wrog

promoting this out of my FailBook comments just because

So, there's this diagram which has been making the rounds now that the James Webb Telescope has been launched. It's good at showing where the Lagrange points are and, if you're used to reading topographic maps, what the gravity well generally looks like, that volcano with the 6000 mile deep crater that I talk about here, if you were having trouble picturing that (modulo the small matter that that was all about the Earth-Moon system, and this picture shows Sun-Earth L1-L5; but they both work the same way).

Granted, with L1, most people seem to be clear about the idea that, with Sun and Earth pulling in opposite directions, there's got to be some point in the middle where the forces are cancelling out, even if that's not quite what L1 is. But for L2-L5, people seem to be completely mystified, and I've been seeing far too many wacky attempts at explanations in the last few days.

Time to sort things out:

First thing that people seem to forget is that, to get this diagram, we are rotating the reference frame.

Meaning we're putting ourselves on this utterly gigantic amusement park carousel, that's spinning at just the right rate, so that, from our point of view, the Earth, normally considered to be whipping around the Sun at around 30 km/sec (in what we'll assume to be a circular orbit, for now), is actually fixed in place. Meaning that white circle you see in the diagram going through L3-L4-L5 that seems to be showing the Earth's orbit — it's really quite misleading — is not actually Earth's orbit … well okay, it is in the sense that everything that would be sharing Earth's orbit around the Sun in the usual view will be somewhere on that circle, but, in this view, the Earth itself is not actually traversing that path.

Because, again, the Earth is not actually moving.

But we still have gravity.

And, for some weird reason, the Earth isn't actually falling into the Sun, as you would expect. Because, as you'll recall if you've ever been on that amusement park ride, there's this other force that was keeping you plastered against the outer wall of the centrifuge while they opened up the bottom. People like to call centrifugal force a "fake" force, but it's plenty real if you're rotating; you have to account for it if you want to get the same answers that the non-rotating people get (Earth doesn't fall into the Sun).

… and it's pretty easy to account for. Centrifugal force is just like gravity in that the acceleration you experience is going to be the same no matter how massive you are, and it's linear in your distance from The Center. (If you want the math, it's ω²r where ω is the rotation rate in radians per unit time. And if you pick the right unit of time, we can even have ω=1 and then it's just r).

"Linear" means, amongst other things, that centrifugal force gets stronger the farther out you go. The white circle is all of the places where it cancels the Sun's gravity. Thus Earth (and anything else placed on that circle) has no net force on it and doesn't move. Inside the circle, the Sun wins and stuff falls in closer; outside the circle centrifugal force wins and stuff flies away. Thus,

L1 is where Earth gravity + centrifugal force (both pointed outward) cancel Sun gravity (inward)
L2 is far enough out so that centrifugal force is now strong enough to cancel both the Sun and Earth pulling together inward.
L3 is where centrifugal force cancels both Sun and Earth pulling together the other way, however since, in this case, the Earth is pulling all the way from the far side of the Sun, its effect is going to be really small. Still, L3, like L2, will be outside the white circle, even if it's only about 600 km outside, a distance way too small to see in this diagram.

The second thing that people fail to mention, and this matters if you want to understand L4 and L5, is that the center of rotation is not the center of the Sun but rather the Sun-Earth center-of-mass.

In other words, if you imagine putting the Earth and Sun on a giant see-saw, then there's the spot where you'd put the fulcrum to make them balance, and that is the point everything revolves around. Admittedly, since the Sun is 1/3 of a million times the mass of the Earth, the Sun will be a whole 450 km left of center while the Earth is hanging out 150 million km to the right, which may not seem like much of a difference from the Sun being in the exact center, but you'll thank me when you see my L4 diagram.

Drumroll… Here, have a diagram:

Note that I have exaggerated the mass of the Earth to be half the mass of the Sun, which is why the center is now 1/3 of the way from the Sun to the Earth rather than being inside the Sun. (and yes, the grey circular dots are probably also the wrong sizes, but who cares?) Now you can see everything that matters.

Yes, all of the triangles that look like they're equilateral are, in fact, equilateral, and that really is a parallelogram in the middle, which is why those vectors at L4 all add to zero. The two blue arrows (acceleration of gravity caused by the Sun on the Earth and whatever's at L4) have to be the same length, and likewise for the two red arrows (acceleration of gravity caused by the Earth on the Sun and whatever is at L4), while the green arrows are proportional to the centrifugal forces (equal, if we do that trick with the time units).

The fun part about this diagram is that all gravitational forces are operating at the same distance (Sun⟷Earth = Sun⟷L4 = Earth⟷L4) so we don't even need to care about whether gravity is inverse-square or anything else; the distance-force relationship is whatever it is. Gravitational acceleration is otherwise proportional to the mass of the thing doing the pulling and completely indifferent to the mass of the thing being pulled. The rest is geometry.

The real magic of L4 (L5) is the way it is stable when the mass of the second body (Earth) is sufficiently small, even though, if you look at the diagram at the top of this post, L4 is clearly a hilltop and, normally, stable means you're at the bottom of a valley. It really does look like if you move even slightly away from L4, the forces accounted for by the diagram (gravity and centrifugal force) will be pushing you outwards from L4, and they will get stronger the farther away you get,…

… which is pretty much the definition of instability.

So, two things:

If the mass of the Earth is small compared to the Sun (has to be *much* smaller for this to work, but a ratio of 1/333,000 is plenty small; even in the Earth-Moon case where the ratio is only 1/100, that's still good enough),
then that hilltop at L4/L5 is extremely large and flat (which you can sort of see in the diagram). Meaning the residual gravity + centrifugal forces that would normally be making L4/L5 unstable are likewise small and can be overwhelmed by Other Stuff, namely:
The force we haven't mentioned yet, … the Coriolis force, which is the other force that comes into play in rotating frames, and which cannot be represented in the diagram because it turns out to be velocity dependent.

Note that if centrifugal force were the whole story, then the moment something gets outside the white circle, it picks up speed, hurtles off to infinity and you never see it again. Which we know isn't what happens: In the non-rotating frame if you launch a probe from Earth that doesn't have solar escape velocity, it goes into an elliptical orbit of its own. Meaning, in the rotating frame, once something is moving away sufficiently fast, there has to be this other force that overwhelms centrifugal force so that it can come back.

You can also get a sense of what Coriolis force does by considering the case where we remove the Earth and the Sun (so that we're back to empty space), and then place a small rock where the Earth used to be, having it be stationary in the non-rotating frame. Once we hop back on board the carousel, we'll see the rock doing a circular orbit around the center (following the white circle) with velocity ωr. Which means something must be double-cancelling the centrifugal acceleration (ω²r outward), to give us ω²r acceleration inwards — what the rock needs to stay on that circular path at that speed — so the magnitude of the Coriolis force needs to be 2ω²r = 2ω(ωr) = 2ωv in this case.

So it shouldn't surprise you too much if I tell you that's what it is generally

a_coriolis = 2 v × ω

where ω is now a vector coming out of the page and we're taking a vector cross product, and, for the rock circularly orbiting in the correct direction, the right hand rule will indeed point us in towards the center. (You can also see this in action by getting on the centrifuge ride and pulling your fist towards your face really quickly).

The rock-in-empty-space scenario is not a bad way to remember how it goes, though, to be sure, if you want to actually derive what the Coriolis force is, there's a bit more work to do.

Anyway, the Coriolus rule along with the centrifugal rule

a_centrifugal = ω²r

turns out to be everything you need to know about the weird forces present in rotating frames.

As for what's happening at L4/L5, the capsule summary is that, with the residual combination of gravity and centrifugal force being so weak over such a wide area, you won't have to be going very fast at all for Coriolis to be the force that matters most.

And if Coriolis is the force that matters most, then you'll be going in circles. Also, velocity dependent means the faster you try to go, the stronger the force will be.

Which then makes L4/L5 rather hard to get away from.

To be sure, the end result is still quite weird+unexpected, and there's no substitute for Actually Doing the Math, which turns out to be kind of like solving the double-jointed pendulum problem. Orbits at L4/L5 have two different modes with different periods (1.05 "months" and 3.35 "months" in the case of Earth-moon, where "month" is the lunar orbital period, if I'm doing this right...) and an orbiting colony there can be in any linear combination of them, so in general there will be this weird Lissajous-figure crap happening …

… meaning if we ever get large numbers of people trying to settle L4/L5 -- not bloody likely, methinks -- there's going to have to be strict regulation of who's in which mode (probably will have to pick one of them and require everybody to be in it), otherwise all of the colonies will be crashing into each other...