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One would hope they’re still in Earth’s orbit - if they’ve achieved escape velocity they’re not coming back.
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So if we’re being technical I think they did achieve escape velocity? But the moon’s gravity and some timed burns will slingshot them back.
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But one could imagine, as a failsafe, arranging things so that Integrity was in fact still in orbit around the Earth, as a sort of "backup" in the event that they somehow missed the moon's slingshot effect.
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