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If an Earthlike planet had two moons, one relatively smaller and closer while the other was bigger and farther away, and one had a prograde orbit while the other had a retrograde orbit, in what ways might this affect the planet's tides?

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First, notice that this happens on earth too, the tidal effect of the moon is about 2.3 times larger than that of the sun. The fact that both orbits go in opposite directions will not be qualitatively important, only quantitatively in terms of the irregular variation in tidal periods and sizes.

If the two moons have the same density and angular size in the sky, and move on the same plane, both tidal effects will have the same strength. This will result in something that does not happen on earth: when the moons are at 90 degrees of each other in the sky (in a reference frame on the center of the planet), the tidal effect will cancel and the entire planet will be momentarily without tides.

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