Solar zenith angle

Formula

\cos \theta_s = \sin \alpha_s = \sin \Phi \sin \delta + \cos \Phi \cos \delta \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.

At solar noon, this reads \cos \theta_s = \sin \Phi \sin \delta + \cos \Phi \cos \delta = \cos(\Phi - \delta) where we have used the difference identity for cosine.

Therefore at solar noon, \theta_s = \Phi - \delta

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.

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

\mathbf{S}=\cos\phi_{s}\cos\lambda_{s}{\mathbf i}+\cos\phi_{s}\sin\lambda_{s}{\mathbf j}+\sin\phi_{s}{\mathbf k}, \mathbf{V}{oz}=\cos\phi{o}\cos\lambda_{o}{\mathbf i}+\cos\phi_{o}\sin\lambda_{o}{\mathbf j}+\sin\phi_{o}{\mathbf k}.

where {\mathbf i}, {\mathbf j} and {\mathbf 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

\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}).

Note that \phi_{s} is the same as \delta, the declination of the Sun, and \lambda_{s}-\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.

Minimum and Maximum

At any given location on any given day, the solar zenith angle, \theta_{s}, reaches its minimum, \theta_\text{min}, at local solar noon when the hour angle h = 0, or \lambda_{s}-\lambda_{o}=0, namely, \cos\theta_\text{min} = \cos(|\phi_{o}-\phi_{s}|), or \theta_\text{min} = |\phi_{o}-\phi_{s}|. If \theta_\text{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_\text{max}, at local midnight when the hour angle h = -180^{\circ}, or \lambda_{s}-\lambda_{o}=-180^{\circ}, namely, \cos\theta_\text{max} = \cos(180^{\circ}-|\phi_{o}+\phi_{s}|), or \theta_\text{max} = 180^{\circ}-|\phi_{o}+\phi_{s}|. If \theta_\text{max} , 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.

Applications

Sunrise/Sunset

Main article: Sunrise equation

Sunset and sunrise occur (approximately) when the zenith angle is 90°, where the hour angle h0 satisfies \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 \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} where Q is the instantaneous irradiance.

Summary of special angles

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.

References

References

1. Jacobson, Mark Z.. (2005). "Fundamentals of Atmospheric Modeling". [[Cambridge University Press]].
2. Hartmann, Dennis L.. (1994). "Global Physical Climatology". [[Academic Press]].
3. (2005). "Ecological climatology: concepts and applications". Cambridge University Press.
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. Woolf, Harold M.. (1968). "On the computation of solar elevation angles and the determination of sunrise and sunset times". NASA Technical Memorandu, X-1646.
6. livioflores-ga. "Equation to know where the Sun is at a given place at a given date-time".