The Truth About "Slingshot" Maneuvers

[wrog]

(This is "Deconstructing The Expanse, Part 2", continued from Part 1, here.)

As luck would have it, I decided to go back and watch part of an episode to check something, mainly to try to pinpoint where exactly the show goes off the rails, and aside from saving myself from a really embarrasing blunder, it also reminded me of a topic I meant to cover but forgot to.

It's Season 3, Episode 7, the one where Maneo, Speed Demon Belter Guy in his tiny racing pinnace decides he's going to be the first one through The Ring and arranges this whole elaborate sequence of "slingshots" to get there first (what happens when he gets there is a fun scene that I liked because I'm actually 12, but I won't spoil it).

And then I realized…

People have multiple misconceptions about gravitational maneuvers. The "slingshot" is badly named. I blame Star Trek.

In "Tommorrow is Yesterday", the original series episode where they accidentally go back to 1967 and grab an Air Force pilot, there's this bit towards the end where they have to do this odd "breakaway" maneuver in close to the Sun, which then propels them to Ludicrous Speed, which then allows them to Do The Time Warp and Fix Everything. Iconic episode; you already knew about it; fine.

I'm assuming D.C. Fontana is responsible for that mess since she wrote it, and even allowing for the possibility that she got the idea from somewhere else, I'm still going to blame her for popularizing it.

Apparently, gravity is supposed to be this kind of rubber band that if you pull hard enough on it, it snaps, and then you have all the velocity you'll ever want. It's a beautiful, clear and simple explanation/analogy, so wonderful that it stuck with people.…

… desite being utterly and completely wrong.

Let's go back to basics:

What is a Slingshot?

Take a (usually) long strip of cloth with a wide patch in the middle, otherwise known as the "sling". Take a rock, otherwise known as the "shot", fold the patch around it, and then take up the other two ends of the strip together in one hand, and pick the whole thing up. You are now holding this pendulum thing with the rock dangling down at the bottom.

You now start swinging the rock in a circle, usually above your head but that part doesn't matter, faster and faster. The rock builds up some velocity.

At this point the Physics 101 teacher will be pointing out that in order for the rock to be moving in a circle, there has to be this force on it aimed in towards the center of the circle — centripetal, they call it,— i.e., towards your shoulder or hips or whatever it is you're swinging the rock around. The force has magnitude $m{v}^2/r$, so if the velocity $v$ is insane, the force is even more insane, and it's your arm and the strip of cloth that are doing that.

At some point, you let go, preferably of just one end of the cloth, but if you don't care that much about a slightly reduced range/release-velocity or getting your slingshot back, letting go of the whole thing is just fine. The rock then flies off at its now insane velocity,…

… hitting the giant in the forehead and killing him instantly. Yay, congratulations, you have just saved your primitive village. Maybe they will tell stories about you and you'll then be able to leverage that into a bid to become King. Good luck with that.

Ok, so, um… Gravity does not do this. Gravity never does this. This is not what the NASA folks are doing.

Yes, if you take an object and drop/throw it in the general direction of a planet/sun/moon/whatever (but not directly at since we don't want it to hit) it will speed up as it falls. Depending on how close it gets to the surface, it can speed up rather a lot. The ideal case is where you have something really massive that you can get really close to, like a dead star or Jupiter, which is sort of almost a dead star (the "dead" part matters because you don't want to be getting burned up when you get in close, though in Jupiter's case there's still a lot of radiation there that can fry you in other ways).

And then we invoke bullshit. Presumably, once your object gets going fast enough, there'd be some way to immediately turn the gravity off, so that it can then go flying away at ludicrous speed, and,… well,… no. Figure, if you could do that, the planet itself would then be immediately flying apart — i.e., if it's rotating, and still not a whole lot of fun for anyone living there even if it's not rotating — which then means you have Death-Star-like capabilities, in which case why are you dicking around with penny-ante "slingshot" maneuvers?

It's Not a Slingshot

What actually happens is that your object does its Jupiter flyby and then, assuming it hasn't grazed the atmosphere, hit one of the ringlets, or done anything else similarly stupid, it comes back out. We can solve for the trajectory: in general, it will be (one branch of) a hyperbola, like this:

We do have choices about what velocity we give it coming in and how close a flyby $r$ we do, which then determines the angle $\theta$ it comes out. But what gravity giveth, gravity taketh away; energy is conserved. Thus, no matter what we do, once our spacecraft gets back out to the same distance it was dropped/thrown from, the velocity will be the same, $\left|{\mathrm{<b>v</b>}}_2\right|=\left|{\mathrm{<b>v</b>}}_1\right|$.

The sharpness of the turn we're doing can be calculated from $$\cos \theta =\frac{1}{1+\frac{r{v}^2}{\mathrm{GM}}}$$ where, for this formula, $v$ is the "from infinity" velocity, i.e., how fast the spacecraft is going when it's essentially out of range of the planet's gravity (strictly speaking, it never is, but there will be a distance from the planet at which its gravity gets smaller than what we can measure).

And,… that's all. If we're just doing a two-body problem, i.e., where there's just Jupiter and the spacecraft, and nobody's firing the engines anywhere, there is nothing to be gained from this.

So, It Doesn't Do Anything?

Actually, it does, and I just dropped two hints in the previous paragraph as to how/why.

The first is that this is not just a two-body problem. What we need to do is look at what's happening from the Sun's point of view, in which Jupiter is now moving in its orbit:

The essential idea here is that having the spacecraft approach Jupiter from the front means that means switching to the Jupiter frame adds Jupiter's velocity to its horizontal component, this new velocity then determines the angle $\theta$ and hence which direction the spacecraft is going to emerge, and if we can keep that angle small enough, it'll emerge in front of Jupiter going forward, so that when switching back to the Sun frame, we're adding in Jupiter's velocity again.

Jupiter in its orbit goes 13.06 km/s. That means we are never going to be adding more than double the Jupiter velocity or a Δv of 26.1 km/s.

Also, the case $\theta =0$ only happens when something falls towards Jupiter at whatever the escape velocity is for that distance (quite small when far away). From the Sun point of view, this will be Jupiter-velocity minus something small. From Jupiter point of view, it'll be a nearly parabolic orbit where the object will be getting flipped around to go forward in front of Jupiter at escape velocity, or from the Sun point of view Jupiter-velocity plus something small. Or a total velocity gain of twice something small.

So for any kind of substantive velocity gain, we're always going to have θ>0 which will then introduce factors of cosθ that reduce our velocity gains accordingly below that promised 26.1 km/s.

If you're NASA trying to do Pioneer 11 and you can't even afford enough fuel for the 2nd half of the Hohmann transfer, you're stuck with doing just the first half (8.8 km/s Δv) using the Big Rocket to get away from Earth, end up approaching Jupiter at a relative velocity of 5.6 km/s, and hoping for the best. The closest flyby you dare is around 110,000 km from the center of the planet (wiki sez Jupiter has a "radius" of around 72,000 km, but when you're talking about a planet where the notion of "surface" is a gentlemen's agreement that gets you quickly down a philosophical rabbit hole if you poke at it too much — maybe it's where the hydrogen gets compressed to something vaguely liquid-like; there's evidently some point lower where it's a fucking metal whatever that means; maybe 72,000 is the cloud tops but you really don't want to be in those clouds, and there's probably also a whole lot of invisible exosphere above that you don't want to be encountering either. Anyway it looks like 110,000 km is what the Pioneer 11 planners went with, the radiation turned out to be nastier than expected, and so that's evidently the closest flyby that anyone has attempted thus far).

And then you crank the formula above to get θ≈13°, which is nice and small, which means you have no problem arranging things to emerge in front of Jupiter with something like 19 km/s, all without spending any additional fuel.

Which NASA was completely happy with in 1973 and it's also good for rather a lot of other solar system maneuvers they couldn't otherwise do.

But it's not Ludicrous Speed.

What's more, once you have your first velocity boost, yes, there's nothing that says you can't go bouncing off the other planets (Pioneer 11 indeed went on to Saturn to get another [small] boost), but that $v^2$ in the formula above will be making your θ angle quite large (cosine going to 0 means θ goes to 90°, i.e., spacecraft not getting deflected at all), and if your angle is too large, then you have to be approaching Jupiter from behind in order to be getting spit out the front, the relative approach velocity will then be a subtraction rather than an addition, and once what Jupiter is subtracting on the way in gets commensurate with what it's adding on the way out, you're really not gaining much of anything at all.

And the only way to make the angle smaller at higher velocities is to do a closer fly-by, but then you start hitting the atmosphere, and, um…, good luck with that.

Is that really all?

Well, all right, there is actually one more general trick — the second hint dropped above — you can use to squeeze extra velocity out of this, and that's to fire your engine during the flyby.

Quick aside about the flyby being this Exciting Time where everybody's all mushed into their seats and Feeling the Speed:  um, NO. Unless you're hitting the atmosphere or firing your engines, you will be in free fall the whole time, and it be just like all of that other boring space travel. At best, you will have nice stuff to look at if you have windows, but that's about it. Also, you probably don't want to have windows because they can break.

The point to firing your engine during flyby goes like this: Adding $\mathrm{\Delta v}$ to your velocity when your velocity is already $v$ increases your kinetic energy per unit mass by $$(v+\mathrm{½}\mathrm{\Delta v})\mathrm{\Delta v}$$ but if you instead dive in close to a planet to bring your velocity up to $V\gg v$ and then do the same burn, your kinetic energy gain will be $$(V+\mathrm{½}\mathrm{\Delta v})\mathrm{\Delta v}$$or, roughly $V/v$ times as much energy, and, once you've popped out of the gravity well back to the distance away you were before and gravity has taken back everything it gave you, you still get to keep that extra energy. It's not quite so straightforward a gain because velocity is the square root of kinetic energy, but it's still something.

In the case of being near Jupiter, diving down to 110,000 km from infinity gets you going at around 48 km/s, so if you're out in deep space going 6 km/s, you want to add another 2 km/s, you could instead dive in close to Jupiter and boost yourself by a mere ¼ km/s to get the same effect. So essentially you're getting an 8-fold savings in fuel expenditure

…except this can only happen at certain points on your trip (i.e. when you're near a planet), and, again, there are diminishing returns: as your $v$ gets large, the multiplier this maneuver applies to your Δv goes down.

Scorecard

In short, these maneuvers are actually really bad at building up speed beyond a certain point (and we'll soon see what kinds of velocities ships are most likely using in The Expanse universe because of mumble-Epstein-Drive-mumble).

The real reason to do this shit is to save fuel.

Which means it's not going to be the racing pinnaces employing these maneuvers. It's going to be the long term Big Cargo haulers that are pushing comets/whatever, where they need to be using the least fuel possible and thus needing to keep their Δv in the toilet, and also where the cargo is stuff like water or minerals or manufactured goods that don't actually care how long it takes to get there. Most likely they'll use ITN and every trick in the book to engineer an entire automated stream of deliveries where the pipeline is years or decades long, which will make sense once you have a colony in a known place that needs to survive and isn't going anywhere.

(... of course, now what I want to see are the scenarios where the colony died decades ago but the supplies keep arriving because it was too much trouble/expense to go hunt down all of the pipeline ships and divert them, and so the places turn into these treasure troves...)

In the next installment, we clarify what we mean by Ludicrous Speed

Comments

[wrog]

And of course it's only after posting this that I come up with a much better analogy.