Insolation - Distribution of Insolation at The Top of The Atmosphere

Distribution of Insolation At The Top of The Atmosphere

The theory for the distribution of solar radiation at the top of the atmosphere concerns how the solar irradiance (the power of solar radiation per unit area) at the top of the atmosphere is determined by the sphericity and orbital parameters of Earth. The theory could be applied to any monodirectional beam of radiation incident onto a rotating sphere, but is most usually applied to sunlight, and in particular for application in numerical weather prediction, and theory for the seasons and the ice ages. The last application is known as Milankovitch cycles.

The derivation of distribution is based on a fundamental identity from spherical trigonometry, the spherical law of cosines:

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, we equate the following for use in the spherical law of cosines:

The distance of Earth from the sun can be denoted R_E, and the mean distance can be denoted R₀, which is very close to 1 AU. The insolation onto a plane normal to the solar radiation, at a distance 1 AU from the sun, is the solar constant, 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:

and

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:

Let h₀ be the hour angle when Q becomes positive. This could occur at sunrise when, or for h₀ as a solution of

or

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 .

is nearly constant over the course of a day, and can be taken outside the integral

Let θ be the conventional polar angle describing a planetary orbit. For convenience, let θ = 0 at the vernal equinox. The declination δ as a function of orbital position is

where ε is the obliquity. The conventional longitude of perihelion ϖ is defined relative to the vernal 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. Note that because of the elliptical orbit, and as a simple consequence of Kepler's second law, θ does not progress exactly 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.