.
Let the sun's latitude and azimuth angle in our special frame rotating once per
year be
and
so that
The sun rotates in an assumed circular orbit as seen from the earth.
The coordinates of the sun's vector
in this orbit are
,
where d = fraction of year elapsed since the winter solstice, i.e.
From our special frame, the orbit is tilted at angle
which is about 23 degrees in the X-Z plane. So we apply a rotation
transformation
as shown below. In addition, our special frame of
reference rotates also once per year but around the earth's axis so that
another rotation transformation is needed. Thus
may be calculated using two coordinate transformations
As a check,
values at winter solstice (d = 0) and at spring equinox (d = 0.25), are
as expected. Explicitly solving for
Note that
so that
(the term responsible for the lag and advance of the sun) is small and
.
Thus
describes a figure of eight motion around the
vector moving mainly up and down by 23 degrees.