Insolation is a measure of solar radiation energy received on a given surface area in a given time. It is commonly expressed as average
irradianceIrradiance is the power of electromagnetic radiation per unit area incident on a surface. Radiant emittance or radiant exitance is the power per unit area radiated by a surface. The SI units for all of these quantities are watts per square meter , while the cgs units are ergs per square centimeter...
in watts per square meter (W/m
^{2}) or kilowatthours per square meter per day (kW·h/(m
^{2}·day)) (or hours/day). In the case of
photovoltaicsPhotovoltaics is a method of generating electrical power by converting solar radiation into direct current electricity using semiconductors that exhibit the photovoltaic effect. Photovoltaic power generation employs solar panels composed of a number of solar cells containing a photovoltaic material...
it is commonly measured as kWh/(kW
_{p}·y) (kilowatt hours per year per kilowatt peak rating).
The object or surface that solar radiation strikes may be a planet, a terrestrial object inside the atmosphere of a planet, or any object exposed to solar rays outside of an atmosphere, including
spacecraftA spacecraft or spaceship is a craft or machine designed for spaceflight. Spacecraft are used for a variety of purposes, including communications, earth observation, meteorology, navigation, planetary exploration and transportation of humans and cargo....
. Some of the solar radiation will be absorbed, while the remainder will be reflected. Usually the absorbed solar radiation is converted to thermal energy, causing an increasing in the object's temperature. Some systems, however, may store or convert a portion of the solar energy into another form of energy, as in the case of photovoltaics or plants. The proportion of radiation reflected or absorbed depends on the object's
reflectivityIn optics and photometry, reflectivity is the fraction of incident radiation reflected by a surface. In general it must be treated as a directional property that is a function of the reflected direction, the incident direction, and the incident wavelength...
or
albedoAlbedo , or reflection coefficient, is the diffuse reflectivity or reflecting power of a surface. It is defined as the ratio of reflected radiation from the surface to incident radiation upon it...
.
Projection effect
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 climateThe amount of heat energy received at any location on the globe is a direct effect of sun angle on climate, as the angle at which sunlight strikes the Earth varies by location, time of day, and season due to the Earth's orbit around the sun and the Earth's rotation around its tilted axis...
.
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.
TrigonometryTrigonometry is a branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves...
tells us that the
sineIn mathematics, the sine function is a function of an angle. In a right triangle, sine gives the ratio of the length of the side opposite to an angle to the length of the hypotenuse.Sine is usually listed first amongst the trigonometric functions....
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
noonNoon is usually defined as 12 o'clock in the daytime. The word noon is also used informally to mean midday regarding the location of the sun not the middle of a persons day. Although this is a time around the middle of the day when people in many countries take a lunch break...
, the northsouth width doubles; the eastwest 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 regionEarth's polar regions are the areas of the globe surrounding the poles also known as frigid zones. The North Pole and South Pole being the centers, these regions are dominated by the polar ice caps, resting respectively on the Arctic Ocean and the continent of Antarctica...
s 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 insolationDirect 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 . Direct insolation is equal to the solar constant minus the atmospheric losses due to absorption and scattering...
is the solar
irradianceIrradiance is the power of electromagnetic radiation per unit area incident on a surface. Radiant emittance or radiant exitance is the power per unit area radiated by a surface. The SI units for all of these quantities are watts per square meter , while the cgs units are ergs per square centimeter...
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 constantThe solar constant, a measure of flux density, is the amount of incoming solar electromagnetic radiation per unit area that would be incident on a plane perpendicular to the rays, at a distance of one astronomical unit...
minus the atmospheric losses due to
absorptionIn physics, absorption of electromagnetic radiation is the way by which the energy of a photon is taken up by matter, typically the electrons of an atom. Thus, the electromagnetic energy is transformed to other forms of energy for example, to heat. The absorption of light during wave propagation is...
and
scatteringLight scattering is a form of scattering in which light is the form of propagating energy which is scattered. Light scattering can be thought of as the deflection of a ray from a straight path, for example by irregularities in the propagation medium, particles, or in the interface between two media...
. While the solar constant varies with the
EarthSun distanceIn astronomy, the Earth's orbit is the motion of the Earth around the Sun, at an average distance of about 150 million kilometers, every 365.256363 mean solar days .A solar day is on average 24 hours; it takes 365.256363 of these to orbit the sun once in the sense of returning...
and
solar cycleThe solar cycle, or the solar magnetic activity cycle, is a periodic change in the amount of irradiation from the Sun that is experienced on Earth. It has a period of about 11 years, and is one component of solar variation, the other being aperiodic fluctuations. Solar variation causes changes in...
s, the losses depend on the time of day (length of light's path through the atmosphere depending on the
Solar elevation angleThe solar elevation angle is the elevation angle of the sun. That is, the angle between the directionof the geometric center of the sun's apparent disk and the horizon...
),
cloud coverCloud cover refers to the fraction of the sky obscured by clouds when observed from a particular location...
,
moistureHumidity is the amount of moisture the air can hold before it rains. Moisture refers to the presence of a liquid, especially water, often in trace amounts...
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 1,366
wattThe watt is a derived unit of power in the International System of Units , named after the Scottish engineer James Watt . The unit, defined as one joule per second, measures the rate of energy conversion.Definition:...
s per square meter (see
solar constantThe solar constant, a measure of flux density, is the amount of incoming solar electromagnetic radiation per unit area that would be incident on a plane perpendicular to the rays, at a distance of one astronomical unit...
). The radiant power is distributed across the entire
electromagnetic spectrumThe electromagnetic spectrum is the range of all possible frequencies of electromagnetic radiation. The "electromagnetic spectrum" of an object is the characteristic distribution of electromagnetic radiation emitted or absorbed by that particular object....
, although most of the power is in the visible light portion of the spectrum. The Sun's rays are
attenuatedIn physics, attenuation is the gradual loss in intensity of any kind of flux through a medium. For instance, sunlight is attenuated by dark glasses, Xrays are attenuated by lead, and light and sound are attenuated by water.In electrical engineering and telecommunications, attenuation affects the...
as they pass through the
atmosphereAn atmosphere is a layer of gases that may surround a material body of sufficient mass, and that is held in place by the gravity of the body. An atmosphere may be retained for a longer duration, if the gravity is high and the atmosphere's temperature is low...
, thus reducing the insolation at the Earth's surface to approximately 1,000 watts per square meter 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
sunlightSunlight, in the broad sense, is the total frequency spectrum of electromagnetic radiation given off by the Sun. On Earth, sunlight is filtered through the Earth's atmosphere, and solar radiation is obvious as daylight when the Sun is above the horizon.When the direct solar radiation is not blocked...
travels through the air, and depending on the extent of atmospheric haze and cloud cover. Ignoring clouds, the average insolation for the Earth is approximately 250 watts per square meter (6 (kW·h/m
^{2})/day), taking into account the lower radiation intensity in early morning and evening, and its nearabsence at night.
The insolation of the sun can also be expressed in Suns, where one Sun equals 1,000 W/m
^{2} at the point of arrival, with kWh/(m
^{2}·day) displayed 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 fluxIn the various subfields of physics, there exist two common usages of the term flux, both with rigorous mathematical frameworks.* In the study of transport phenomena , flux is defined as flow per unit area, where flow is the movement of some quantity per time...
, 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, for example, a different planet, such as Mars.
Distribution of insolation at the top of the atmosphere
The theory for the
distribution of solar radiation at the top of the atmosphere concerns how the solar
irradianceIrradiance is the power of electromagnetic radiation per unit area incident on a surface. Radiant emittance or radiant exitance is the power per unit area radiated by a surface. The SI units for all of these quantities are watts per square meter , while the cgs units are ergs per square centimeter...
(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 predictionNumerical weather prediction uses mathematical models of the atmosphere and oceans to predict the weather based on current weather conditions. Though first attempted in the 1920s, it was not until the advent of computer simulation in the 1950s that numerical weather predictions produced realistic...
, and theory for the seasons and the ice ages. The last application is known as
Milankovitch cyclesMilankovitch theory describes the collective effects of changes in the Earth's movements upon its climate, named after Serbian civil engineer and mathematician Milutin Milanković, who worked on it during First World War internment...
.
The derivation of distribution is based on a fundamental identity from
spherical trigonometrySpherical trigonometry is a branch of spherical geometry which deals with polygons on the sphere and the relationships between the sides and the angles...
, the spherical law of cosines:
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 angleZenith Angle can refer to:* In astronomy, the angle made between the surface of the Earth and a line between the observer and the observed * The Zenith Angle is a science fiction novel authored by Bruce Sterling...
Θ, we equate the following for use in the spherical law of cosines:
The distance of Earth from the sun can be denoted R
_{E}, and the mean distance can be denoted R
_{0}, which is very close to 1
AUAn astronomical unit is a unit of length equal to about or approximately the mean Earth–Sun distance....
. The insolation onto a plane normal to the solar radiation, at a distance 1 AU from the sun, is the
solar constantThe solar constant, a measure of flux density, is the amount of incoming solar electromagnetic radiation per unit area that would be incident on a plane perpendicular to the rays, at a distance of one astronomical unit...
, denoted S
_{0}. 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:
and
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:
Let
h_{0} be the hour angle when Q becomes positive. This could occur at sunrise when
, or for
h_{0} as a solution of
or
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
.
is nearly constant over the course of a day, and can be taken outside the integral
Let θ be the conventional polar angle describing a planetary
orbitIn physics, an orbit is the gravitationally curved path of an object around a point in space, for example the orbit of a planet around the center of a star system, such as the Solar System...
. For convenience, let
θ = 0 at the vernal
equinoxAn equinox occurs twice a year, when the tilt of the Earth's axis is inclined neither away from nor towards the Sun, the center of the Sun being in the same plane as the Earth's equator...
. The
declinationIn astronomy, declination is one of the two coordinates of the equatorial coordinate system, the other being either right ascension or hour angle. Declination in astronomy is comparable to geographic latitude, but projected onto the celestial sphere. Declination is measured in degrees north and...
δ as a function of orbital position is
where ε is the obliquity. The conventional longitude of perihelion ϖ is defined relative to the vernal equinox, so for the elliptical orbit:
or
With knowledge of ϖ, ε and
e from astrodynamical calculations and S
_{o} from a consensus of observations or theory,
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
for a particular time of year, and particular latitude, is a useful application in the theory of
Milankovitch cyclesMilankovitch theory describes the collective effects of changes in the Earth's movements upon its climate, named after Serbian civil engineer and mathematician Milutin Milanković, who worked on it during First World War internment...
. For example, at the summer solstice, the declination δ is simply equal to the obliquity ε. The distance from the sun is
For this summer solstice calculation, the role of the elliptical orbit is entirely contained within the important product
,
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
spacecraftA spacecraft or spaceship is a craft or machine designed for spaceflight. Spacecraft are used for a variety of purposes, including communications, earth observation, meteorology, navigation, planetary exploration and transportation of humans and cargo....
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.
The projection effect can be used in
architectureArchitecture is both the process and product of planning, designing and construction. Architectural works, in the material form of buildings, are often perceived as cultural and political symbols and as works of art...
to design buildings that are cool in summer and warm in winter, by providing large vertical windows on the equatorfacing side of the building (the south face in the
northern hemisphereThe Northern Hemisphere is the half of a planet that is north of its equator—the word hemisphere literally means “half sphere”. It is also that half of the celestial sphere north of the celestial equator...
, or the north face in the
southern hemisphereThe Southern Hemisphere is the part of Earth that lies south of the equator. The word hemisphere literally means 'half ball' or "half sphere"...
): 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 pathIn astronomy, an analemma is a curve representing the angular offset of a celestial body from its mean position on the celestial sphere as viewed from another celestial body relative to the viewing body's celestial equator...
through the sky spans 47 degrees through the year).
Insolation figures are used as an input to worksheets to size
solar power systemsSolar energy, radiant light and heat from the sun, has been harnessed by humans since ancient times using a range of everevolving technologies. Solar radiation, along with secondary solarpowered resources such as wind and wave power, hydroelectricity and biomass, account for most of the available...
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 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
NorwayNorway , officially the Kingdom of Norway, is a Nordic unitary constitutional monarchy whose territory comprises the western portion of the Scandinavian Peninsula, Jan Mayen, and the Arctic archipelago of Svalbard and Bouvet Island. Norway has a total area of and a population of about 4.9 million...
to up to 2,900 in
AustraliaAustralia , officially the Commonwealth of Australia, is a country in the Southern Hemisphere comprising the mainland of the Australian continent, the island of Tasmania, and numerous smaller islands in the Indian and Pacific Oceans. It is the world's sixthlargest country by total area...
.
In the fields of
civil engineeringCivil engineering is a professional engineering discipline that deals with the design, construction, and maintenance of the physical and naturally built environment, including works like roads, bridges, canals, dams, and buildings...
and
hydrologyHydrology is the study of the movement, distribution, and quality of water on Earth and other planets, including the hydrologic cycle, water resources and environmental watershed sustainability...
, 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
pyranometerA pyranometer is a type of actinometer used to measure broadband solar irradiance on a planar surface and is a sensor that is designed to measure the solar radiation flux density from a field of view of 180 degrees...
.
Conversion factor (multiply top row by factor to obtain side column) 

W/m^{2} 
kW·h/(m^{2}·day) 
sun hours/day 
kWh/(m^{2}·y) 
kWh/(kWp·y) 
W/m^{2} 
1 
41.66666 
41.66666 
0.1140796 
0.1521061 
kW·h/(m^{2}·day) 
0.024 
1 
1 
0.0027379 
0.0036505 
sun hours/day 
0.024 
1 
1 
0.0027379 
0.0036505 
kWh/(m^{2}·y) 
8.765813 
365.2422 
365.2422 
1 
1.333333 
kWh/(kWp·y) 
6.574360 
273.9316 
273.9316 
0.75 
1 
See also
 Albedo
Albedo , or reflection coefficient, is the diffuse reflectivity or reflecting power of a surface. It is defined as the ratio of reflected radiation from the surface to incident radiation upon it...
 Flux
In the various subfields of physics, there exist two common usages of the term flux, both with rigorous mathematical frameworks.* In the study of transport phenomena , flux is defined as flow per unit area, where flow is the movement of some quantity per time...
 Power density
Power density is the amount of power per unit volume....
 Sunlight
Sunlight, in the broad sense, is the total frequency spectrum of electromagnetic radiation given off by the Sun. On Earth, sunlight is filtered through the Earth's atmosphere, and solar radiation is obvious as daylight when the Sun is above the horizon.When the direct solar radiation is not blocked...
 Sun chart
A Sun chart is a graph of the ecliptic of the Sun through the sky throughout the year at a particular latitude.Most sun charts plot azimuth versus altitude throughout the days of the winter solstice and summer solstice, as well as a number of intervening days...
 Earth's energy budget
The Earth can be considered as a physical system with an energy budget that includes all gains of incoming energy and all losses of outgoing energy. The planet is approximately in equilibrium, so the sum of the gains is approximately equal to the sum of the losses.Note on accompanying images:...
External links