A note on Dhuhr

From Pray Times

The time for Zuhr has been defined in several ways in the literature:

1. When the Sun begins to decline (Zawaal) after reaching its highest point in the sky.
2. When the shadow of an indicator (a vertical stick) reaches its minimum length and starts to increase.
3. 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:

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, Zuhr time is 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 Zuhr starts. We need the following information to calculate this time:

• Sun-Earth distance (d)
  • Minimum: 147,098,074 km
  • Maximum: 152,097,701 km

• Sun radius (r):
  • Average: 695,500 km

Using the above data, the time t needed for Sun to pass its zenith can be computed by the following formula:

The maximum value obtained by the above formula (which corresponds to the minimum Sun-Earth distance) is 65.0 seconds. Therefore, it takes approximately 1.1 minutes until Sun's disk comes out of its zenith that should be considered into consideration for calculating Zuhr, if the third definition is used.

Conclusion

There are three definitions for the Zuhr time as described above. According to the first two definitions, Zuhr = midday, and according to the third definition, Zuhr = midday + 1.1 minutes.

References

• Sun, from Wikipedia, the free encyclopedia.
• Sun-Earth Distance, from Wiki Answers.