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If I spacecraft was leaving Callisto to return to Earth, would there by any situation where it would use a gravity assist from Jupiter to increase its velocity?

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No, there wouldn't be any way to use Jupiter's gravity as for an assist because the spacecraft is already around orbit around it, or around Callisto which amounts to the same thing. In a gravity assist energy must be exchanged between two gravitational bodies, for example when the Voyager probes got assists the forces acted upon the probes and Jupiter equally. This changed Jupiter's velocity infinitesimally because it's such a massive object, but changed the probes' velocity substantially because they have little mass. A spacecraft in orbit around Jupiter has no relative difference in velocity to Jupiter, so there no energy to be exchanged.

A spacecraft could potentially take advantage of Jupiter's high gravity to increase its rocket impulse using the Oberth Effect, where the spacecraft would maneuver as deeply as possible into Jupiter's gravity well, and then fire its engine at periapsis, where velocity is highest. This would expose the spacecraft to the worst of Jupiter's radiation environment, so may not be practical, although if it was visiting Jupiter's moons it would need to be pretty hardy to begin with.

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  • $\begingroup$ The Oberth Effect, that is what I was thinking of, just didn't remember the term for it. Thanks. $\endgroup$
    – Bob516
    Nov 24 '20 at 18:17
  • $\begingroup$ It could possibly use a gravity assist from Jupiter roughly 12 years later. It would have to leave the Jupiter system onto an elliptical orbit and meet up with Jupiter one orbit later. If you're not in a hurry that's probably the cheapest to get from Callisto to more or less anywhere. Also a gravity assist from Ganymede or Europa might work. $\endgroup$ Nov 24 '20 at 19:25

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