Solar zenith angle explained

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.^{[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] 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

$$\backslash cos\; \backslash theta\_s\; =\; \backslash sin\; \backslash alpha\_s\; =\; \backslash sin\; \backslash Phi\; \backslash sin\; \backslash delta\; +\; \backslash cos\; \backslash Phi\; \backslash cos\; \backslash delta\; \backslash cos\; h$$

where

\theta _{s} is the solar zenith angle\alpha _{s} is the solar altitude angle, \alpha _{s} = 90 ^{\circ} - \theta _{s} h is the hour angle, in the local solar time.\delta is the current declination of the Sun\Phi 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.

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.^[4]

In the Earth-Centered Earth-Fixed (ECEF) geocentric Cartesian coordinate system, let

(\phi _{s} , λ _{s} ) and (\phi _{o} , λ _{o} ) be the latitudes and longitudes, or coordinates, of the subsolar point and the observer's point, then the upward-pointing unit vectors at the two points, S and V _{oz} , are

$$\backslash mathbf=\backslash cos\backslash phi\_\backslash cos\backslash lambda\_+\backslash cos\backslash phi\_\backslash sin\backslash lambda\_+\backslash sin\backslash phi\_,$$$$\backslash mathbf\_=\backslash cos\backslash phi\_\backslash cos\backslash lambda\_+\backslash cos\backslash phi\_\backslash sin\backslash lambda\_+\backslash sin\backslash phi\_.$$

where

{ i} , { j} and { k} are the basis vectors in the ECEF coordinate system.

Now the cosine of the solar zenith angle,

\theta _{s} , is simply the dot product of the above two vectors

$$\backslash cos\backslash theta\_\; =\; \backslash mathbf\backslash cdot\backslash mathbf\_\; =\; \backslash sin\backslash phi\_\backslash sin\backslash phi\_\; +\; \backslash cos\backslash phi\_\backslash cos\backslash phi\_\backslash cos(\backslash lambda\_-\backslash lambda\_).$$

Note that

\phi _{s} is the same as \delta , the declination of the Sun, and λ _{s} -λ _{o} is equivalent to - h , where h 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

At any given location on any given day, the solar zenith angle,

\theta _{s} , reaches its minimum, \theta _{min} , at local solar noon when the hour angle h = 0 , or λ _{s} -λ _{o} =0 , namely, \cos\theta _{min} = \cos(|\phi _{o} - \phi _{s} |) , or \theta _{min} = |\phi _{o} - \phi _{s} | . If \theta _{min} > 90 ^{\circ} , it is polar night.

And at any given location on any given day, the solar zenith angle,

\theta _{s} , reaches its maximum, \theta _{max} , at local midnight when the hour angle h = -180 ^{\circ} , or λ _{s} -λ _{o} =-180 ^{\circ} , namely, \cos\theta _{max} = \cos( 180 ^{\circ} - |\phi _{o} + \phi _{s} |) , or \theta _{max} = 180 ^{\circ} - |\phi _{o} + \phi _{s} | . If \theta _{max} < 90 ^{\circ} , 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.^[5]

Applications

Sunrise/Sunset

See main article: articles and Sunrise equation. Sunset and sunrise occur (approximately) when the zenith angle is 90°, where the hour angle h₀ satisfies$$\backslash cos\; h\_0\; =\; -\backslash tan\; \backslash Phi\; \backslash tan\; \backslash delta.$$

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$$\backslash overline\; =\; \backslash frac$$where Q is the instantaneous irradiance.

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.^[6]

See also

• Azimuth
• Solar azimuth angle
• Horizontal coordinate system
• List of orbits
  • Position of the Sun
• Sun path
• Sunrise
• Sunset
• Sun transit time

Notes and References

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