Mostly, I’m interested in L4 and L5, but the more I try to explain it to myself the more I realize I have no idea how any of them work except maybe L1- just an equilibrium of the two forces?
In: 5
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First we need [our trusty Lagrange point map](https://upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Lagrange_points2.svg/1202px-Lagrange_points2.svg.png)
Lagrange points are about having the same angular speed as the body of interest(We’ll stick with Earth here) so that items at that point complete an orbit in the same amount of time and don’t move relative to Earth
An object trying to complete an orbit of the Sun in 365 days inside the orbit of Earth would be moving too slowly and would fall into a lower orbit picking up speed on the way. Trying to complete an orbit in 365 days outside the orbit of Earth results in too much speed which pushes the object into a higher(longer) orbit so it takes more than 365 days
Okay, since we’ve got those two states covered, how do you get an object to orbit closer to the sun in the same time as the Earth? You need to reduce the inward force! Since you can’t shrink the Sun its about finding the point where Earth’s gravity provides the right balance to give the outward tug and keep things stable. This is L1
L2 is further out than the Earth so you need more force so Earth’s gravity gets combined with the Sun’s gravity to make a spot where the extra speed is perfectly compensated by the extra Gravity
L3 is pretty similar to L2 in principle. Its on the exact opposite side of the Sun from Earth so the gravities combine, but since Earth is sooo far away its only a *tiny* bit out past Earth’s orbit
L4 and L5 are the odd balls. They’re actually orbits around the center of gravity(barycenter) of the Earth-Sun system not around the Sun. They’re an equal distance to the Sun and Earth and are actually stable. If something in L4 (behind Earth) shifts closer to Earth then it’ll pick up speed, shift out from the sun a bit, losing speed, then shift further away from Earth so it loses its gravity tug and falls inwards, picking up speed, and then shifting closer to Earth again. L4 and L5 pick up little asteroids called Trojans, Jupiter has a ton of them
It’s an equilibrium of forces, but there’s more than 2. Sort of.
L1, L2, and L3 think of as in a saddle. To go further clockwise or counterclockwise they have to gain energy, so if they drifted a tiny bit in that way, that would come back to the middle. BUT, if they drifted any closer in our out (towards or away from the sun or the earth) they would start drifting faster in that direction and then end up somewhere different.
L4 and L5 are at the top of a hill, if they drift in ANY direction, they keep going faster.
As for the forces, there’s the gravitational pull of the sun and earth (or whatever two bodies you’re looking at the orbits of), but there’s also a Coriolis effect that pulls and pushes the satellite (or whatever) away from or towards earth the further you get from the L4 and L5.
Dunno if that was LI5 enough…
The basic principle of the Lagrange points is that the net force of gravity on an object determines the orbital speed of the object. The harder gravity pulls, the faster the orbit must be, and the closer two objects are to each other, the stronger the pull of gravity.
The secondary principle of Lagrange points is that the points are “stationary” relative to the orbital body (Earth), which means that they have to complete an orbit at the same time as the orbital body does.
Since the circumference of the orbit (and therefore the distance needed to complete an orbit) increases with diameter, that means that objects farther from the Sun than the Earth must move faster, while objects closer to the Sun than Earth must move slower.
If you put these principles together, you get a paradox; an object farther from the Sun than Earth must move faster than Earth to maintain its “stationary” position, but since it’s farther from the Sun, it moves more slowly. The reverse is also true; an object close to the Sun most orbit more slowly, but gravity will cause it to orbit faster.
The Lagrange points arise because both the Sun and Earth exert gravitational pull. These can add and subtract from each other to change the orbital speed, which then changes how it moves relative to the Earth, if it moves at all (and if it doesn’t, then it’s a Lagrange point).
L1: Between Sun and Earth: The Sun dominates the gravitational pull, but Earth’s gravity negates some of the Sun’s gravity depending on how close the satellite is to the Sun or the Earth. This causes the orbital speed to decrease. At a specific point, the orbital speed matches what’s needed to be stationary relative to Earth: L1.
L2: Beyond Sun and Earth: The Sun and Earth are both pulling the satellite inwards, increasing the gravitational pull. This increases the orbital speed. Since it’s closer to Earth, Earth’s gravity varies more with distance than the Sun’s gravity, so an object that’s too far away will feel very little of Earth’s gravity but about the same Solar gravity. If you pick the right point, Earth and Solar gravity combine to get the exact orbital speed necessary to get the satellite to be stationary relative to Earth: L2.
L3: Opposite Sun and Earth: At this distance, Earth’s pull is not very significant relative to the Sun, so the satellite is acting a lot like Earth. However, Earth is still pulling a bit on it, so the satellite is a bit closer to the Sun than Earth is.
L4/L5: 60 Degrees Ahead of or Behind Earth, on Earth’s Orbit: Because the Sun and Earth are pulling in different directions (but also not directly opposite directions like L1), the forces are not as simple to describe. Satellites at both of these points are being pulled slightly inwards by the Earth, as Earth is not directly in front of or behind the satellites. However, the forward/backwards pull of the Earth is counteracted by the slight angle at which the Sun is pulling on the satellites; L4, ahead of Earth, is pulled backwards by the Earth and slightly forwards by the Sun, while L5, behind Earth, is pulled forwards by Earth and slightly backwards by the Sun.
[Scott Manly video explains.](https://youtu.be/7PHvDj4TDfM)
3:45 to start the explanation
5:30 to see the combination of forces added up to make stable areas.
Alright. Let me take a shot (and if I’m saying something wrong, somebody correct me). I’m gonna focus on L4 and L5.
You know how gravity works, right? When you’re sitting high up there in space, you’ve got plenty of gravitational potential energy. But then you start to drift towards a high-mass object, like the Sun. You *start to gain speed* towards that high-mass object.
A Lagrange orbit works ’cause the same thing happens at the L4 and L5 Lagrange points. Specifically, it works because of the way gravity interacts between objects when they’re moving.
So say that there’s an object at L4 or L5 of the Sun-Earth orbit. Why isn’t it crashing towards the Sun or towards the Earth? Well, it’s not crashing towards Earth because like the Earth, it’s got some speed to it. It has to have speed in order to be orbiting the Sun at all. Also, at the L4 and L5 point, it’s equally far apart from the Sun or the Earth. Point is, all the forces balance out there and it can stay where it is, orbiting the Sun, just, in the same orbit as the Earth. It’s like any other orbit in that way.
Now. Why are L4 and L5 *stable?*
Okay, let’s say that suddenly, for whatever reason, our object at L4 or L5 starts to drift sunward.
Well, when it drifts sunward, it starts to *gain speed*, right? Because it’s getting closer to the sun, and that’s how gravity works.
Here’s the problem: it already had a lot of speed to start with. When it starts to drift sunward from the L4 or L5 point, it’s on the inside of the Earth’s orbit. That speed that it just gained, causes it to go *too* fast, deflecting outward, breaking out of the circle of the Earth’s orbit for a different reason, but with a similar result, to a gravitational slingshot.
By ending up on the outside of Earth’s orbit, now it’s *even farther* from the Sun than it started as… and in order to “climb” that gravitational gradient, it had to expend gravitational energy, just like a ball slowing down when it’s thrown upwards. So it slows down.
But the Earth? The *Earth didn’t slow down* in *its* orbit. The Earth starts to catch up. And as the Earth catches up, it starts to pull the object back toward itself.
But here’s the key: because of the angles of the circle, the Earth pulls the object *back over the line, to the inside of the circle*. As the Earth catches up, it pulls the object back not just towards the Earth, but *also towards the Sun too*. And as that object starts to go back over the line, it gains gravitational energy again, speeding up again; and the angle that it’s speeding up at *causes it to leave the circle again*.
And then the cycle repeats. In other words, the object starts orbiting around the Lagrange point.
Now, the whole orbit around the Lagrange point thing has to meet a lot of conditions in order to work. It works *because* the object has enough speed to be orbiting the Sun. The two objects, such as the Sun and the Earth, can’t be too equal in their masses; if the Earth’s gravitational pull were too strong relative to the Sun’s, it’d just attract the object instead of creating a Lagrange point orbit. The object has to be near a Lagrange point; it can’t be too far advanced along Earth’s orbit, or Earth wouldn’t be able to affect it enough, and it can’t be so close to the Earth, that Earth pulls the object all the way down into an orbit around itself. It has to hit that sweet spot.
But as long as all of those conditions are met, the forces all balance and the object stays in a stable orbit around the L4 or L5 Lagrange point: speeding up when it moves in, moving out because of the speed almost like a gravitational slingshot, slowing down as it moves out, and then getting pulled back in as the Earth creeps up behind it.
That’s my understanding anyway.
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