoretical daily-average insolation at the top of the atmosphere, where θ is the polar angle of the Earth's orbit, and θ = 0 at the vernal equinox, and θ = 90° at the summer solstice; φ is the latitude of the Earth. The calculation assumed conditions appropriate for 2000 A.D.: a solar constant of S0 = 1367 W m−2, obliquity of ε = 23.4398°, longitude of perihelion of ϖ = 282.895°, eccentricity e = 0.016704. Contour labels (green) are in units of W m−2.

The theory for the distribution of solar radiation at the top of the atmosphere concerns how the solar irradiance (the power of solar radiation per unit area) at the top of the atmosphere is determined by the sphericity and orbital parameters of Earth. The theory could be applied to any monodirectional beam of radiation incident onto a rotating sphere, but is most usually applied to sunlight, and in particular for application in numerical weather prediction, and theory for the seasons and the ice ages. The last application is known as Milankovitch cycles.

The derivation of distribution is based on a fundamental identity from spherical trigonometry, the spherical law of cosines:

$\cos(c)=\cos(a)\cos(b)+\sin(a)\sin(b)\cos(C)\,$ 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, we equate the following for use in the spherical law of cosines:

$C=h\,$ $c=\Theta \,$ $a={\tfrac {1}{2}}\pi -\phi \,$ $b={\tfrac {1}{2}}\pi -\delta \,$ $\cos(\Theta )=\sin(\phi )\sin(\delta )+\cos(\phi )\cos(\delta )\cos(h)\,$ The distance of Earth from the sun can be denoted RE, and the mean distance can be denoted R0, which is very close to 1 AU. The insolation onto a plane normal to the solar radiation, at a distance 1 AU from the sun, is the solar constant, 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:

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

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

${\overline {Q}}^{\text{day}}=-{\frac {1}{2\pi }}{\int _{\pi }^{-\pi }Q\,dh}$ Let h0 be the hour angle when Q becomes positive. This could occur at sunrise when $\Theta ={\tfrac {1}{2}}\pi$ , or for h0 as a solution of

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

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

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

$\int _{\pi }^{-\pi }Q\,dh=\int _{h_{o}}^{-h_{o}}Q\,dh=S_{o}{\frac {R_{o}^{2}}{R_{E}^{2}}}\int _{h_{o}}^{-h_{o}}\cos(\Theta )\,dh$ $\int _{\pi }^{-\pi }Q\,dh=S_{o}{\frac {R_{o}^{2}}{R_{E}^{2}}}\left[h\sin(\phi )\sin(\delta )+\cos(\phi )\cos(\delta )\sin(h)\right]_{h=h_{o}}^{h=-h_{o}}$ $\int _{\pi }^{-\pi }Q\,dh=-2S_{o}{\frac {R_{o}^{2}}{R_{E}^{2}}}\left[h_{o}\sin(\phi )\sin(\delta )+\cos(\phi )\cos(\delta )\sin(h_{o})\right]$ ${\overline {Q}}^{\text{day}}={\frac {S_{o}}{\pi }}{\frac {R_{o}^{2}}{R_{E}^{2}}}\left[h_{o}\sin(\phi )\sin(\delta )+\cos(\phi )\cos(\delta )\sin(h_{o})\right]$ Let θ be the conventional polar angle describing a planetary orbit. For convenience, let θ = 0 at the vernal equinox. The declination δ as a function of orbital position is

$\sin \delta =\sin \varepsilon ~\sin(\theta -\varpi )\,$ where ε is the obliquity. The conventional longitude of perihelion ϖ is defined relative to the vernal equinox, so for the elliptical orbit:

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

${\frac {R_{o}}{R_{E}}}={1+e\cos(\theta -\varpi )}$ With knowledge of ϖ, ε and e from astrodynamical calculations  and So from a consensus of observations or theory, ${\overline {Q}}^{\mathrm {day} }$ can be calculated for any latitude φ and θ. Note that because of the elliptical orbit, and as a simple consequence of Kepler's second law, θ does not progress exactly 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 ${\overline {Q}}^{\mathrm {day} }$ 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 simply equal to the obliquity ε. The distance from the sun is

${\frac {R_{o}}{R_{E}}}=1+e\cos(\theta -\varpi )=1+e\cos({\tfrac {\pi }{2}}-\varpi )=1+e\sin(\varpi )$ For this summer solstice calculation, the role of the elliptical orbit is entirely contained within the important product $e\sin(\varpi )$ , which is known as the precession index, the variation of which 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 will be dominant.

## Applications

In spacecraft design and planetology, it is the primary variable affecting equilibrium temperature.

In construction, insolation is an important consideration when designing a building for a particular climate. It is one of the most important climate variables for human comfort and building energy efficiency. Insolation variation by month; 1984-1993 averages for January (top) and April (bottom)

The projection effect can be used in architecture to design buildings that are cool in summer and warm in winter, by providing large 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 noonday Sun is high in the sky. (The Sun's north/south path through the sky spans 47 degrees through the year).

Insolation figures are used as an input to worksheets to size solar power systems for the location where they will be installed. This can be misleading since insolation figures assume the panels are parallel with the ground, when in fact, except in the case of asphalt solar collectors, they are almost always mounted at an angle to face towards the sun. This gives inaccurately low estimates for winter. The figures can be obtained from an insolation map or by city or region from insolation tables that were generated with historical data over the last 30–50 years. Photovoltaic panels are rated under standard conditions to determine the Wp rating (watts peak), which can then be used with the insolation of a region to determine the expected output, along with other 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.

In the fields of 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.

Conversion factor (multiply top row by factor to obtain side column)
W/m2 kW·h/(m2·day) sun hours/day kWh/(m2·y) kWh/(kWp·y)
W/m2 1 41.66666 41.66666 0.1140796 0.1521061
kW·h/(m2·day) 0.024 1 1 0.0027379 0.0036505
sun hours/day 0.024 1 1 0.0027379 0.0036505
kWh/(m2·y) 8.765813 365.2422 365.2422 1 1.333333
kWh/(kWp·y) 6.574360 273.9316 273.9316 0.75 1