The general expression for , the time of sunrise and sunset, can be given an alternative rearrangement which shows two components to the variation - changing lengths of morning and afternoon which are symmetrical about the time of the Sun's highest elevation , and a changing .
Point P will experience the Sun at highest elevation when the azimuths of P and the Sun are the same, i.e.
Writing the equation for and substituting in the polar and azimuthal angles for the Sun's vector, and , we get
Thus ( ) is the (symmetrical) length of morning and afternoon about the time of Sun's highest elevation .
Evaluating with approximations:
Changing to hours with = 45 degrees latitude
Now evaluating the time of Sun's highest elevation, i.e. evaluating the azimuthal angle of the Sun, , we find
So putting both components together,
This formula of sunrise time throughout the year is the red solid line in the graph , and the variation of time of Sun's highest elevation each day is shown as the blue dotted line.
Astronomers know the more general form of this calculation as
the Equation of Time.