Is the Moon always visible during winter on the North Pole?
At the North Pole during summer when there is permanent daylight, does the Moon rise and set, and if not, does that mean that during winter with no sunrise the Moon is always up?
The Moon does rise and and set during both summer and winter on the North Pole (or South Pole). The exact movement is complicated, but can be understood the combination of two separate movements:
1) Rotation of the Earth on its axis, which results in movements that change over the course of one day.
2) Orbit of the Moon around the earth, which results in movements that change over the course of one lunar month (about 29 days).
While the Moon does rise during the summer at the North Pole, since the Sun is always up, you generally can't see it, so I'll focus on the movement of the Moon during the winter.
The daily movement from Earth's rotation causes the Moon to circle once around the sky. If you spent the entire day staring at it, you'd have to turn around exactly once. This movement is also the same that the Sun makes during the summer. To give you a better idea of how this looks, here is a video showing how the Sun moves in the sky at the North Pole: Arctic Midnight Sun
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.
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