The solar zenith angle is the zenith angle of the Sun, i.e., the angle between the sun’s rays and the vertical direction. It is the complement to the solar altitude or solar elevation, which is the altitude angle or elevation angle between the Sun’s rays and a horizontal plane. Near (but usually not exactly at) solar noon, the zenith angle is at a minimum and is equal to the absolute value of latitude minus solar declination angle. This is the basis by which ancient mariners navigated the oceans. Solar zenith angle is normally used in combination with the solar azimuth angle to determine the position of the Sun as observed from a given location on the surface of the Earth.

Formula

where

• is the solar zenith angle

• is the solar altitude angle

• is the hour angle, in the local solar time.

• is the current declination of the Sun

• is the local latitude.

This equation is equivalent to the formula for the angle between vectors in spherical coordinates applied to the local normal and the direction to the sun, with the adjustment that the latitude is the complement to the polar angle and that the hour angle is the difference of azimuths.

where we have used the difference identity for cosine.

Derivation of the formula using the subsolar point and vector analysis

While the formula can be derived by applying the cosine law to the zenith-pole-Sun spherical triangle, the spherical trigonometry is a relatively esoteric subject.

By introducing the coordinates of the subsolar point and using vector analysis, the formula can be obtained straightforward without incurring the use of spherical trigonometry.

In the Earth-Centered Earth-Fixed (ECEF) geocentric Cartesian coordinate system be the latitudes and longitudes, or coordinates, of the subsolar point, are

are the basis vectors in the ECEF coordinate system.

, is simply the dot product of the above two vectors

is the hour angle defined earlier. So the above format is mathematically identical to the one given earlier.

Additionally, Ref. also derived the formula for solar azimuth angle in a similar fashion without using spherical trigonometry.

Minimum and Maximum

, it is polar night.

, it is polar day.

Caveats

The calculated values are approximations due to the distinction between common/geodetic latitude and geocentric latitude. However, the two values differ by less than 12 minutes of arc, which is less than the apparent angular radius of the Sun.

The formula also neglects the effect of atmospheric refraction.

Applications

Sunrise/Sunset

Sunset and sunrise occur (approximately) when the zenith angle is 90°, where the hour angle h0 satisfies

Precise times of sunset and sunrise occur when the upper limb of the Sun appears, as refracted by the atmosphere, to be on the horizon.

Albedo

A weighted daily average zenith angle, used in computing the local albedo of the Earth where Q is the instantaneous irradiance.

Summary of special angles

For example, the solar elevation angle is:

• 90° at the subsolar point, which occurs, for example, at the equator on a day of equinox at solar noon

• near 0° at the sunset or at the sunrise

• between −90° and 0° during the night (midnight)

An exact calculation is given in position of the Sun. Other approximations exist elsewhere.

See also

• Azimuth

• Solar azimuth angle

• Horizontal coordinate system

• List of orbits

• Position of the Sun

• Sun path

• Sunrise

• Sunset

• Sun transit time

References