Solar irradiance is the power per unit area (surface power density) 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 (W/m2) 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 (joule per square metre, J/m2) 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 (TSI) is a measure of the solar power over all wavelengths per unit area incident on the Earth's upper atmosphere. It is measured facing (pointing at / parallel to) the incoming sunlight (i.e. the flux through a surface perpendicular to the incoming sunlight; other angles would not be TSI). The solar constant is a conventional measure of mean TSI at a distance of one astronomical unit (AU). Direct normal irradiance (DNI), or beam radiation, is measured perpendicularly to the Sun direction. It excludes diffuse solar radiation (radiation that is scattered or reflected by atmospheric components). 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 (length of light's path through the atmosphere depending on the solar elevation angle), cloud cover, moisture content and other contents. The irradiance above the atmosphere also varies with time of year (because the distance to the Sun varies), although this effect is generally less significant compared to the effect of losses on DNI. Direct horizontal irradiance (DirHI), or beam horizontal irradiance (BHI), is the direct component of irradiance received on a horizontal surface as opposed to a surface perpendicular to the direct sunlight. Diffuse horizontal irradiance (DHI), 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 (radiation coming from the sun disk). There would be almost no DHI in the absence of atmosphere. Global horizontal irradiance (GHI) is the total irradiance from the Sun on a horizontal surface on Earth. For instantaneous measurement, it is the sum of direct irradiance (after accounting for the solar zenith angle, z {\displaystyle z} , of the Sun) and diffuse horizontal irradiance: GHI = DHI + DNI × cos ( z ) = DHI + DirHI {\displaystyle {\text{GHI}}={\text{DHI}}+{\text{DNI}}\times \cos(z)={\text{DHI}}+{\text{DirHI}}}
Global tilted irradiance (GTI) 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 (GNI) 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 (e.g. spectral TSI, spectral DNI, etc.) are any of the above with units divided either by meter or nanometer (for a spectral graph as function of wavelength), or per-Hz (for a spectral function with an x-axis of frequency). When one plots such spectral distributions as a graph, the integral of the function (area under the curve) will be the (non-spectral) irradiance. e.g.: Say one had a solar cell on the surface of the earth facing straight up, and had DNI in units of Wm−2nm−1, graphed as a function of wavelength (in nm). Then, the unit of the integral (Wm−2) is the product of those two units.
Units The SI unit of irradiance is watts per square metre (W/m2 = Wm−2). The unit of insolation often used in the solar power industry is kilowatt hours per square metre (kWh/m2). The langley is an alternative unit of insolation. One langley is one thermochemical calorie per square centimetre or 41,840 J/m2.
At the top of Earth's atmosphere
The average annual solar radiation arriving at the top of the Earth's atmosphere is about 1361 W/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 (1 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 4 π 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.
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:
cos ( c ) = cos ( a ) cos ( b ) + sin ( a ) sin ( b ) cos ( C ) {\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:
C = h c = Θ a = 1 2 π − φ b = 1 2 π − δ cos ( Θ ) = sin ( φ ) sin ( δ ) + cos ( φ ) cos ( δ ) cos ( h ) {\displaystyle {\begin{aligned}C&=h\\c&=\Theta \\a&={\tfrac {1}{2}}\pi -\varphi \\b&={\tfrac {1}{2}}\pi -\delta \\\cos(\Theta )&=\sin(\varphi )\sin(\delta )+\cos(\varphi )\cos(\delta )\cos(h)\end{aligned}}}
This equation can be also derived from a more general formula:
cos ( Θ ) = sin ( φ ) sin ( δ ) cos ( β ) + sin ( δ ) cos ( φ ) sin ( β ) cos ( γ ) + cos ( φ ) cos ( δ ) cos ( β ) cos ( h ) − cos ( δ ) sin ( φ ) sin ( β ) cos ( γ ) cos ( h ) − cos ( δ ) sin ( β ) sin ( γ ) sin ( h ) {\displaystyle {\begin{aligned}\cos(\Theta )=\sin(\varphi )\sin(\delta )\cos(\beta )&+\sin(\delta )\cos(\varphi )\sin(\beta )\cos(\gamma )+\cos(\varphi )\cos(\delta )\cos(\beta )\cos(h)\\&-\cos(\delta )\sin(\varphi )\sin(\beta )\cos(\gamma )\cos(h)-\cos(\delta )\sin(\beta )\sin(\gamma )\sin(h)\end{aligned}}}
where β is an angle from the horizontal and γ is the solar azimuth angle. The derivation of the cosine of solar zenith angle, cos ( Θ ) {\displaystyle \cos(\Theta )} , 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 RE 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:
Q = { S o R o 2 R E 2 cos ( Θ ) cos ( Θ ) > 0 0 cos ( Θ ) ≤ 0 {\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}}}
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:
Q ¯ day = − 1 2 π ∫ π − π Q d h {\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 Θ = 1 2 π {\displaystyle \Theta ={\tfrac {1}{2}}\pi } , or for h0 as a solution of
sin ( φ ) sin ( δ ) + cos ( φ ) cos ( δ ) cos ( h o ) = 0 {\displaystyle \sin(\varphi )\sin(\delta )+\cos(\varphi )\cos(\delta )\cos(h_{o})=0}
or
cos ( h o ) = − tan ( φ ) tan ( δ ) {\displaystyle \cos(h_{o})=-\tan(\varphi )\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 Q ¯ day = 0 {\displaystyle {\overline {Q}}^{\text{day}}=0} .
R o 2 R E 2 {\displaystyle {\frac {R_{o}^{2}}{R_{E}^{2}}}} is nearly constant over the course of a day, and can be taken outside the integral
∫ π − π Q d h = ∫ h o − h o Q d h = S o R o 2 R E 2 ∫ h o − h o cos ( Θ ) d h = S o R o 2 R E 2 [ h sin ( φ ) sin ( δ ) + cos ( φ ) cos ( δ ) sin ( h ) ] h = h o h = − h o = − 2 S o R o 2 R E 2 [ h o sin ( φ ) sin ( δ ) + cos ( φ ) cos ( δ ) sin ( h o ) ] {\displaystyle {\begin{aligned}\int _{\pi }^{-\pi }Q\,dh&=\int _{h_{o}}^{-h_{o}}Q\,dh\\[5pt]&=S_{o}{\frac {R_{o}^{2}}{R_{E}^{2}}}\int _{h_{o}}^{-h_{o}}\cos(\Theta )\,dh\\[5pt]&=S_{o}{\frac {R_{o}^{2}}{R_{E}^{2}}}{\Bigg [}h\sin(\varphi )\sin(\delta )+\cos(\varphi )\cos(\delta )\sin(h){\Bigg ]}_{h=h_{o}}^{h=-h_{o}}\\[5pt]&=-2S_{o}{\frac {R_{o}^{2}}{R_{E}^{2}}}\left[h_{o}\sin(\varphi )\sin(\delta )+\cos(\varphi )\cos(\delta )\sin(h_{o})\right]\end{aligned}}}
Therefore:
Q ¯ day = S o π R o 2 R E 2 [ h o sin ( φ ) sin ( δ ) + cos ( φ ) cos ( δ ) sin ( h o ) ] {\displaystyle {\overline {Q}}^{\text{day}}={\frac {S_{o}}{\pi }}{\frac {R_{o}^{2}}{R_{E}^{2}}}\left[h_{o}\sin(\varphi )\sin(\delta )+\cos(\varphi )\cos(\delta )\sin(h_{o})\right]}
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
δ = ε sin ( θ ) {\displaystyle \delta =\varepsilon \sin(\theta )}
where ε is the obliquity. (Note: The correct formula, valid for any axial tilt, is sin ( δ ) = sin ( ε ) sin ( θ ) {\displaystyle \sin(\delta )=\sin(\varepsilon )\sin(\theta )} .) The conventional longitude of perihelion ϖ is defined relative to the March equinox, so for the elliptical orbit:
R E = R o ( 1 − e 2 ) 1 + e cos ( θ − ϖ ) {\displaystyle R_{E}={\frac {R_{o}(1-e^{2})}{1+e\cos(\theta -\varpi )}}}
or
R o R E = 1 + e cos ( θ − ϖ ) 1 − e 2 {\displaystyle {\frac {R_{o}}{R_{E}}}={\frac {1+e\cos(\theta -\varpi )}{1-e^{2}}}}
With knowledge of ϖ, ε and e from astrodynamical calculations and So from a consensus of observations or theory, Q ¯ day {\displaystyle {\overline {Q}}^{\text{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 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:
Q ≈ S 0 ( 1 + 0.034 cos ( 2 π n 365.25 ) ) {\displaystyle Q\approx S_{0}\left(1+0.034\cos \left(2\pi {\frac {n}{365.25}}\right)\right)}
where n is a number of a day of the year.
Variation Total solar irradiance (TSI) changes slowly on decadal and longer timescales. The variation during solar cycle 21 was about 0.1% (peak-to-peak). 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 (EUV) 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% (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 Q ¯ d a y {\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
R o R E = 1 + e cos ( θ − ϖ ) = 1 + e cos ( π 2 − ϖ ) = 1 + e sin ( ϖ ) {\displaystyle {\frac {R_{o}}{R_{E}}}=1+e\cos(\theta -\varpi )=1+e\cos \left({\frac {\pi }{2}}-\varpi \right)=1+e\sin(\varpi )}
For this summer solstice calculation, the role of the elliptical orbit is entirely contained within the important product e sin ( ϖ ) {\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 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 0.05–0.15 W/m2 per century.
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. 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 Earth Radiation Budget Satellite (ERBS), VIRGO on the Solar Heliospheric Observatory (SoHO) and the ACRIM instruments on the Solar Maximum Mission (SMM), Upper Atmosphere Research Satellite (UARS) and ACRIMSAT. Pre-launch ground calibrations relied on component rather than system-level measurements since irradiance standards at the time lacked sufficient absolute accuracies. Measurement stability involves exposing different radiometer cavities to different accumulations of solar radiation to quantify exposure-dependent degradation effects. These effects are then compensated for in the final data. Observation overlaps permits corrections for both absolute offsets and validation of instrumental drifts. Uncertainties of individual observations exceed irradiance variability (~0.1%). Thus, instrument stability and measurement continuity are relied upon to compute real variations. Long-term radiometer drifts can potentially be mistaken for irradiance variations which can be misinterpreted as affecting climate. Examples include the issue of the irradiance increase between cycle minima in 1986 and 1996, evident only in the ACRIM composite (and not the model) and the low irradiance levels in the PMOD composite during the 2008 minimum. Despite the fact that ACRIM I, ACRIM II, ACRIM III, VIRGO and TIM all track degradation with redundant cavities, notable and unexplained differences remain in irradiance and the modeled influences of sunspots and faculae.
Persistent inconsistencies Disagreement among overlapping observations indicates unresolved drifts that suggest the TSI record is not sufficiently stable to discern solar changes on decadal time scales. Only the ACRIM composite shows irradiance increasing by ~1 W/m2 between 1986 and 1996. It is noteworthy that the most accurate TSI reconstructions with empirical and physics-based semi-empirical models using independent inputs consistently disfavor this increase during the ACRIM-gap. Recommendations to resolve the instrument discrepancies include validating optical measurement accuracy by comparing ground-based instruments to laboratory references, such as those at National Institute of Science and Technology (NIST); NIST validation of aperture area calibrations uses spares from each instrument; and applying diffraction corrections from the view-limiting aperture. For ACRIM, NIST determined that diffraction from the view-limiting aperture contributes a 0.13% signal not accounted for in the three ACRIM instruments. This correction lowers the reported ACRIM values, bringing ACRIM closer to TIM. In ACRIM and all other instruments but TIM, the aperture is deep inside the instrument, with a larger view-limiting aperture at the front. Depending on edge imperfections this can directly scatter light into the cavity. This design admits into the front part of the instrument two to three times the amount of light intended to be measured; if not completely absorbed or scattered, this additional light produces erroneously high signals. In contrast, TIM's design places the precision aperture at the front so that only desired light enters. Variations from other sources likely include an annual systematics in the ACRIM III data that is nearly in phase with the Sun-Earth distance and 90-day spikes in the VIRGO data coincident with SoHO spacecraft maneuvers that were most apparent during the 2008 solar minimum.
TSI Radiometer Facility TIM's high absolute accuracy creates new opportunities for measuring climate variables. TSI Radiometer Facility (TRF) is a cryogenic radiometer that operates in a vacuum with controlled light sources. L-1 Standards and Technology (LASP) designed and built the system, completed in 2008. It was calibrated for optical power against the NIST Primary Optical Watt Radiometer, a cryogenic radiometer that maintains the NIST radiant power scale to an uncertainty of 0.02% (1σ). As of 2011 TRF was the only facility that approached the desired <0.01% uncertainty for pre-launch validation of solar radiometers measuring irradiance (rather than merely optical power) at solar power levels and under vacuum conditions. TRF encloses both the reference radiometer and the instrument under test in a common vacuum system that contains a stationary, spatially uniform illuminating beam. A precision aperture with an area calibrated to 0.0031% (1σ) determines the beam's measured portion. The test instrument's precision aperture is positioned in the same location, without optically altering the beam, for direct comparison to the reference. Variable beam power provides linearity diagnostics, and variable beam diameter diagnoses scattering from different instrument components. The Glory/TIM and PICARD/PREMOS flight instrument absolute scales are now traceable to the TRF in both optical power and irradiance. The resulting high accuracy reduces the consequences of any future gap in the solar irradiance record.
2011 reassessment The most probable value of TSI representative of solar minimum is 1360.9±0.5 W/m2, lower than the earlier accepted value of 1365.4±1.3 W/m2, established in the 1990s. The new value came from SORCE/TIM and radiometric laboratory tests. Scattered light is a primary cause of the higher irradiance values measured by earlier satellites in which the precision aperture is located behind a larger, view-limiting aperture. The TIM uses a view-limiting aperture that is smaller than the precision aperture that precludes this spurious signal. The new estimate is from better measurement rather than a change in solar output. A regression model-based split of the relative proportion of sunspot and facular influences from SORCE/TIM data accounts for 92% of observed variance and tracks the observed trends to within TIM's stability band. This agreement provides further evidence that TSI variations are primarily due to solar surface magnetic activity. Instrument inaccuracies add a significant uncertainty in determining Earth's energy balance. The energy imbalance has been variously measured (during a deep solar minimum of 2005–2010) to be +0.58±0.15 W/m2, +0.60±0.17 W/m2 and +0.85 W/m2. Estimates from space-based measurements range +3–7 W/m2. SORCE/TIM's lower TSI value reduces this discrepancy by 1 W/m2. This difference between the new lower TIM value and earlier TSI measurements corresponds to a climate forcing of −0.8 W/m2, which is comparable to the energy imbalance.
On Earth's surface
Average annual solar radiation arriving at the top of the Earth's atmosphere is roughly 1361 W/m2. The Sun's rays are attenuated as they pass through the atmosphere, leaving maximum normal surface irradiance at approximately 1000 W/m2 at sea level on a clear day. When 1361 W/m2 is arriving above the atmosphere (when the Sun is at the zenith in a cloudless sky), direct sun is about 1050 W/m2, and global radiation on a horizontal surface at ground level is about 1120 W/m2. The latter figure includes radiation scattered or reemitted by the atmosphere and surroundings. The actual figure varies with the Sun's angle and atmospheric circumstances. Ignoring clouds, the daily average insolation for the Earth is approximately 6 kWh/m2 = 21.6 MJ/m2. The o