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I wonder if there is a stable star-moon-planet system where a specific area of the planet would be permanently covered by shadow. I'm not looking for a common tidally locked planet with a roughly 50% dark zone. This is the desired order of preference for my world:

  1. Most of the planet has a traditional day-night cycle, except for some area which is always covered by night (like the Australia continent, or China).

  2. Most of the planet has a traditional day-night cycle, except for one or more vertical stripes of it (ex: GMT-3 is always covered by shadow).

  3. Instead of a tidally locked 50% covered by shadow, say, 25% is permanently covered by shadow, 25% is always bathed by sunlight, and the other 50% has a day-night cycle.

  4. Ok, I give up. Just a tidally locked planet between two stars, so there are two 25% areas covered by shadow and 25% in-between with light. Is this possibly stable?

I'm willing to allow any non-conventional (but non-magical) feature to this system: large moons, more than one sun or even a torus planet topology.

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    $\begingroup$ What's a landlocked planet? Is that like Russian nesting dolls, but with planets? $\endgroup$ – Frostfyre Sep 13 '16 at 19:28
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    $\begingroup$ I believe he means tidally locked. $\endgroup$ – Ross Sep 13 '16 at 19:28
  • $\begingroup$ I wonder how feasible a geostationary moon would be. $\endgroup$ – Twelfth Sep 13 '16 at 19:28
  • $\begingroup$ @Twelfth Charon and Pluto are tidally locked to each other, Pluto's day is Charon's orbital period. $\endgroup$ – Ross Sep 13 '16 at 19:30
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    $\begingroup$ @RossC - Ah yes....but that still leaves morning and evening sun. This gets harder to have the moon always in the way to the sun. hmmm back to drawing board $\endgroup$ – Twelfth Sep 13 '16 at 19:41
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While this doesn't meet any of your stated parameters (specifically the permanent requirement) and I have zero background in this field, I figured I'd post anyway :)

Uranus has a massive axial tilt that gives its poles 42 years of darkness followed by 42 years of sunlight. Someone else may want to comment if speed of rotation can affects the time, what a moon system might do, etc.

Source: http://www.universetoday.com/18955/tilt-of-uranus/

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If you are willing to have large-scale artificial structures, it gets a lot easier. For example, a planet whose equator is coincident with the plane of its orbit, and has a large solid, artificial ring in geosynchronous orbit would have its equator in permanent shadow.

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As mentioned in the comments:

A moon or satellite (artificial or no) is tidally locked to your planet, and your planet to it. So the system turns in a "day" and the two bodies always show the same face to each other. The system needs to be exactly in the plane of the solar system so that a solar eclipse will happen over the same place every day.

Unfortunately the sun would be visible in the morning and evening.

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I believe a rogue planet could find orbit a binary system where said planet rotates in a way where it's equator is dark and poles get plenty of light (because the orbit is equidistant from the two stars masses in a small orbit). This pic shows the concept (sizes not depicted well :P ): enter image description here

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How about planetary rings? If the planet and ring system had only a small axial tilt then a horizontal band near the equator would be in perpetual shadow.

enter image description here

Other more complex configurations would also be possible, but wouldn't quite result in 100% permanent shadow, or at least I can't picture any specific configurations that would.

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Earth's moon does have some small areas like that. Requirements are: 1. Rotation axis near zero tilt to the orbit. 2. Craters or large mountains near the north or south pole.

The tilt means the sun never rises much above the horizon at each pole, and some areas there could have a mountain in the way at any given time of day. If you want to arrange it so, you could also have some nearby areas get sunlight only for a few minutes a day with an appropriately positioned canyon. The only issue would be the atmosphere, if the planet has one, would still supply twilight illumination to the area. Some of our moon's southern craters have areas that can hold surface ice because they never get direct light, so it never warms up to melt or sublimate the ice.

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