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After learning what apogee and perigee are, I'm curious to know if/when these stages align with particular phases of the Moon? I'm obviously assuming that the orbit has a repeating pattern, but maybe I'm wrong?

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  • $\begingroup$ When the periapsis coincides with Full Moon, you're facing the Supermoon phenomenon. $\endgroup$ – SF. Dec 7 '15 at 8:07
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In general, no. Most lunar cycles won't align with its orbital distance to Earth (perigee at its closest or apogee at its farthest). There is however an orbital pattern, or a resonance if you will, that combines both periodically and it's called a full moon cycle which is roughly 14 lunations long period when the Moon will be full at roughly its perigee. When the Moon is both the closest to Earth (perigee) and also in opposition (on the other side of the Earth than the Sun) at exactly the same time, a supermoon eclipse happens. This is another such cycle and it is called saros. It occurs every 18 years and a bit, or 223 synodic months (time between one full moon and another).

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