Two bodies, rigid theory.
Three bodies, chaos theory.
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dart board;; science bs
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Two bodies, rigid theory.
Three bodies, chaos theory.
You can thank the three body problem
... Two things can happen: Either the planet gets very, very close to the binary, suffering tidal disruption or being engulfed by one of the stars, or its orbit gets significantly perturbed by the binary to be eventually ejected from the system ...
This could be described intuitively in my opinion as a repeated slingshot effect ... the same way that would make so that tides on Earth push the Moon gradually further away while slowing down Earths' rotation ...
In these cases, would each star orbiter each other and the planet have an elliptical orbit around both them or would be other type of arrangements like the planet doing an eight shaped orbit between both stars?
At least intuitively it seems to me that 8-shaped paths would be extremely unlikely and unstable. Wikipedia describes two types of orbits:

So the planet can orbit both stars elliptically, or just one of them - but the latter can be the case only when the other star is a dwarf that can't affect the planet too much.