The solar zenith angle is the angle between the sun’s rays and the vertical direction. It is closely related to the solar altitude angle, which is the angle between the sun’s rays and a horizontal plane. Since these two angles are complementary, the cosine of either one of them equals the sine of the other. They can both be calculated with the same formula, using results from spherical trigonometry. At solar noon, the zenith angle is at a minimum and is equal to latitude minus solar declination angle. This is the basis by which ancient mariners navigated the oceans.
- is the solar zenith angle
- is the solar altitude angle, = 90° –
- is the hour angle, in the local solar time.
- is the current declination of the Sun
- is the local latitude.
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.
In the Earth-Centered Earth-Fixed (ECEF) geocentric Cartesian coordinate system, let and be the longitudes and latitudes, or coordinates, of the subsolar point and the observer's point, then the upward-pointing unit vectors at the two points, and , are
where , and are the basis vectors in the ECEF coordinate system.
Now the cosine of the solar zenith angle, , is simply the dot product of the above two vectors
Note that is the same as , the declination of the Sun, and is equivalent to , where is the hour angle defined earlier. So the above format is mathematically identical to the one given earlier.
Minimum and Maximum
At any given location on any given day, the solar zenith angle, , reaches its minimum, , at local solar noon when the hour angle , or , namely, , or . If , it is polar night.
And at any given location on any given day, the solar zenith angle, , reaches its maximum, , at local midnight when the hour angle , or , namely, , or . If , it is polar day.
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.
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.
A weighted daily average zenith angle, used in computing the local albedo of the Earth, is given by
Summary of special angles
For example, the solar elevation angle is :
- 90° if you are on the equator, a day of equinox, at a solar hour of twelve
- near 0° at the sunset or at the sunrise
- between -90° and 0° during the night (midnight)
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- Zhang, T., Stackhouse, P.W., Macpherson, B., and Mikovitz, J.C., 2021. A solar azimuth formula that renders circumstantial treatment unnecessary without compromising mathematical rigor: Mathematical setup, application and extension of a formula based on the subsolar point and atan2 function. Renewable Energy, 172, 1333-1340. DOI: https://doi.org/10.1016/j.renene.2021.03.047
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