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How would the phases of the moon change depending on the inclination of its orbit with respect to its primary's equatorial plane? Earth's moon has an orbit that is rather close to the ecliptic, but most natural satellites seem to orbit closer to their primary's equatorial plane.

(Initially I wanted to know what the moon would look like if it had a polar orbit, but it turns out this configuration is unstable due to the Kozai-Lidov mechanism. Is there a "most stable orbit" for natural satellites, some sort of orbital inclination where this effect won't occur? If a natural satellite is on a highly inclined orbit, and if these orbits are unusual, what kind of mechanism can put them there?)

For the purposes of this question, I'm assuming everything else is fairly equal (e.g. Earth-sized primary, Luna-sized moon, and Sol-sized parent star).

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This question asks for hard science. All answers to this question should be backed up by equations, empirical evidence, scientific papers, other citations, etc. Answers that do not satisfy this requirement might be removed. See the tag description for more information.

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    $\begingroup$ You are asking two different questions here: the one in the title and the one in the body. $\endgroup$ – L.Dutch Mar 19 at 9:08
  • $\begingroup$ "sort of orbital inclination where this effect won't occur?" — link you posted has link to en.m.wikipedia.org/wiki/Kozai_mechanism that in turn has equation for timescale of oscillation. Have you read it? Why isn't it sufficient? After all, in cosmology "stable" = with time scale long enough. $\endgroup$ – Mołot Mar 19 at 9:17
  • $\begingroup$ @L.Dutch I'm more interested in the title question, expanded by the first question in the body. Asking what the moon would look like is approximately equal to how would the phases of the moon appear, at least to me. The bracketed text is additional supplementary information not necessarily requiring an answer. $\endgroup$ – Robbie Mar 19 at 9:43
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Addressing the question in the title: From the equator the Moon would still rise roughly in the east and set roughly in the west as the Earth’s rotation is much greater than the period of the Moon’s orbit. But over a period of 28 days the timing of Moon rise would shift through 24 hours.

From the poles the Moon would travel across the sky from horizon to horizon over a period of roughly 14 days and would then disappear for 14 days until reappearing again.

The phases of the Moon would progress in the same way that we currently see them except for 2 things. The full cycle would take an entire calendar year. The Moon’s orbit would be roughly at right angles to the plane of the ecliptic and its orientation with respect to the Sun would only change with the seasons.

The orientation of the crescent and half Moons would also be shifted by roughly 90 degrees so the points of a crescent Moon would tend to point roughly up or down depending on the hemisphere viewed from and the degree of the inclination of the Moon’s orbit from the ecliptic.

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