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# Solar constant

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### Solar constant

Solar irradiance spectrum at top of atmosphere, on a linear scale and plotted against wavenumber.

The solar constant, a measure of flux density, is the conventional name for the mean solar electromagnetic radiation (the solar irradiance) per unit area that would be incident on a plane perpendicular to the rays, at a distance of one astronomical unit (AU) from the Sun (roughly the mean distance from the Sun to the Earth). The solar constant includes all types of solar radiation, not just the visible light. It is measured by satellite as being 1.361 kilowatts per square meter (kW/m²) at solar minimum and approximately 0.1% greater (roughly 1.362 kW/m²) at solar maximum.[1] The solar "constant" is not a physical constant in the modern CODATA scientific sense; it varies in value, and has been called a "misconception".[2] It has been shown to vary historically in the past 400 years over a range of less than 0.2 percent.[2]

## Contents

• Calculation 1
• Historical measurements 2
• Relationship to other measurements 3
• Apparent magnitude 3.2
• The Sun's total radiation 3.3
• Past variations in solar irradiance 4
• References 6

## Calculation

Solar irradiance is measured by satellite above Earth's atmosphere,[3] and is then adjusted using the inverse square law to infer the magnitude of solar irradiance at one Astronomical Unit (AU) to evaluate the solar constant.[4] The approximate average value cited,[1] 1.3608 ± 0.0005  kW/m², which is 81.65 kJ/m² per minute, is equivalent to approximately 1.951 calories per minute per square centimeter, or 1.951 langleys per minute.

Solar output is nearly, but not quite, constant. Variations in total solar irradiance (TSI) were small and difficult to detect accurately with technology available before the satellite era (+/- 2% in 1954). Total solar output is now measured as varying (over the last three 11-year sunspot cycles) by approximately 0.1%;[5] see solar variation for details.

## Historical measurements

In 1838, Claude Pouillet made the first estimate of the solar constant. Using a very simple pyrheliometer he developed, he obtained a value of 1.228 kW/m²,[6] close to the current estimate.

In 1875, Jules Violle resumed the work of Pouillet and offered a somewhat larger estimate of 1.7 kW/m² based, in part, on a celebrated measurement that he made from Mont Blanc in France.

In 1884, Samuel Pierpont Langley attempted to estimate the solar constant from Mount Whitney in California. By taking readings at different times of day, he tried to correct for effects due to atmospheric absorption. However, the final value he proposed, 2.903 kW/m², was much too large.

A 1903 Langley bolograph with an erroneous solar constant of 2.54 calories/minute/square centimeter.

Between 1902 and 1957, measurements by Charles Greeley Abbot and others at various high-altitude sites found values between 1.322 and 1.465 kW/m². Abbot showed that one of Langley's corrections was erroneously applied. Abbot's results varied between 1.89 and 2.22 calories (1.318 to 1.548  kW/m²), a variation that appeared to be due to the Sun and not the Earth's atmosphere.[7]

In 1954 the solar constant was evaluated as 2.00 cal/min/sq cm ± 2%.[8] Current results are about 2.5 percent lower.

## Relationship to other measurements

The actual direct solar irradiance at the top of the atmosphere fluctuates by about 6.9% during a year (from 1.412 kW/m² in early January to 1.321 kW/m² in early July) due to the Earth's varying distance from the Sun, and typically by much less than 0.1% from day to day. Thus, for the whole Earth (which has a cross section of 127,400,000 km²), the power is 1.730×1017 W (or 173,000 terawatts),[9] plus or minus 2 W/m2.[10] The solar constant does not remain constant over long periods of time (see Solar variation), but over a year the solar constant varies much less than the solar irradiance measured at the top of the atmosphere. This is because the solar constant is evaluated at a fixed distance of 1 Astronomical Unit (AU) while the solar irradiance will be affected by the ellipticity of the Earth's orbit.

The Earth receives a total amount of radiation determined by its cross section (π·RE²), but as it rotates this energy is distributed across the entire surface area (4·π·RE²). Hence the average incoming solar radiation, taking into account the angle at which the rays strike and that at any one moment half the planet does not receive any solar radiation, is one-fourth the solar constant (approximately 340 W/m²). The amount reaching the Earth's surface (as insolation) is further reduced by atmospheric attenutation, which varies. At any given moment, the amount of solar radiation received at a location on the Earth's surface depends on the state of the atmosphere, the location's latitude, and the time of day.

### Apparent magnitude

The solar constant includes all wavelengths of solar electromagnetic radiation, not just the visible light (see Electromagnetic spectrum). It is positively correlated with the apparent magnitude of the Sun which is −26.8. The solar constant and the magnitude of the Sun are two methods of describing the apparent brightness of the Sun, though the magnitude is based on the Sun's visual output only.

The angular diameter of the Earth as seen from the Sun is approximately 1/11,700 radians (about 18 arc-seconds), meaning the solid angle of the Earth as seen from the Sun is approximately 1/175,000,000 of a steradian. Thus the Sun emits about 2.2 billion times the amount of radiation that is caught by Earth, in other words about 3.86×1026 watts.[11]

## Past variations in solar irradiance

Space-based observations of solar irradiance started in 1978. These measurements show that the solar constant is not constant. It varies with the 11-year sunspot solar cycle. When going further back in time, one has to rely on irradiance reconstructions, using sunspots for the past 400 years or cosmogenic radionuclides for going back 10,000 years. Such reconstructions show that solar irradiance varies with distinct periodicities. These cycles are: 11 years (Schwabe), 88 years (Gleisberg cycle), 208 years (DeVries cycle) and 1,000 years (Eddy cycle).[12][13][14][15][16]

## References

1. ^ a b Kopp, G.; Lean, J. L. (2011). "A new, lower value of total solar irradiance: Evidence and climate significance" (PDF). Geophysical Research Letters 38: n/a.
3. ^ Satellite observations of total solar irradiance
5. ^ Willson, Richard C.; H.S. Hudson (1991). "The Sun's luminosity over a complete solar cycle". Nature 351 (6321): 42–4.
6. ^ The measurement of the solar constant by Claude Pouillet, by J-L Dufresne, La Météorologie, No. 60, pp. 36-43, Feb. 2008.
7. ^
8. ^ Francis S. Johnson (December 1954). "The Solar Constant". Journal of Meteorology 11 (6): 431–439.
9. ^ Archer, D. (2012). Global Warming: Understanding the Forecast.
10. ^ Wild, Martin; Folini, Doris; Schär, Christoph; Loeb, Norman; et al. (2013). The Earth's radiation balance and its representation in CMIP5 models. Copernicus.
11. ^ The Sun at nine planets.org
12. ^ Wang; et al. (2005). "Modeling the Sun’s Magnetic Field and Irradiance since 1713". The Astrophysical Journal 625 (1): 522–538.
13. ^ Steinhilber et al. (2009), Geophysical Research Letters, Volume 36, L19704, doi:10.1051/0004-6361/200811446
14. ^ Vieira; et al. (2011). "Evolution of the solar irradiance during the Holocene". Astronomy&Astrophysics 531: A6.
15. ^ Steinhilber; et al. (2012). "9,400 years of cosmic radiation and solar activity from ice cores and tree rings". Proceedings of the National Academy of Sciences 109: 5967–5971.
16. ^ Vieira, L. E. A.; Norton, A.; Kretzschmar, M.; Schmidt, G. A.; Cheung, M. C. M. (2012). "How the inclination of Earth's orbit affects incoming solar irradiance". Geophys. Res. Lett. 39: L16104.
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