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I need to find when the sun reaches the Zenith at a given latitude.

What I've done so far:

$L=23.5 \cdot \sin(\frac{2\pi}{365.25}\cdot D) $

Here L is the latitude (<23.5) and D is number of days from March 21.

This is based on my understanding that the sun executes SHM around the celestial equator. I then solve this equation and get two roots, which are the required days.

My questions are:

  1. Since the orbit of the earth is not exactly spherical, how much deviation actually occurs?

  2. Am I right in understanding that since the declination is continuously shifting, the sun may actually never reach the Zenith? In such a scenario, I am only calculating the day when the little circle that the sun is instantaneously travelling on, meets the Zenith.

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