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This answer is meant as a supplement to notovny's. I agree with their conclusions (the scenario is impossible because of the instability of this Lagrange point, and the fact that the Hill sphere is too small), and I just want to derive the "curious relation" they came up with. We start with Kepler's third law. $T_M$ and $T_p$ are the periods of the planet ...

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In you scenario the planet's recapture looks much more questionable to me. First, the space is really big. So the probability of such a massive planet to be so close to a star so that it would capture it is extremely low. Even if it happens, the most likely orbit would be very elongated, which makes life challenging by going in and out of the habitability ...

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Not unintentionally Let us take a look at our solar system. A body in Earth orbit (ie in the Goldilocks zone where life is most likely to evolve) needs an additional 11 km/s in order to escape into interstellar space. Note that this is the best case scenario, ie where a fast-moving, sufficiently massive body ploughs into the planet from "behind" to ...

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Alternative: the moon-to-be was gladly orbiting its star until a rogue jupiter, coming from behind, "gently" pulled it out of the orbit and wandered off. Yeah, gently is sorta relatively speaking, some tidal sloshing may have occurred in the process, their Venice wasn't quite happy about it. Ah, yes, it has been rediscovered in 2013

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