Currently, we use gravity assist to speed up spacecraft to help get to father orbits like Mars. But our return plans seem to rely on atmospheric braking as the primary deceleration.
I know the science says anything that we can use to speed up can be used to slow down.
What are the practical limitations and obstacles to using gravity assist?
Is this something that could be used at Mars, say by Phobos and Deimos, as part of a crewed mission?
Yes, gravity assist can be used to increase or decrease a spacecraft’s momentum. This article from NASA explains the basics of it.
Something similar could be done for a mission to Mars.
The gotcha is that the influence on the spacecraft by the other body depends on how massive it is versus the spacecraft. Doing a drive-by with Jupiter will give you a lot of delta-V. A drive by of teeny Phobos or Deimos, not so much.
My spacecraft design class students are working on designing a mission to Neptune, and hope to shed some of their speed at arrival through a fly-by of Neptune’s moon Triton. Their initial estimation of the possible change in speed was suspiciously high, on the order of 7 km/s. I’m imagining their final value will be a lot lower.
So, yes, you can definitely slow down through a gravity assist (as explained upthread).
Using Deimos / Phobos? These little moons are so small and low mass… OTOH, recent missions have specified using a Mars flyby, which used to be rare, as Mars is also relatively small in mass and doesn’t help that much. But in some cases it can. For instance, the Europa Clipper, on its way to Jupiter, just did a flyby of Mars two weeks ago.
NASA planned a mission to the Sun that would have used a gravity assist from Jupiter to drastically reduce the probe’s speed so it would fall toward the Sun in a polar orbit getting very close to the Sun. This was the Solar Probe Plus. Later the trajectory was redesigned to use multiple gravity assists from the Earth. The name was changed to the Parker Solar Probe.
The greatest of all
Yes, gravity assists can transfer momentum from the spacecraft to the celestial body to “slow down” or put the spacecraft into a lower energy orbit; the study of this is called “capture dynamics” and there are several distinct ways in which a spacecraft or other body can be captured by a carefully plotted trajectory to progressively transfer momentum with different planets. It is also possible to plot exchanges with multiple bodies within a planetary system to capture a spacecraft; this can be done provided that the kinetic energy of the spacecraft is not so great that that it doesn’t at least periodically interact with and transfer momentum to the bodies in question. The Cassini-Huygens mission is a prime example of this, where interactions with various moons allowed the Cassini spacecraft to change orbits and even make substantial planar changes in order to explore all of the large moons in the Saturnian system:
Such a trajectory could also, in theory, be used to send a spacecraft to Mars but the gravitational spheres of influence of Phobos and Deimos are so small that they would be basically ineffective in keeping a spacecraft from leaving the vicinity of Mars in all but a very marginal set of cases. This would also likely take decades or centuries, and thus would not be suited for a crewed mission.
That seems really high given that Triton’s escape speed is less than 1.5 km/s, although given its mean orbital speed of 4.39 km/s it is just within the realm of possibility given a really favorable set of conditions. I don’t know what text your students are using—I assume Danby or Prussing as a basic text, and maybe Vallado if they are really advanced—but I’ve been working through this text as time and enthusiasm allow and it is interesting if quite dense for such a small tome, going well beyond standard KAM theory. Unfortunately, it is pretty expensive and difficult to find used but if they are interested in modeling really complex celestial interactions it is invaluable.
Stranger
Our department is really strong on Astrodynamics, so they use fairly advanced theory (often compared with Vallado but we use course notes from instructors) and lots of numerical simulations. But they do make mistakes, LOL. 
(Some of my other teams are investigating CR3BP type orbits around the moon for a lunar commsat mission thanks to the stability of some of these orbits for long term stationkeeping.)
You can’t use gravity assist from flying by a planet to change your speed with respect to that planet. The benefit comes from the fact that the planet is moving with respect to other solar system objects, so by changing your direction relative to that planet, you’re changing your speed relative to other objects.
So if you’re looking to send a craft from Earth to land on Mars, say, you can use flybys of the Moon or other planets (or even Earth itself, after some other flyby) to get to the vicinity of Mars. But once you reach Mars’s location, the only planet available to do a flyby of is Mars, and you can’t use it to slow yourself down. The only options are rockets (which are expensive), aerobraking, or lithobraking (which tends to be hard on the payload).
Right. That’s why I was asking about Phobos and Deimos.
Right. That’s why I was trying to think of anything else. I figured if I can think of it, someone already has, so I was curious. Basically the two moons are much too small and too close in speed to Mars to help.
Aerobraking was to take 6 months and several orbits around Mars to reduce the apoapsis (highest point in orbit) and align the MGS spacecraft in the proper polar orbit . The duration of the aerobraking phase is directly related to how fast Mars’ relatively thin atmosphere reduces the spacecraft’s velocity.
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Not with a pure gravity assist. But the Oberth effect makes it so that you can achieve a dramatic amplification in your delta-V if you fire your thrusters deep in a gravity well. It’s a distinct effect but both involve a momentum transfer with the large body. And every Oberth maneuver requires performing a gravity assist as well.
How about for travel to a distant star that has a massive companion? How slow would an interstellar spacecraft need to be traveling to use gravity assist to decelerate enough for a long visit?
Tricky question. You want the companion to not just be massive, but dense, so you can get close to it. Which means that a black hole or neutron star is the ideal target. You can use one of those even if your craft is going a meaningful fraction of the speed of light. But then, there are tidal effects that’ll rip your craft apart if you aren’t careful. And your math had better be very good, since frame dragging and such will play a role. The Oberth effect isn’t going to help you much, unfortunately, since you’ll spend so little time near the body that (assuming the craft is compatible with human biology) you just won’t get much delta-V out of it.
Would be fun to actually do the math here. Maybe someone’s already done so.
This makes me think of the Kim Stanley Robinson book Aurora, where the “distant star” is our sun, approached by failed interstellar colonists. KSR used some bad math to get the coloniats home.
At the end of the novel, the Ship returns to Earth, decelerating mostly using what is called the ‘Oberth Manoeuvre’, invented by Hermann Oberth in 1928. This is a two-burn orbital manoeuvre that would, on the first burn, drop an orbiting spacecraft down into a central body’s gravity well, followed by a second burn deep in the well, to accelerate the spacecraft to escape the gravity well. A ship can gain energy by firing its engines to accelerate at the periapsis of its elliptical path.
Robinson wants to use this to decelerate from 3% of light speed down to Earth orbital velocity. 3% of lightspeed is 9,000 km/s. For reference, Earth’s orbital velocity is 30 km/s. Several deceleration mechanisms are referred to in the book. An unpowered gravity assist, passing by the sun and reversing direction, can steal energy from the sun’s rotational motion around the centre of the galaxy. That’s worth about 440 km/s. Other unpowered gravity assists can be used once the ship is in a closed orbit in the sun’s gravitational well. Flybys for aerobraking in the atmospheres of the gas giants are referred to as well. Altogether, these can get you <100 km/s.
But the key problem with using the Oberth Manoeuvre for deceleration of this returning starship is that this craft is on an unbound orbit. That means that, on entering the Solar System its trajectory can be bent by the sun’s gravity, but will then exit the System because it has not lost enough velocity to be bound to the Solar System. To be bound would require velocity decreased down to perhaps 100 km/sec, which is 1% of the incoming velocity. Therefore 99% of the deceleration has to take place in the first pass. And you can’t get that much from an Oberth Manoeuvre.
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To a rough approximation, the most you can get out of a gravity assist (either trying to speed up or slow down, since everything’s time-symmetric) is a few times the escape speed of the system’s primary body, as measured at the distance of the closest object orbiting it.
From time to time I have wondered what the possibilities are in using a hypervelocity star that “just happens to be passing by” to get a real boost (and then target another one at the other end to slow down).
Maybe do a leapfrog thing. Use Barnard’s star to get to a faster moving star … to get to a hypervelocity star.