Solar zenith angle

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

where

• $\backslash theta\_s$ is the solar zenith angle
• $\backslash alpha\_s$ is the solar altitude angle, $\backslash alpha\_s\; =\; 90^\backslash circ\; -\; \backslash theta\_s$
• $h$ is the hour angle, in the local solar time.
• $\backslash delta$ is the current declination of the Sun
• $\backslash Phi$ is the local latitude.

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.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

In the Earth-Centered Earth-Fixed (ECEF) geocentric Cartesian coordinate system, let $(\backslash phi\_\{s\},\; \backslash lambda\_\{s\})$ and $(\backslash phi\_\{o\},\; \backslash lambda\_\{o\})$ 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, $\backslash mathbf\{S\}$ and $\backslash mathbf\{V\}\_\{oz\}$, are

$$\backslash mathbf\{S\}=\backslash cos\backslash phi\_\{s\}\backslash cos\backslash lambda\_\{s\}\{\backslash mathbf\; i\}+\backslash cos\backslash phi\_\{s\}\backslash sin\backslash lambda\_\{s\}\{\backslash mathbf\; j\}+\backslash sin\backslash phi\_\{s\}\{\backslash mathbf\; k\},$$

$$\backslash mathbf\{V\}\_\{oz\}=\backslash cos\backslash phi\_\{o\}\backslash cos\backslash lambda\_\{o\}\{\backslash mathbf\; i\}+\backslash cos\backslash phi\_\{o\}\backslash sin\backslash lambda\_\{o\}\{\backslash mathbf\; j\}+\backslash sin\backslash phi\_\{o\}\{\backslash mathbf\; k\}.$$

where $\{\backslash mathbf\; i\}$, $\{\backslash mathbf\; j\}$ and $\{\backslash mathbf\; k\}$ are the basis vectors in the ECEF coordinate system.

Now the cosine of the solar zenith angle, $\backslash theta\_\{s\}$, is simply the dot product of the above two vectors

$$\backslash cos\backslash theta\_\{s\}\; =\; \backslash mathbf\{S\}\backslash cdot\backslash mathbf\{V\}\_\{oz\}\; =\; \backslash sin\backslash phi\_\{o\}\backslash sin\backslash phi\_\{s\}\; +\; \backslash cos\backslash phi\_\{o\}\backslash cos\backslash phi\_\{s\}\backslash cos(\backslash lambda\_\{s\}-\backslash lambda\_\{o\}).$$

Note that $\backslash phi\_\{s\}$ is the same as $\backslash delta$, the declination of the Sun, and $\backslash lambda\_\{s\}-\backslash lambda\_\{o\}$ 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.

File:Solar Zenith Angle min.png

File:Solar Zenith Angle max.png

At any given location on any given day, the solar zenith angle, $\backslash theta\_\{s\}$, reaches its minimum, $\backslash theta\_\backslash text\{min\}$, at local solar noon when the hour angle $h\; =\; 0$, or $\backslash lambda\_\{s\}-\backslash lambda\_\{o\}=0$, namely, $\backslash cos\backslash theta\_\backslash text\{min\}\; =\; \backslash cos(|\backslash phi\_\{o\}-\backslash phi\_\{s\}|)$, or $\backslash theta\_\backslash text\{min\}\; =\; |\backslash phi\_\{o\}-\backslash phi\_\{s\}|$. If $\backslash theta\_\backslash text\{min\}\; >\; 90^\{\backslash circ\}$, it is polar night.

And at any given location on any given day, the solar zenith angle, $\backslash theta\_\{s\}$, reaches its maximum, $\backslash theta\_\backslash text\{max\}$, at local midnight when the hour angle $h\; =\; -180^\{\backslash circ\}$, or $\backslash lambda\_\{s\}-\backslash lambda\_\{o\}=-180^\{\backslash circ\}$, namely, $\backslash cos\backslash theta\_\backslash text\{max\}\; =\; \backslash cos(180^\{\backslash circ\}-|\backslash phi\_\{o\}+\backslash phi\_\{s\}|)$, or $\backslash theta\_\backslash text\{max\}\; =\; 180^\{\backslash circ\}-|\backslash phi\_\{o\}+\backslash phi\_\{s\}|$. 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.

The formula also neglects the effect of atmospheric refraction.{{cite journal | title = On the computation of solar elevation angles and the determination of sunrise and sunset times | page = 3 | first = Harold M. | last = Woolf | journal = NASA Technical Memorandu, X-1646 | date = 1968 | location = Washington, D.C.}}

{{Main articles|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.

A weighted daily average zenith angle, used in computing the local albedo of the Earth, is given by

$$\backslash overline\{\backslash cos\; \backslash theta\_s\}\; =\; \backslash frac\{\backslash displaystyle\; \backslash int\_\{-h\_0\}^\{h\_0\}\; Q\; \backslash cos\; \backslash theta\_s\; \backslash ,\; \backslash text\{d\}h\}\{\backslash displaystyle\; \backslash int\_\{-h\_0\}^\{h\_0\}\; Q\; \backslash ,\; \backslash text\{d\}h\}$$

where Q is the instantaneous irradiance.

{{Subsolar point date graph|300px|float=right}}

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.{{cite web|last=livioflores-ga|url=http://answers.google.com/answers/threadview/id/782886.html|title=Equation to know where the Sun is at a given place at a given date-time|accessdate=9 March 2013}}

• Azimuth
• Solar azimuth angle
• Horizontal coordinate system
• List of orbits
• {{slink|Photovoltaic mounting system#Orientation and inclination}}
• Position of the Sun
• Sun path
• Sunrise
• Sunset
• Sun transit time

{{Reflist|30em}}

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Category:Horizontal coordinate system

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Category:Solar energy