Solar zenith angle

Template:Short description

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.<ref>Template:Cite book</ref><ref name="hartmann">Template:Cite book</ref> At solar noon, the altitude angle (complement of the solar 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.<ref>Template:Cite book</ref><templatestyles src="Template:TOC_right/styles.css" />{{#if:|<templatestyles src="Template:TOC limit/styles.css" />}}

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.

FormulaEdit

<math display="block"> \cos \theta_s = \sin \alpha_s = \sin \Phi \sin \delta + \cos \Phi \cos \delta \cos h</math>

where

<math>\theta_s</math> is the solar zenith angle
<math>\alpha_s</math> is the solar altitude angle, <math>\alpha_s = 90^\circ - \theta_s</math>
<math>h</math> is the hour angle, in the local solar time.
<math>\delta</math> is the current declination of the Sun
<math>\Phi</math> is the local latitude.

Derivation of the formula using the subsolar point and vector analysisEdit

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.<ref name="Zhangetal">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</ref>

In the Earth-Centered Earth-Fixed (ECEF) geocentric Cartesian coordinate system, let <math>(\phi_{s}, \lambda_{s})</math> and <math>(\phi_{o}, \lambda_{o})</math> 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, <math>\mathbf{S}</math> and <math>\mathbf{V}_{oz}</math>, are

<math display="block">\mathbf{S}=\cos\phi_{s}\cos\lambda_{s}{\mathbf i}+\cos\phi_{s}\sin\lambda_{s}{\mathbf j}+\sin\phi_{s}{\mathbf k},</math> <math display="block">\mathbf{V}_{oz}=\cos\phi_{o}\cos\lambda_{o}{\mathbf i}+\cos\phi_{o}\sin\lambda_{o}{\mathbf j}+\sin\phi_{o}{\mathbf k}.</math>

where <math>{\mathbf i}</math>, <math>{\mathbf j}</math> and <math>{\mathbf k}</math> are the basis vectors in the ECEF coordinate system.

Now the cosine of the solar zenith angle, <math>\theta_{s}</math>, is simply the dot product of the above two vectors

<math display="block">\cos\theta_{s} = \mathbf{S}\cdot\mathbf{V}_{oz} = \sin\phi_{o}\sin\phi_{s} + \cos\phi_{o}\cos\phi_{s}\cos(\lambda_{s}-\lambda_{o}).</math>

Note that <math>\phi_{s}</math> is the same as <math>\delta</math>, the declination of the Sun, and <math>\lambda_{s}-\lambda_{o}</math> is equivalent to <math>-h</math>, where <math>h</math> is the hour angle defined earlier. So the above format is mathematically identical to the one given earlier.

Additionally, Ref. <ref name="Zhangetal" /> also derived the formula for solar azimuth angle in a similar fashion without using spherical trigonometry.

Minimum and MaximumEdit

File:Solar Zenith Angle min.png

The daily minimum of the solar zenith angle as a function of latitude and day of year for the year 2020.

File:Solar Zenith Angle max.png

The daily maximum of the solar zenith angle as a function of latitude and day of year for the year 2020.

At any given location on any given day, the solar zenith angle, <math>\theta_{s}</math>, reaches its minimum, <math>\theta_\text{min}</math>, at local solar noon when the hour angle <math>h = 0</math>, or <math>\lambda_{s}-\lambda_{o}=0</math>, namely, <math>\cos\theta_\text{min} = \cos(|\phi_{o}-\phi_{s}|)</math>, or <math>\theta_\text{min} = |\phi_{o}-\phi_{s}|</math>. If <math>\theta_\text{min} > 90^{\circ}</math>, it is polar night.

And at any given location on any given day, the solar zenith angle, <math>\theta_{s}</math>, reaches its maximum, <math>\theta_\text{max}</math>, at local midnight when the hour angle <math>h = -180^{\circ}</math>, or <math>\lambda_{s}-\lambda_{o}=-180^{\circ}</math>, namely, <math>\cos\theta_\text{max} = \cos(180^{\circ}-|\phi_{o}+\phi_{s}|)</math>, or <math>\theta_\text{max} = 180^{\circ}-|\phi_{o}+\phi_{s}|</math>. If <math>\theta_\text{max} < 90^{\circ}</math>, it is polar day.

CaveatsEdit

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.<ref>Template:Cite journal</ref>

ApplicationsEdit

Sunrise/SunsetEdit

Template:Main articles Sunset and sunrise occur (approximately) when the zenith angle is 90°, where the hour angle h₀ satisfies<ref name="hartmann" /> <math display="block">\cos h_0 = -\tan \Phi \tan \delta.</math>

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.

AlbedoEdit

A weighted daily average zenith angle, used in computing the local albedo of the Earth, is given by <math display="block">\overline{\cos \theta_s} = \frac{\displaystyle \int_{-h_0}^{h_0} Q \cos \theta_s \, \text{d}h}{\displaystyle \int_{-h_0}^{h_0} Q \, \text{d}h}</math> where Q is the instantaneous irradiance.<ref name="hartmann" />

Summary of special anglesEdit

Template:Subsolar point date graph 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.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

ReferencesEdit

Template:Reflist Template:Portal bar