Solar zenith angle

The solar zenith angle is the angle between the zenith and the centre of the sun's disc. The solar elevation angle is the altitude of the sun, the angle between the horizon and the centre of the sun's disc. 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]

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 elevation angle, $\alpha_s$ = 90° – $\theta_s$
$h$ is the hour angle, in the local solar time.
$\delta$ is the current declination of the Sun
$\Phi$ is the local latitude.

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 neglects the effect of atmospheric refraction.^[3]

Applications

Sunrise/Sunset

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

\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{\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 insolation.^[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

An exact calculation is given in position of the Sun. Other approximations exist elsewhere.^[4]

References

↑ Jacobson, Mark Z. (2005). Fundamentals of Atmospheric Modeling (2nd ed.). Cambridge University Press. p. 317. ISBN 0521548659.
1 2 3 Hartmann, Dennis L. (1994). Global Physical Climatology. Academic Press. p. 30. ISBN 0080571638.
↑ 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.
↑ livioflores-ga. "Equation to know where the Sun is at a given place at a given date-time". Google. Retrieved 9 March 2013.

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