##### From Wikipedia, the free encyclopedia

**Insolation** is the power received on Earth per unit area on a horizontal surface.^{[3]} Not to be confused with `insulation.’ It depends on the height of the Sun above the horizon.^{[1]}

## Insolation[edit]

The solar irradiance integrated over time is called solar irradiation, solar exposure or insolation. However, insolation is often used interchangeably with irradiance in practice.

## Units[edit]

The SI unit of irradiance is watt per square meter (W/m^{2}).

An alternate unit of measure is the Langley (1 thermochemical calorie per square centimeter or 41,840 J/m^{2}) per unit time.

The solar energy industry uses watt-hour per square metre (Wh/m^{2}) divided by the recording time. 1 kW/m^{2} = 24 kWh/(m^{2} day).

Irradiance can also be expressed in Suns, where one Sun equals 1000 W/m^{2} at the point of arrival.^{[4]}

## Absorption and reflection[edit]

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[edit]

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’sCosine; 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 (1.6 km) 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. At a lower angle the light must travel through more atmosphere. This attenuates it (by absorption and scattering) further reducing insolation at the surface.

## Categories[edit]

Direct insolation is measured at a given location with a surface element perpendicular to the Sun. It excludes diffuse insolation (radiation that is scattered or reflected by atmospheric components). Direct insolation is equal to the irradiance above the atmosphere minus the atmospheric losses due to absorption and scattering. While the irradiance above the atmosphere varies with time of year (because the distance to the sun varies), 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. (See values for direct and total insolation further down.) Insolation affects plant metabolism and animal behavior.^{[5]}

Diffuse insolation is the contribution of light scattered by the atmosphere to total insolation.

## Earth[edit]

Average annual solar radiation arriving at the top of the Earth’s atmosphere is roughly 1366 W/m^{2}.^{[6]}^{[7]} The radiation is distributed across the electromagnetic spectrum. About half is infrared light.^{[8]} The Sun’s rays areattenuated as they pass through the atmosphere, leaving maximum normal surface irradiance at approximately 1000 W /m^{2} at sea level on a clear day. When 1367 W/m^{2} is arriving above the atmosphere (as when the earth is one astronomical unit from the sun), direct sun is about 1050 W/m^{2}, and global radiation on a horizontal surface at ground level is about 1120 W/m^{2}.^{[9]} The latter figure includes radiation scattered or reemitted by 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/m^{2} = 21.6 MJ/m^{2}.

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.^{[10]}

### Solar potential maps[edit]

### Top of the atmosphere[edit]

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 climate 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:

- {\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:

- {\displaystyle C=h\,}

- {\displaystyle c=\Theta \,}

- {\displaystyle a={\tfrac {1}{2}}\pi -\phi \,}

- {\displaystyle b={\tfrac {1}{2}}\pi -\delta \,}

- {\displaystyle \cos(\Theta )=\sin(\phi )\sin(\delta )+\cos(\phi )\cos(\delta )\cos(h)\,}

The separation of Earth from the sun can be denoted R_{E} and the mean distance can be denoted R_{0}, approximately 1 AU. The solar constant is 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:

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

and

- {\displaystyle 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:

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

Let *h*_{0} be the hour angle when Q becomes positive. This could occur at sunrise when {\displaystyle \Theta ={\tfrac {1}{2}}\pi }, or for *h*_{0} 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 {\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 \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}
- {\displaystyle \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}}}

- {\displaystyle \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]}

- {\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

- {\displaystyle \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:

- {\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^{[11]} 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.

## Variation[edit]

### Total irradiance[edit]

Total solar irradiance (TSI)^{[12]} changes slowly on decadal and longer timescales. The variation during solar cycle 21 was about 0.1% (peak-to-peak).^{[13]} In contrast to older reconstructions,^{[14]} most recent TSI reconstructions point to an increase of only about 0.05% to 0.1% between the Maunder Minimum and the present.^{[15]}^{[16]}^{[17]}

### Ultraviolet irradiance[edit]

Ultraviolet irradiance (EUV) varies by approximately 1.5 percent from solar maxima to minima, for 200 to 300 nm wavelengths.^{[18]} However, a proxy study estimated that UV has increased by 3.0% since the Maunder Minimum.^{[19]}

### Milankovitch cycles[edit]

Some variations in insolation are not due to solar changes but rather due to the Earth moving between its perigee and apogee, or changes in the latitudinal distribution of radiation. These orbital changes orMilankovitch 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

- {\displaystyle {\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 {\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[edit]

The space-based TSI record comprises measurements from more than ten radiometers spanning three solar cycles.

### Technique[edit]

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 precisionaperture 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 m−2 per century.^{[20]}

### Intertemporal calibration[edit]

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.^{[20]}

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 lacked absolute accuracies.^{[20]}

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 final data. Observation overlaps permits corrections for both absolute offsets and validation of instrumental drifts.^{[20]}

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 be mistaken for irradiance variations that 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[edit]

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 m−2 between 1986 and 1996; this change is also absent in the model.^{[20]}

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.^{[20]}

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, 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 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.^{[20]}

Variations from other sources likely include an annual cycle that is nearly in phase with the Sun-Earth distance in ACRIM III data 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[edit]

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% (1*σ*). 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.^{[20]}

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 area calibrated to 0.0031% (1*σ*) 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.^{[20]}

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.^{[20]}

Instrument | Irradiance: View-Limiting Aperture Overfilled | Irradiance: Precision Aperture Overfilled | Difference Attributable To Scatter Error | Measured Optical Power Error | Residual Irradiance Agreement | Uncertainty |
---|---|---|---|---|---|---|

SORCE/TIM ground | NA | −0.037% | NA | −0.037% | 0.000% | 0.032% |

Glory/TIM flight | NA | −0.012% | NA | −0.029% | 0.017% | 0.020% |

PREMOS-1 ground | −0.005% | −0.104% | 0.098% | −0.049% | −0.104% | ∼0.038% |

PREMOS-3 flight | 0.642% | 0.605% | 0.037% | 0.631% | −0.026% | ∼0.027% |

VIRGO-2 ground | 0.897% | 0.743% | 0.154% | 0.730% | 0.013% | ∼0.025% |

### 2011 reassessment[edit]

The most probable value of TSI representative of solar minimum is 1360.8 ± 0.5 W m−2, lower than the earlier accepted value of 1365.4 ± 1.3 W m−2, 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 precision aperture that precludes this spurious signal. The new estimate is from better measurement rather than a change in solar output.^{[20]}

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.^{[20]}

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),^{[21]} +0.60 ± 0.17 W/m²^{[22]} and +0.85 W m−2. Estimates from space-based measurements range from +3 to 7 W m−2. SORCE/TIM’s lower TSI value reduces this discrepancy by 1 W m−2. This difference between the new lower TIM value and earlier TSI measurements corresponds to a climate forcing of −0.8 W m−2, which is comparable to the