known as Milankovitch cycles. 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, the following applies to the spherical law of cosines: This equation can be also derived from a more general formula: where β is an angle from the horizontal and γ is an 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: 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 ho = 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 Therefore: 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 where ε is the obliquity. (Note: The correct formula, valid for any axial tilt, is .) 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 So from a consensus of observations or theory, 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