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Sun's Vector in Frame

Call the unit vector pointing from the center of the earth to the sun tex2html_wrap_inline228. Let the sun's latitude and azimuth angle in our special frame rotating once per year be tex2html_wrap_inline230 and tex2html_wrap_inline232 so that

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The sun rotates in an assumed circular orbit as seen from the earth. The coordinates of the sun's vector in this orbit are tex2html_wrap_inline33 , where d = fraction of year elapsed since the winter solstice, i.e.

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From our special frame, the orbit is tilted at angle tex2html_wrap_inline234 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 tex2html_wrap_inline228 may be calculated using two coordinate transformations

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As a check, tex2html_wrap_inline228 values at winter solstice (d = 0) and at spring equinox (d = 0.25), are

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as expected. Explicitly solving for tex2html_wrap_inline228

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Note that tex2html_wrap_inline248 so that tex2html_wrap_inline250 (the term responsible for the lag and advance of the sun) is small and tex2html_wrap_inline252 . Thus tex2html_wrap_inline228 describes a figure of eight motion around the tex2html_wrap_inline200 vector moving mainly up and down by 23 degrees.