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Components of Sunrise Variation

The general expression for tex2html_wrap_inline262 , 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 tex2html_wrap_inline302 , and a changing tex2html_wrap_inline302 .

Point P will experience the Sun at highest elevation when the azimuths of P and the Sun are the same, i.e.

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Writing the equation for tex2html_wrap_inline262 and substituting in the polar and azimuthal angles for the Sun's vector, tex2html_wrap_inline230 and tex2html_wrap_inline232 , we get

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where

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Thus ( tex2html_wrap_inline314 ) is the (symmetrical) length of morning and afternoon about the time of Sun's highest elevation tex2html_wrap_inline232 .

Evaluating with approximations:

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Changing to hours with tex2html_wrap_inline210 = 45 degrees latitude

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Now evaluating the time of Sun's highest elevation, i.e. evaluating the azimuthal angle of the Sun, tex2html_wrap_inline232 , we find

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So putting both components together,

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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.