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I'll be honest here; I don't know much about the physics of planets and planetary systems, etc. However, I've been looking into ellipsoidal moons, and most of what I've found is about Nix, one of Pluto's moons. The problem with this is that Pluto is very different from Earth, and Charon changes Nix's orbit drastically.

So, without another moon half the size of the planet, how would an ellipsoidal moon orbit and spin? What would phases look like? And would a different moon shape have any effect on larger aspects of everyday life?

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The tides and orbit only depend on the centre of gravity. If you had some infeasibly strong substance from which you could make an ellipsoidal moon with the same mass as our current moon, and you were to remove our current moon and replace it with the new moon with its centre of gravity in the same place and moving with the same velocity, then nothing would change. It would stay in exactly the same orbit and the tides would be exactly the same. It would just look a bit different.

An ellipsoidal moon would experience tides differently, and would probably be more likely to tidally lock to its primary -- but our moon is already tidally locked to the Earth, so nothing would change there either.

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As long as you use the tag, a notably ellipsoid moon has to be small. Small enough that it isn't pulled into a round shape. There are some effects due to the rotation, but if you seriously want an egg, it has to be smaller than Ceres.

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