# Solar Angles

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The axis about which the earth rotates is tilted at an angle of 23.45 degrees to the plane of the earth’s orbital plane and the sun’s equator.
The earth’s axis results in a day-by-day variation of the angle between the earth–sun line and the earth’s equatorial plane called the solar declination δ. This angle may be estimated by the following equation:

where N = year day, with January 1 + 1.
The position of the sun can be defined in terms of its altitude β above the horizon and its azimuth ϕ measured in horizontal plane (Fig.).
To determine the angle of incidence θ between a direct solar beam and the normal to the surface, the surface azimuth ψ and the surface-solar azimuth γ must be known. The surface-solar azimuth is designated by γ and is the angular difference between the solar azimuth ϕ and the surface azimuth ψ. For a surface facing the east of south, γ = ϕ − ψ in the morning, and γ = ϕ + ψ in the afternoon. For surfaces facing the west of south, γ = ϕ + ψ in the morning and γ = ϕ − ψ in the afternoon. For south-facing surfaces, ψ = 0 degree, so γ = ϕ for all conditions. The angles δ, β, and ϕ are always positive.
For a surface with tilt angle Σ (measured from the horizontal), the angle of incidence θ is given by

For vertical surfaces, Σ = 90 degrees, cosΣ = 0, and sinΣ = 1.0, so Eq. above becomes

For horizontal surfaces, Σ = 0 degree, sinΣ = 0, and cosΣ = 1.0, so Eq. leads to

Latitude φ is the angular location north or south of the equator, north positive; −90 degrees ≤ φ ≤ 90 degrees.
Zenith angle θz, the angle between the vertical and the line to the sun, is the angle of incidence of direct (beam) radiation on a horizontal surface (θz = θ).
Hour angle ω is the angular displacement of the sun east or west of the local meridian due to rotation of the earth on its axis at 15 degrees per hour; morning negative (−ωs) and afternoon positive (+ωs) (Fig.). The sun position at any hour τ can be expressed as follows: