Using a planetary flyby to reduce speed

Question

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?

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Edit:

Hey, everybody. Thank you for your answers. I didn't expect such an active community. This is great.

In response to your answers:

Top answer is "Just handwave it." Ultimately that's what I would be doing because the characters working out the mathematics are not POV characters. The POV characters are only capable of describing what the technically savvy characters are doing in dumbed down terms because that's all the author is capable of. Still I didn't want to be so far off that even the handwavy explanation didn't fly.

Good points: Return on investment for getting it right is low. Agreed.

To folks who mentioned Kerbal Space Program (KSP), thank you. I wasn't aware of this tool. It seems like it might be a time trap, but a rewarding one.

To the person who said that in a setting with Dyson spheres there are a lot more options available than what I've laid out; In universe, the ASI inside the Dyson sphere doesn't share technology, energy or material resources with the human Kuiper Belt population. Otherwise, I would agree with your assessment.

Answer 1

The important thing here is that spacecraft trajectory planning is complex. There's a reasonable amount of math involved in solving your problem and I suspect that the readers/viewers/players of your creation seem unlikely to be there for the mathematics. You might consider therefore conserving your details, because this is stuff that you can get wrong, and getting it right doesn't necessarily add any more to your story than handwaving in the right trajectory changes to come to a satisfyingly dramatic conclusion. Everyone is happy to suspend their disbelief in exchange for a good story, and plenty of good scifi authors fudge their figures, deliberately or accidentally.

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The biggest issue you'll have is the sheer slowness of everything. Solar escape velocity at Uranus' distance from the sun is about 9.6 km/s, and Uranus is nearly 3 billion kilometers away. It'll take years. You can't be going much faster than solar escape velocity, because gravitational assists are extremely limited in the amount of change they can impart, and they have an optimal approach velocity vector and the assist is much lower if you're going too fast. If you go barelling out system at hundreds or thousands of kilometers per second (which is the sort of speed you'll want to be travelling if you aren't interested in multiple-year-travel-times) then a gravity assist will remove too little velocity from your spacecraft to keep you from shooting into interstellar space. If you're going out at 10 or 20 km/s... well. You can look at Voyager 2's planetary schedule to get an idea of how long things can take.

Take a look at this space.SE answer to give you an idea of how powerful a gravity assist can be: How much delta-v can we squeeze out of a gravitational slingshot and what factors limit it?.

The $\Delta V$ you can get is:

$$2\,v_\infty\over 1+{r\,v_\infty^2\over\mu}$$

Where $v_\infty$ is the relative velocity of your spacecraft relative to the body you're getting an assist from, $r$ is the point of closest approach, and $\mu$ is the standard gravitational parameter which is the gravitational constant $G$ multiplied by the mass of the assisting body. The linked answer contains a nice graph that shows that if you're too slow or too fast, the amount of assist you can get is limited

The very best assist Uranus can offer you is when your incoming velocity matches $\sqrt{\mu/r}$. At a cloud-skimming altitude of 26000 km, that gives you a delta-V of nearly 15 km/s. That's a lot by the standards of modern space probes, but it is peanuts when looking at how fast you need to be going to reach Uranus in less time than it takes to settle down and raise a family.

For a more detailed worked example, you could have a read of the flyby section of Orbital Mechanics and Astrodynamics. Its a fiddly process, and as such I'm not going to do it for you this time.

Answer 2

I am getting serious flashbacks to playing Kerbal Space Program...

So Starfish Prime has a very good answer - but...

AeroBraking

This is something that we used to do in KSP. In short - if you were running low on fuel and needed to slow down, you would buzz a planet with an Atmosphere - not enough for full re-entry, but enough for the drag of the Atmosphere to slow you down - not doing a gravity assisted slingshot.

Depending on Vessel design, heat resistance, dissapation etc. you can get some very drastic changes in delta V by using this technique.

Here is a video about how to do it in KSP - you can borrow the equations/maths if needed or if you want a higher level 'this is what they did' you can do that too.