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Rotation of the Earth

The earth rotates uniformly once per 24 hours in this frame (mean solar time). Let a point on the earth's surface P define a unit vector tex2html_wrap_inline204 at the origin of the frame. The azimuth angle tex2html_wrap_inline206 of this vector is measured as usual in the X-Y plane from the X-axis, so in a day, tex2html_wrap_inline206 covers the complete range of 0 to 360 degrees. Generally, time zones are chosen so that at some longitude in the zone, tex2html_wrap_inline208 at noon. To simplify, let P have this longitude, at latitude tex2html_wrap_inline210. Then tex2html_wrap_inline206 is a measure of time of day at P with respect to noon. If we can find where vector tex2html_wrap_inline204 is at sunrise, then that value of tex2html_wrap_inline206 will give us the time of sunrise.

The explicit relationship between the vector xyz components and angles tex2html_wrap_inline210 and tex2html_wrap_inline206 is

displaymath202

If P were in a different place in the time zone, i.e. P had a different longitude from that assumed above, a constant shift would occur in the times of sunrise, sunset, and sun's highest elevation.

We will take the latitude of the observer to be tex2html_wrap_inline214.