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If a probe entered a highly elliptical orbit around Uranus after firing its engine (similar to what Cassini did at Saturn), could it use Uranus' moon Titania as a gravity assist to circularize its orbit?

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    $\begingroup$ Wait, what does "within five years of arriving at Uranus" mean? If you are there for five years then you are already in orbit. $\endgroup$ – uhoh Nov 23 '18 at 0:23
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    $\begingroup$ @uhoh Right. You mean you could not read my mind? Hope it makes a bit more sense now. $\endgroup$ – Bob516 Nov 23 '18 at 5:45
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    $\begingroup$ I've adjusted the language, I hope you don't mind. Please feel free to roll-back or edit further. I think this expresses what you are asking, have I got it right? My mind-reading takes a nose-dive in the afternoon, I'm going to make some more coffee now... $\endgroup$ – uhoh Nov 23 '18 at 5:55
  • $\begingroup$ Would the question be more open to an answer it the were no restrictions on the approach? $\endgroup$ – Bob516 Nov 23 '18 at 6:05
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    $\begingroup$ I guess there are two things here. One is the ways that Cassini used Titan's gravity as part of the original capture by Saturn, and all of the other times to modify it's orbit after capture. Second is the similarities and differenced to that dance that might also be available in the Uranus/Titan system. There could be enough here for two questions possibly. I can't judge the best way to proceed though. $\endgroup$ – uhoh Nov 23 '18 at 6:13
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I don't think you can circularise your elliptical orbit by only using Titania's gravity, or at least you can't get into a stable circular orbit.

Let's ignore all other stuff orbiting Uranus and assume a three object problem for simplicity. If you are only using Titania's gravity to enter a circular orbit, at some point your new circular orbit will need to diverge from Titania's enough so that it's gravity won't alter it anymore.

Let's start with a coplanar orbit: as Titania's own orbit is already circular, your new circular orbit would be near syncronous with it, and you would be nearly static as seen from Titania. You'll end up falling onto it.

If you want to orbit in a different plane, you'll still have two near misses each orbit, which will again disturb your orbit.

Adding all other Uranus' satellites will only complicate things further.

You could, however, use Titania to lower your apoapsis and then do a short burn to circularise later on when you are far away from it.

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Assuming you approach Titania from the appropriate direction, there is nothing stopping you from using it for a gravity-assist either to accelerate or decelerate.

Actually performing the maneuver is a matter of timing and good aim.
So yes, in principle you can absolutely use Titania to alter your orbit with a little planning.

One caveat is that you will probably need a small amount of fuel/thrust to fine-tune your trajectory, especially after the maneuver, because you don't want to get into a roughly circular orbit that grazes the orbit of Titania and in a few months get deorbited onto Uranus by an unintended grav-assist from the moon.

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