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Given that eclipses occur, why aren't they more regular? If the sun and the moon are in the same plane, shouldn't the moon transect the line from the earth to the sun every month? I guess order of magnitude, if the moon at 10^-2 radians was in a random place every day, then an eclipse once / 18 months is about right.

This is a genuine question, not a troll, this question has bothered me for years.

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The sun and moon are not in the same plane there is about a 5 degree tilt. You are right that if they were then there would be an eclipse every month. I encourage you to read this article which details on this phenomenon. Also, see the video for better illustration.

Video

Eclipses

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Probably not a full answer, but sadly too long for a comment.

The Solar system isn't 2-dimensional. If it were, every time the Moon passes over a point on Earth's surface it would block the Sun there (if it were the daytime in the first place). However, this scenario works predict one other blatantly untrue idea: that all solar eclipses would occur around a single Equator-like great circle on Earth. In truth, the orbits are in different planes.

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