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I would like to know when the moon is directly overhead 0° S, 166° E. Are there any websites that will show me any of those dates?

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Unlike the sun, which follows exactly the ecliptic and so is directly on the zenith on the date of the equinoxes, the moon is tilted by 5 degrees relative to the eclipic. And the direction of this tilt changes over 18 years.

Moreover the motion of the moon is quite fast so even if it is rising due East, by the time it culminates it will have moved and so not be exactly on the zenith.

As you see the moon's orbit is actually one of the more complex orbits in the solar system. It will be quite close to the zenith once a day, and very close once a month, The only way to find if the moon is (within 0.5 degrees) of the zenith is to use planetatium software to model its motion. Possiblities include "stellarium" or if you have programming skills then "astropy".

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