Solar zenith angle
The solar zenith angle z is a function of time, day number and latitude. It can be calculated using the relation:
in which is the declination of the sun, the latitude (defined as positive in the northern hemisphere) and the hour angle. The latter is a measure of the local time, i.e. it is defined as the angle through which the earth must turn to bring the meridian of the location of observation directly under the sun. The solar declination is a function of the day of the year and is independent of the location. It varies from 23o27' on June 21 to -23o27' on December 22.
At noon at any latitude cos( ) = 0 and z= - . At sunrise or sunset at any latitude except the North or South Poles cos( z ) = 0, and = Nd/2 = half-day length (expressed in rad, i.e., 2 corresponds to 24 h), and
The day length will be 12 hours if either tan( ) = 0 (the equator on all days) or tan( ) = 0 (the equinoxes at all latitudes except the Poles). The latitude of the polar night may be found by setting Nd = 0 in the above equation, leading to 90o-| |= in the winter hemisphere.
At the Poles cos( ) = 0, sin( ) = 1,
and cos( z ) = sin( ) or (in degrees):
90o - z = .
Hence, at these points the elevation angle of the sun always
equals the declination angle and during 6 months of daylight
the sun simply circles around the horizon, never rising more than
23o27' above it.