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I was experimenting with various star system designs with a gravity simulator, and one of the patterns I ended up with is best described as a planet with a very elongated elliptic orbit that itself rotates around an axis located at/near one of the orbit's foci. This gif illustrates it pretty accurately:

Orbit

I believe I created this by simulating a low-mass body moving near the large-mass body and falling into orbit, but at a slight angle so as to create an offset that shifted the orbit every revolution. Would this be possible with an actual planet and a star? If so, would there be any noteworthy side effects aside from the obvious (e.g. having extremely varying climates at different points during the year)?

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Not only is this possible, it happens with real planets including the Earth. It is called Apsidal precession. This cycle and a similar cycle relating to the Earth's axis Axial precession are known as the Milankovitch cycles. They are believed to be a factor in long-term climate changes including ice ages, though how significant they are is not well understood. If a planet had a large eccentricity (like Mercury or, taken to an extreme, your example), the effects of that would dwarf the effects from the precessions and likely make the planet uninhabitable.

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    $\begingroup$ Here is a Khan Academy video discussing these topics (I couldn't put it in my answer because I am restricted to 2 links). $\endgroup$ – Tom Anderson Feb 26 '15 at 5:56
  • $\begingroup$ It depends how far, and for how long the planet is out of the habitable zone. Place it around a star with a wide habitable zone, and perhaps it might be okay. $\endgroup$ – user3082 Feb 26 '15 at 7:56

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