watt

An alternative unit of measure is the Langley (1 thermochemical calorie per square centimeter or 41,840 J/m²) per unit time.

The solar energy industry uses watt-hour per square metre (Wh/m²) per unit time^{[citation needed]}. The relation to the SI unit is thus:

1 kW/m² x (24 h/day) = (24 kWh/m²)/day

(24kWh/m²)/day*(365days/year) = (8760 kWh/m²)/year.

Irradiation at the top of the atmosphere ( c ) = cos ⁡ ( a ) cos ⁡ ( b ) + sin ⁡ ( a ) sin ⁡ ( b ) cos ⁡ ( C ) {\displaystyle \cos(c)=\cos(a)\cos(b)+\sin(a)\sin(b)\cos(C)} spherical trigonometry, 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 angle Θ, the following applies to the spherical law of cosines:

${\displaystyle C=h}$

${\displaystyle c=\Theta }$

${\displaystyle {\begin{aligned}\cos(\Theta )&=\sin(\phi )\sin(\delta )\cos(\beta )+\sin(\delta )\cos(\phi )\sin(\beta$

where β is an angle from the horizontal and γ is an azimuth angle.

${\displaystyle {\overline {Q}}^{\mathrm {day} }}$, the theoretical daily-average irradiation 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 separation of Earth from the sun can be denoted R_E and the mean distance can be denoted R₀, approximately 1 astronomical unit (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:

${\displaystyle Q={\begin{cases}S_{o}{\frac {R_{o}^{2}}{R_{E}^{2}}}\cos(\Theta )&\cos(\Theta )>0\\0&\cos(\Theta )\leq 0\end{cases}}}$astronomical unit (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:

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

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

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 $$

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

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

Therefore:

${\displaystyle {\overline {Q}}^{\text{day}}={\frac {S_{o}}{\pi }}{\frac {R_{o}^{2}}{R_{E}^{2}}}\left[h_{o}\sin(\phi )\sin(\delta )+\cos(\phi )\cos(\delta )\sin(h_{o})\right]}$

Let θ be the conventional polar angle describing a planetary orbit. Let θ = 0 at the vernal equinox. The declination δ as a function of orbital position is^[12][13]

${\displaystyle \delta =\varepsilon \sin(\theta )}$

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^[14] 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.

A simplified equation for irradiance on a given day is:^[15]

${\displaystyle Q=S_0(1+0.034\cos <!--\; \; -->\left(2\mathrm{\pi <!--\; \pi \; -->}\frac{n}{365.25}\right)}$

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^[12][13]

${\displaystyle \delta =\varepsilon \sin(\theta )}$

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

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

${\displaystyle R_E=\frac{R_o}{1+e\cos <!--\; \; -->(\mathrm{\theta <!--\; \theta \; -->}-<!--\; -\; -->\mathrm{\varpi <!--\; \varpi \; -->}}}$

or

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

With knowledge of ϖ, ε and e from astrodynamical calculations^[14] 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.

A simplified equation for irradiance on a given day is:^[15]

${\displaystyle Q=S_{0}\left(1+0.034\cos \left(2\pi {\frac {n}{365.25}}\right)\right)}$[15]

where n is a number of a day of the year.

Variation

Total solar irradiance (TSI)^[16] changes slowly on decadal and longer timescales. The variation during solar cycle 21 was about 0.1% (peak-to-peak).^[17] In contrast to older reconstructions,^[18] most recent TSI reconstructions point to an increase of only about 0.05% to 0.1% between the Maunder Minimum and the present.^[19][20][21] Ultraviolet irradiance (EUV) varies by approximately 1.5 percent from solar maxima to minima, for 200 to 300 nm wavelengths.^[22] However, a proxy study estimated that UV has increased by 3.0% since the Maunder Minimum.^[23]

[16] changes slowly on decadal and longer timescales. The variation during solar cycle 21 was about 0.1% (peak-to-peak).^[17] In contrast to older reconstructions,^[18] most recent TSI reconstructions point to an increase of only about 0.05% to 0.1% between the Maunder Minimum and the present.^[19][20][21] Ultraviolet irradiance (EUV) varies by approximately 1.5 percent from solar maxima to minima, for 200 to 300 nm wavelengths.^[22] However, a proxy study estimated that UV has increased by 3.0% since the Maunder Minimum.^[23]

Some variations in insolation are not due to solar changes but rather due to the Earth moving between its perihelion and aphelion, or changes in the latitudinal distribution of radiation. These orbital changes or Milankovitch cycles have caused radiance variations of as much as 25% (locally; global average changes are much smaller) over long periods. The most recent significant event was an axial tilt of 24° during boreal summer near the Holocene climatic optimum. 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

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.

Measurement

The space-based TSI record comprises measurements from more than ten radiometers spanning three solar cycles. All modern TSI satellite instruments employ active cavity electrical substitution radiometry. This technique applies measured electrical heating to maintain an absorptive blackened cavity in thermal equilibrium while incident sunlight passes through a precision aperture of calibrated area. The aperture is modulated via a shutter. Accuracy uncertainties of <0.01% are required to detect long term solar irradiance variations, because expected changes are in the range 0.05 to 0.15 W/m² per century.^[24]

Intertemporal calibration

In orbit, radiometric calibrations drift for reasons including solar degradation of the cavity, electronic degradation of the heater, surface degradation of the precision aperture and varying surface emissions and temperatures that alter thermal backgrounds. These calibrations require compensation to preserve consistent measurements.^[24]

For various reasons, the sources do not always agree. The Solar Radiation and Climate Experiment/Total Irradiance Measurement (SORCE/TIM) TSI values are lower than prior measurements by the Earth Radiometer Budget Experiment (ERBE) on the active cavity electrical substitution radiometry. This technique applies measured electrical heating to maintain an absorptive blackened cavity in thermal equilibrium while incident sunlight passes through a precision aperture of calibrated area. The aperture is modulated via a shutter. Accuracy uncertainties of <0.01% are required to detect long term solar irradiance variations, because expected changes are in the range 0.05 to 0.15 W/m² per century.^[24]

Intertemporal calibration r 2 {\displaystyle 4\pi r^{2}} , meaning that the solar radiation arriving at the top of the atmosphere, averaged over the entire surface of the Earth, is simply divided by four to get 340 W/m². In other words, averaged over the year and the day, the Earth's atmosphere receives 340 W/m² from the sun. This figure is important in radiative forcing.

The output of, for example, a photovoltaic panel, partly depends on the angle of the sun relative to the panel. One Sun is a unit of AU). This means that the approximately circular disc of the Earth, as viewed from the sun, receives a roughly stable 1361 W/m² at all times. The area of this circular disc is πr², in which r is the radius of the Earth. Because the Earth is approximately spherical, it has total area ${\displaystyle 4\pi r^{2}}$, meaning that the solar radiation arriving at the top of the atmosphere, averaged over the entire surface of the Earth, is simply divided by four to get 340 W/m². In other words, averaged over the year and the day, the Earth's atmosphere receives 340 W/m² from the sun. This figure is important in radiative forcing.

The output of, for example, a photovoltaic panel, partly depends on the angle of the sun relative to the panel. 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.^[30]

Part of the radiation reaching an object 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

Projection effect: One sunbeam one mile wide shines on the ground at a 90° angle, and another at a 30° angle. The 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 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, and moreover receive no insolation at all for the six months of their respective winters.

Absorption effect

Solar radiation maps are built using databases derived from satellite imagery, as for example using visible images from Meteosat Prime satellite. A method is applied on the images to determine solar radiation.

Solar irradiation figures are used to plan the deployment of solar power systems.^[32] 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. Different solar power technologies are able to use different component of the total irradiation. While solar photovoltaics panels are able to convert to electricity both direct irradiation and diffuse irradiation, concentrated solar power is only able to operate efficiently with direct irradiation, thus making these systems suitable only in locations with relatively low cloud cover.

Because solar collectors panels are almost always mounted at an angle^[33] towards the sun, insolation must be adjusted to prevent estimates that are inaccurately low for winter and inaccurately high for summer.^[34] This also means that the amount of sun falling on a solar panel at high latitude is not as low compared to one at the equator as would appear from just considering insolation on a horizontal surface.

Photovoltaic panels are rated under standard conditions to determine the Wp (watt peak) rating,^[35] 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).^[36] Insolation values range from 800 to 950 kWh/(kWp·y) in Norway to up to 2,900 kWh/(kWp·y) in Australia.

Buildings

In construction, insolation is an important consideration when designing a building for a particular site.^[37]

Insolation variation by month; 1984–1993 averages for January (top) and April (bottom)

The projection effect can b

Because solar collectors panels are almost always mounted at an angle^[33] towards the sun, insolation must be adjusted to prevent estimates that are inaccurately low for winter and inaccurately high for summer.^[34] This also means that the amount of sun falling on a solar panel at high latitude is not as low compared to one at the equator as would appear from just considering insolation on a horizontal surface.

Photovoltaic panels are rated under standard conditions to determine the Wp (watt peak) rating,^[35] 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).^[36] Insolation values range from 800 to 950 kWh/(kWp·y) in Norway to up to 2,900 kWh/(kWp·y) in Australia.

In construction, insolation is an important consideration when designing a building for a particular site.^[37]

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).

Civil engineering

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 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.

Climate research

Irradiance plays a part in climate modeling and weather forecasting. A non-zero average global net radiation at the top of the atmosphere is indicative of Earth's thermal disequilibrium as imposed by climate forcing.

The impact of the lower 2014 TSI value on climate models is unknown. A few tenths of a percent change in the absolute TSI level is typically considered to be of minimal consequence for climate simulations. The new measurements require climate model parameter adjustments.

Experiments with GISS Model 3 investigated the sen

The impact of the lower 2014 TSI value on climate models is unknown. A few tenths of a percent change in the absolute TSI level is typically considered to be of minimal consequence for climate simulations. The new measurements require climate model parameter adjustments.

Experiments with GISS Model 3 investigated the sensitivity of model performance to the TSI absolute value during present and pre-industrial epochs, and describe, for example, how the irradiance reduction is partitioned between the atmosphere and surface and the effects on outgoing radiation.^[24]

Assessing the impact of long-term irradiance changes on climate requires greater instrument stability^[24] combined with reliable global surface temperature observations to quantify climate response processes to radiative forcing on decadal time scales. The observed 0.1% irradiance increase imparts 0.22 W/m² climate forcing, which suggests a transient climate response of 0.6 °C per W/m². This response is larger by a factor of 2 or more than in the IPCC-assessed 2008 models, possibly appearing in the models' heat uptake by the ocean.^[24]

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

Solar activity and irradiance measurement is a concern for space travel. For example, the American space agency, NASA, launche

Solar activity and irradiance measurement is a concern for space travel. For example, the American space agency, NASA, launched its Solar Radiation and Climate Experiment (SORCE) satellite with Solar Irradiance Monitors.^[1]

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