At the end of my novel, the surviving crew of a spaceship is hurtling past a Dyson swarm that encloses the sun where the asteroid belt used to be. They are traveling too fast for orbital insertion and have lost most of the water they use for reaction mass. They are using a less powerful backup propulsion system to try to slow down enough to acheive some sort of orbital insertion, even if it is similar to a long-period comet that would take them out near the Oort cloud before heading back in.
They have a separate water supply for crew needs, and they are considering whether to re-purpose this supply for reaction mass at the beginning of the sequel.
They are hoping to use a Uranus fly-by to reduce their speed. They won't be able to get as close to the planet as they'd like, but if they use the last of their water for reaction mass, they can get closer than they would otherwise.
How does one calculate how much a flyby would reduce their velocity including the variable of distance from the planet?