Solar zenith angle

The solar zenith angle is the angle between the sun’s rays and the vertical. 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.[1][2] 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.[3]

Formula

${\displaystyle \cos \theta _{s}=\sin \alpha _{s}=\sin \Phi \sin \delta +\cos \Phi \cos \delta \cos h}$

where

• ${\displaystyle \theta _{s}}$ is the solar zenith angle
• ${\displaystyle \alpha _{s}}$ is the ' or solar altitude angle, ${\displaystyle \alpha _{s}}$ = 90° – ${\displaystyle \theta _{s}}$
• ${\displaystyle h}$ is the hour angle, in the local solar time.
• ${\displaystyle \delta }$ is the current declination of the Sun
• ${\displaystyle \Phi }$ is the local latitude.

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.[4]

Applications

Sunrise/Sunset

Sunset and sunrise occur (approximately) when the zenith angle is 90°, where the hour angle h₀ satisfies[2]

${\displaystyle \cos h_{0}=-\tan \Phi \tan \delta .}$

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, is given by

${\displaystyle {\overline {\cos \theta _{s}}}={\frac {\int _{-h_{0}}^{h_{0}}Q\cos \theta _{s}{\text{d}}h}{\int _{-h_{0}}^{h_{0}}Q{\text{d}}h}}}$

where Q is the instantaneous irradiance.[2]

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)

An exact calculation is given in position of the Sun. Other approximations exist elsewhere.[5]

See also

• Azimuth
• Horizontal coordinate system
• Position of the Sun
• Sun path
• Sunrise
• Sunset
• Sun transit time

References

1. Jacobson, Mark Z. (2005). Fundamentals of Atmospheric Modeling (2nd ed.). Cambridge University Press. p. 317. ISBN 0521548659.
2. Hartmann, Dennis L. (1994). Global Physical Climatology. Academic Press. p. 30. ISBN 0080571638.
3. Bonan, Gordon (2005). Ecological climatology: concepts and applications. Cambridge University Press. p. 62. ISBN 9781316425190. Retrieved 13 November 2019.
4. Woolf, Harold M. (1968). "On the computation of solar elevation angles and the determination of sunrise and sunset times". NASA Technical Memorandu, X-1646. Washington, D.C.: 3.
5. livioflores-ga. "Equation to know where the Sun is at a given place at a given date-time". Retrieved 9 March 2013.