Solar irradiance

Solar irradiance is the power per unit area received from the Sun in the form of electromagnetic radiation in the wavelength range of the measuring instrument.
Solar irradiance is measured in watts per square metre in SI units.
Solar irradiance is often integrated over a given time period in order to report the radiant energy emitted into the surrounding environment during that time period. This integrated solar irradiance is called solar irradiation, solar radiation, solar exposure, solar insolation, or insolation.
Irradiance may be measured in space or at the Earth's surface after atmospheric absorption and scattering. Irradiance in space is a function of distance from the Sun, the solar cycle, and cross-cycle changes.
Irradiance on the Earth's surface additionally depends on the tilt of the measuring surface, the height of the Sun above the horizon, and atmospheric conditions.
Solar irradiance affects plant metabolism and animal behavior.
The study and measurement of solar irradiance has several important applications, including the prediction of energy generation from solar power plants, the heating and cooling loads of buildings, climate modeling and weather forecasting, passive daytime radiative cooling applications, and space travel.

Types

There are several measured types of solar irradiance.

Total solar irradiance is a measure of the solar power over all wavelengths per unit area incident on the Earth's upper atmosphere. It is measured facing the incoming sunlight. The solar constant is a conventional measure of mean TSI at a distance of one astronomical unit.
Direct normal irradiance , or beam radiation, is measured perpendicularly to the Sun direction. It excludes diffuse solar radiation. Direct irradiance is equal to the extraterrestrial irradiance above the atmosphere minus the atmospheric losses due to absorption and scattering. Losses depend on time of day, cloud cover, moisture content and other contents. The irradiance above the atmosphere also varies with time of year, although this effect is generally less significant compared to the effect of losses on DNI.
Direct horizontal irradiance , or beam horizontal irradiance , is the direct component of irradiance received on a horizontal surface as opposed to a surface perpendicular to the direct sunlight.
Diffuse horizontal irradiance , or diffuse sky radiation, is the radiation at the Earth's surface from light scattered by the atmosphere. It is measured on a horizontal surface with radiation coming from all points in the sky excluding circumsolar radiation. There would be almost no DHI in the absence of atmosphere.
Global horizontal irradiance is the total irradiance from the Sun on a horizontal surface on Earth. For instantaneous measurement, it is the sum of direct irradiance and diffuse horizontal irradiance:
Global tilted irradiance is the total radiation received on a surface with defined tilt and azimuth, fixed or Sun-tracking. GTI can be measured or modeled from GHI, DNI, DHI. It is often a reference for photovoltaic power plants, while photovoltaic modules are mounted on the fixed or tracking constructions.
Global normal irradiance is the total irradiance from the Sun at the surface of Earth at a given location with a surface element perpendicular to the Sun.

Spectral versions of the above irradiances are any of the above with units divided either by meter or nanometer, or per-Hz. When one plots such spectral distributions as a graph, the integral of the function will be the irradiance. e.g.: Say one had a solar cell on the surface of the earth facing straight up, and had DNI in units of Wmnm, graphed as a function of wavelength. Then, the unit of the integral is the product of those two units.

Units

The SI unit of irradiance is watts per square metre. The unit of insolation often used in the solar power industry is kilowatt hours per square metre.
The langley is an alternative unit of insolation. One langley is one thermochemical calorie per square centimetre or 41,840J/m².

At the top of Earth's atmosphere

The average annual solar radiation arriving at the top of the Earth's atmosphere is about 1361W/m². This represents the power per unit area of solar irradiance across the spherical surface surrounding the Sun with a radius equal to the distance to the Earth. This means that the approximately circular disc of the Earth, as viewed from the Sun, receives a roughly stable 1361W/m² at all times. The area of this circular disc is, in which is the radius of the Earth. Because the Earth is approximately spherical, it has total area, 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 340W/m². In other words, averaged over the year and the day, the Earth's atmosphere receives 340W/m² from the Sun. This figure is important in radiative forcing.

Derivation

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 climatic 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:
where, and are arc lengths, in radians, of the sides of a spherical triangle. is the angle in the vertex opposite the side which has arc length. Applied to the calculation of solar zenith angle, the following applies to the spherical law of cosines:
This equation can be also derived from a more general formula:
where is an angle from the horizontal and is the solar azimuth angle.
The derivation of the cosine of solar zenith angle,, based on vector analysis instead of spherical trigonometry is also available in the article about solar azimuth angle.
The separation of Earth from the Sun can be denoted and the mean distance can be denoted, approximately The solar constant is denoted. The solar flux density onto a plane tangent to the sphere of the Earth, but above the bulk of the atmosphere is:
The average of over a day is the average of over one rotation, or the hour angle progressing from to :
Let be the hour angle when becomes positive. This could occur at sunrise when, or for as a solution of
or
If, then the sun does not set and the sun is already risen at, so. If, the sun does not rise and.
is nearly constant over the course of a day, and can be taken outside the integral
Therefore:
Let θ be the conventional polar angle describing a planetary orbit. Let θ = 0 at the March equinox. The declination δ as a function of orbital position is
where is the obliquity. The conventional longitude of perihelion ϖ is defined relative to the March 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. 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 March equinox, θ = 90° is exactly the time of the June solstice, θ = 180° is exactly the time of the September equinox and θ = 270° is exactly the time of the December solstice.
A simplified equation for irradiance on a given day is:
where n is a number of a day of the year.

Variation

Total solar irradiance changes slowly on decadal and longer timescales. The variation during solar cycle 21 was about 0.1%. In contrast to older reconstructions, most recent TSI reconstructions point to an increase of only about 0.05% to 0.1% between the 17th century Maunder Minimum and the present.
However, current understanding based on various lines of evidence suggests that the lower values for the secular trend are more probable. In particular, a secular trend greater than 2 Wm⁻² is considered highly unlikely. Ultraviolet irradiance varies by approximately 1.5 percent from solar maxima to minima, for 200 to 300 nm wavelengths. However, a proxy study estimated that UV has increased by 3.0% since the Maunder Minimum.
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% 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 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, 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 and spans three solar cycles.
All modern TSI satellite instruments employ active cavity electrical substitution radiometry. This technique measures the electrical heating needed to maintain an absorptive blackened cavity in thermal equilibrium with the incident sunlight which 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 0.05–0.15W/m² per century.