A note on Dhuhr

Dhuhr has been defined in several ways in the fiqh literature:

# When the Sun begins to decline (Zawaal) after reaching its highest point in the sky.
# When the shadow of an indicator (a vertical stick) reaches its minimum length and starts to increase. 
# When the Sun's disk comes out of its zenith line, which is a line between the observer and the center of the Sun when it is at the highest point.

The first and the second definitions are obviously equivalent, as the shadow length has a direct correlation to the Sun's elevation in the sky via the following formula:

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Shadow Length = Object Height &times; cot(Sun Angle).
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The Sun's angle is a continuous function over time and has only one peak point (the point at which the tangent to its curve has zero slope) which is realized exactly at midday. Therefore, according to the first two definitions, Dhuhr begins immediately after the midday.

The third definition is slightly different from the previous two definitions. According to this definition, Sun must passes its zenith line before Dhuhr starts. We need the following information to calculate this time:

* Sun radius (r): 695,500 km
* Sun-Earth distance (d): 147,098,074 km to 152,097,701 km

Having r and d, the time t needed for Sun to pass its zenith line can be computed using the following formula:

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t = arctan(r/d) / 2&pi; &times; 24 &times; 60 &times; 60.
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The maximum value obtained by the above formula (which corresponds to the minimum Sun-Earth distance) is 65.0 seconds. Therefore, it takes approximately 1 minute until Sun's disk comes out of its zenith that should be considered into consideration for calculating Dhuhr, if the third definition is used.

== Conclusion ==

There are three definitions for the Dhuhr time as described above. According to the first two definitions, Dhuhr = midday, and according to the third definition, Dhuhr = midday + 65 sec.

== References ==

* [http://en.wikipedia.org/wiki/Sun Sun], from Wikipedia, the free encyclopedia.
* [http://wiki.answers.com/Q/What_is_the_distance_from_the_Sun_to_the_Earth Sun-Earth Distance], from Wiki Answers.