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I'm wondering if it is possible for a planet to have one pole always facing its star, the other always facing away (by rotating as if a tidal-locked planet), but style have a day and night cycle on some part of it (by having another axis of rotation ?).

In case I'm not clear enough, I drew this wonderful paint schema to illustrate : Illustration

Is it possible for a planet to rotate in a way that would allow this ?

Would there be another natural way to achieve the same result outside of the rotation ?

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You did say "on some part of it", so both of these possibilities are pretty restricted.

Eclipses

The planet has a large, low moon that regularly eclipses part of the surface.

Wobble

Although the planet is tidally locked, it only got there in the (geologically) recent past, and it still has some wobble, aka nutation. This will provide a day-night cycle near the twilight area.

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  • $\begingroup$ I have been reflecting on your answer and the one below yours, and it made me realize I asked the wrong question. I was too narrow-minded when visualizing my problem, as the tidal-locking is not mandatory for my need. I will ask another question, the right one this time, without merging my hypotheses with my real need. Sorry for the inconvience. On a side note, I liked your moon idea, I didn't think of using a moon this way and it opens some nice possibilities. Thank you ! $\endgroup$ – Yutreza Aug 29 '16 at 16:57
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No.

What you describe is actually a planet with two axes of rotation, which is unstable short-term and would violate the conservation of angular momentum long-term, as momentum is additive.

The closest natural phenomenon to what you've described is called precession, but it occurs on a time scale measured in tens of thousands years and not, too my knowledge, at orthogonal angles.

See also essentially a duplicate question.

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