# 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

${\displaystyle \cos \theta _{s}=\sin \alpha _{s}=\sin \Phi \sin \delta +\cos \Phi \cos \delta \cos h}$

where

• ${\displaystyle \theta _{s}}$ is the solar zenith angle
• ${\displaystyle \alpha _{s}}$ is the solar elevation angle or solar altitude angle, ${\displaystyle \alpha _{s}}$ = 90° – ${\displaystyle \theta _{s}}$
• ${\displaystyle h}$ is the hour angle, in the local solar time.
• ${\displaystyle \delta }$ is the current declination of the Sun
• ${\displaystyle \Phi }$ 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 also 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]

${\displaystyle \cos h_{0}=-\tan \Phi \tan \delta .}$

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

${\displaystyle {\overline {\cos \theta _{s}}}={\frac {\int _{-h_{0}}^{h_{0}}Q\cos \theta _{s}{\text{d}}h}{\int _{-h_{0}}^{h_{0}}Q{\text{d}}h}}}$

where Q is the instantaneous irradiance.[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 (midnight)

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