how do satellites orbit around LaGrangian points?
In: Engineering
Well, it only really happens at L4 and L5.
If you look at the following image, if you’re orbiting L4 or L5, you’ll be following one of those oblong contours around it. This is only in a frame of reference where the other two bodies are fixed:
https://en.wikipedia.org/wiki/Lagrange_point#/media/File:Lagrange_points2.svg
Basically, when you’re orbiting L4 the planet pulls you closer to the sun, which speeds you up, which flings you back out past the planet’s orbit, which slows you down again. If you’re watching the satellite orbit around the sun, it has the same period as the planet, but it goes both above and below the planet’s orbit.
L5 is the opposite, the satellite approaches the planet from a lower, faster orbit, then gets flung out where it slows down.
Caveat: that image is very much not-to-scale. If you want to play around with a simulation, I remember there was a mod called Principia for kerbal space program. Here’s a screenshot of a lagrange point orbit in that: https://www.reddit.com/r/KerbalSpaceProgram/comments/57dfw2/found_out_principia_was_getting_stable_so_i_sent/
[This graphic will help, it shows the gravity gradients of the Earth-Sun system](https://upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Lagrange_points2.svg/1202px-Lagrange_points2.svg.png)
If you look at L1 or L2 you’ll see the blueish arrows that point outwards from the LaGrange point, these tell you that if you drift closer to the Sun or Earth you’ll get pulled away from it, so instead we tilt the orbit up 90 degrees so it doesn’t get closer/farther from the Earth/Sun. It forms this big ellipse that’s rotated upwards and we call that a [halo orbit](https://en.wikipedia.org/wiki/Halo_orbit)
Then those little red arrows help you out. As the satellite drifts ahead/behind the Earth’s orbit or above/below, the combined gravity of the Sun/Earth pull it back towards that central point, this gives you a gravitational pull that can be orbited around.
If you had a very very simple system with just the Earth and just the Sun then you could park your satellite perfectly at L1 or L2 and it’d hang out there. Unfortunately there’s more mass than just those two and Jupiter is big enough to muck with the exact location of L1 which makes satellites at these points unstable so they need a bit more fuel allocation for regular station keeping maneuvers with thrusters.
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