A photo in the NYTimes article Ice Surveys and Neckties at Dinner: Here’s Life at an Arctic Outpost has got me thinking.

During the winter months at either the north or south pole, when it is constantly dark, how does the Moon behave?

Would a nearly-full moon sometimes circle for days without setting, or would it still rise and set, or would it be all together missing, or some combination of all of the above?

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The moon is quite interesting as you get near the poles. It still moves round due to the Earth's rotation once a day, but it also orbits with a period of a month so what you get is two weeks of moon above the horizon followed by two weeks where you cannot see it.

What changes through the year is what phase you see when the moon is visible, and in the middle of winter you have two weeks around full moon, whereas in summer you have two weeks around the new moon - not so exciting. Spring and autumn obviously have two weeks around first and last quarter, respectively.

This simulation using Stellarium shows the effect quite well.

From astro.cornell.edu - this wording may help:

The second movement caused by the Moon's orbit around the Earth is analogous to the movement of the Sun over the course of a year only it repeats over the course of a lunar month. Near the new Moon phase, the Moon is near the Sun and therefore never rises during the winter. As the Moon approaches full, it will start to pop up above the horizon. Eventually near the full Moon phase it will be high enough in the sky to stay up all day and circle like the Sun in the video above. The elevation of the circle will rise as the Moon becomes completely full and then start to decrease until it begins to dip below the horizon. Eventually the Moon will stop rising at all as it gets close enough to the new phase. The cycle then repeats.

And another useful reference at planetarium.madison.k12.wi.us:

At the beginning of Winter, when it's nighttime all of the time, the moon would be in the sky for the 2 weeks closest to Full Moon, and then below the horizon for the next 2 weeks. And at the beginning of Spring (click on the graphic), when the sun is at sunrise all of the time, the moon would be up in the sky for the 2 weeks closest to First Quarter (waxing), and then below the horizon for the next 2 weeks.

The animated graphic above shows what we would see from the North Pole if we went out every day at noon, for 14 days in a row, from March 1st to March 14, 2006. We start with a thin crescent moon near the horizon, and end with a full moon near the horizon. Halfway through, the First Quarter moon would be when the moon is highest above the horizon.

Keep in mind, that if you were observing the moon constantly, throughout a 24 hour period, the moon would seem to move to the right in the sky along with the sun, stars, and planets due to the Earth's rotation. Nothing would seem to rise and set: they would just seem to circle around you.

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    $\begingroup$ @uhoh At full moon the moon is opposite the sun so visible in winter not in summer. At new moon it is near the sun, so visible in summer (in principle) not in winter. There will be a secondary effect caused by the moon's orbit being inclined 5 degrees to the plane of the ecliptic, which may well change as you describe, but that's dominated by the fact that Earth's equator is inclined 23 degrees $\endgroup$ Sep 24 '18 at 15:13
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    $\begingroup$ @uhoh There are three planes of interest: the Earth's equator, the Moons orbit and the ecliptic. What makes the Moon (and Sun) intermittently visible from the poles is mostly the 23 degreed between the Ecliptic and the Equator. The 5 degrees between the ecliptic and the Moons orbit should show up as a variation in that basic pattern $\endgroup$ Sep 24 '18 at 15:46
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    $\begingroup$ @SteveLinton oh my goodness, I don't know where the Moon is! I'd assumed it would have found it's way close to the Earth's equatorial plane since the Earth's bulge and J2 are at play, but yes, at 400,000 km the quadrupole moment's $1/r^3$ potential drops off so fast compared to the monopole's $1/r$ that it has little effect, and the solar system's perturbations take over. Ha! my Moon has just moved to where everyone else's has been all along. Thanks! $\endgroup$
    – uhoh
    Sep 24 '18 at 15:54
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    $\begingroup$ @RoryAlsop very nice answer; I learned a lot today, thanks! $\endgroup$
    – uhoh
    Sep 24 '18 at 16:05
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    $\begingroup$ @uhoh The term "ecliptic" should've been a clue. ;) But anyway, our Moon is a bit odd, since it's so big relative to the Earth. In some ways it's more like a planet that shares our orbit than a moon, so it's not surprising that its orbital plane is close to the ecliptic (but precesses rather quickly due to the complex combination of influences from the Earth and the Sun). Most other moons in the solar system orbit in their primary's equatorial plane, that might not apply to the extremely distant moons. $\endgroup$
    – PM 2Ring
    Sep 25 '18 at 2:23

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