To calculate the amount of sunlight reaching the ground, both the elliptical orbit of the Earth and the attenuation by the Earth's atmosphere have to be taken into account. The extraterrestrial solar illuminance (Eext), corrected for the elliptical orbit by using the day number of the year (dn), is:
![E_{\rm ext}=E_{\rm sc}\left[1+0.034 \cdot \cos\left(2\pi\frac{{\rm dn}-3}{365}\right)\right],](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_seabQfdiD1UmyAxO255IWdsaArNkkeGYg2f9avaqTCXk8QZVF_Kv95a4IF5v2nZLCr0urmrrLnKY9D45Mcc7o7j-nnluA4Z5DGMNWCYZv54Tn0lroHGP0q4or2MLY3aD6rafbJiLEqyFfvrSmXMw=s0-d)
where dn=1 on January 1; dn=2 on January 2; dn=32 on February 1, etc. In this formula dn−3 is used, because in modern times Earth's perihelion, the closest approach to the Sun and therefore the maximum Eext, occurs around January 3 each year.
The solar illuminance constant (Esc), is equal to 128×103 lx. The direct normal illuminance (Edn), corrected for the attenuating effects of the atmosphere is given by:

where c is the atmospheric extinction coefficient and m is the relative optical airmass.
Solar constant
A 1903 Langley bolograph with an erroneous solar constant of 2.54 calories/minute/square centimeter.
Solar irradiance spectrum at top of atmosphere, on a linear scale and plotted against wavenumber.
The solar constant, a measure of flux, is the amount of incoming solar electromagnetic radiation per unit area that would be incident on a plane perpendicular to the rays, at a distance of one astronomical unit (AU) (roughly the mean distance from the Sun to the Earth). When solar irradiance is measured on the outer surface of Earth's atmosphere,[2] the measurements can be adjusted using the inverse square law to infer the magnitude of solar irradiance at one AU and deduce the solar constant.[3]
The solar constant includes all types of solar radiation, not just the visible light. It is measured by satellite to be roughly 1.366 kilowatts per square meter (kW/m²).[4][5] 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 one part per thousand from day to day. Thus, for the whole Earth (which has a cross section of 127,400,000 km²), the power is 1.740×1017 W, plus or minus 3.5%. The solar constant does not remain constant over long periods of time (see Solar variation), but over a year varies much less than the variation of direct solar irradiance at the top of the atmosphere arising from the ellipticity of the Earth's orbit. The approximate average value cited,[4] 1.366 kW/m², is equivalent to 1.96 calories per minute per square centimeter, or 1.96 langleys (Ly) per minute.
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 342 W/m²). 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 and the location's latitude.
The solar constant includes all wavelengths of solar electromagnetic radiation, not just the visible light (see Electromagnetic spectrum). It is linked to the apparent magnitude of the Sun, −26.8, in that 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.
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 attempted to remove effects due to atmospheric absorption. However, the value he obtained, 2.903 kW/m², was still too great. 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. Abbott proved that one of Langley's corrections was erroneously applied. His 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.[6]
The angular diameter of the Earth as seen from the Sun is approximately 1/11,000 radians, meaning the solid angle of the Earth as seen from the Sun is approximately 1/140,000,000 of a steradian. Thus the Sun emits about two billion times the amount of radiation that is caught by Earth, in other words about 3.86×10
26 watts.