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Lagrange points are the points in a multi-body gravitational system in which the gravitational force and centrifugal force sum to zero. The image below from this Wikipedia article shows the 5 Lagrange points in a two body system. The L1, L2, and L3 Lagrange points are stable in one direction but unstable in the others while the L4 and L5 Lagrange points are unstable in both directions. My question is: Is there a buildup of space trash at these stable (in one direction) Lagrange points? Or does their instability in the other dimension prevent such a buildup?

Edit: The L4 and L5 Lagrange points are stable in both directions "provided that the ratio of M1/M2 is greater than 24.96," to quote the Wikipedia article linked above, which is the case for the Earth-Moon system. In light of this fact, my updated question is: Is there a buildup of space trash at any of the Earth-Moon Lagrange points?

Lagrange points in a two-body system

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The gravitational force and the centrifugal force of the orbit are zero. –  Cephalopod Mar 20 at 22:10
    
You are correct. My wording of the question was a bit misleading, I've edited it. Thanks. –  Chris Mueller Mar 21 at 14:56
    
"shows the 5 Lagrange points in a two body system". I would rather say this is a three body system, where m3 << m2 and m3 << m1. –  Cedric H. Mar 22 at 12:38

3 Answers 3

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To add to other answers, the L1-L2 Lagrange points are unstable because they need to follow radial velocity of the two parent bodies as they orbit each other, in our case the Earth in a heliocentric orbit around the Sun, but none of them are really at the orbital altitude matching their radial velocity (too slow in L1 and too fast in L2). Since the Earth's orbit isn't exactly circular (orbital eccentricity of ~ 0.017), their altitude will also slightly change during one orbital period. This is less of a case with L4-L5, and arguably also L3, depending on orbital eccentricity of parent bodies, and is where we might find Trojan and Hilda family of asteroids, respectively. All these Lagrange points can also be perturbed by gravitational influence of other celestials in the system, for example Jupiter's and even Moon's orbit in case of L1-L2 Sun-Earth points. And this is of course about instabilities in the velocity vector along the M1 parent body. Orthogonal to it and towards the M1 and M2 bodies, it's just the tipping point in direction towards M1 or M2.

Slightly simplifying, what I'm talking about is that the heliocentric velocity (using $v_o \approx {2 \pi a \over T}$) will be $\text{SEL1} \approx 29.49\ \text{km/s}\ \ $ and $\text{SEL2} \approx 30.08\ \text{km/s}\ \ $, where Earth's orbital velocity is $v_o \approx 29.78\ \text{km/s}\ \ $. This difference will be maintained by the L1-L3 saddle points, not too dissimilar to surfing at the tip of a wave. Any lateral movement will tip your balance towards one of the two parent bodies (M1 or M2).

So Lagrange point satellite orbits need to be managed, what's usually referred to as orbital station-keeping. Effects of these perturbations and instabilities can be somewhat offset by placing Lagrange point satellites into Halo or Lissajous orbits and using precision orbital insertion, but not even these will be stable without using onboard propellants and corrections to their orbits. Any orbital debris or whole defunct satellites will eventually spiral towards the Earth (see update to this answer, but they hold too much orbital momentum to really fall towards the Sun, as their orbital period actually matches that of the Earth's, but their semi-major axis towards the Sun doesn't for about ± 1.5 million kilometers or roughly ± 1%).

TL;DR: All this means that the L1 and L2 points would essentially be free from long-term orbital debris. And with L3-L5, unless the bodies there formed from the same protoplanetary disk and share same radial velocity that's required to stay at that altitude, there is nearly no chance that any other body with significantly different orbital energy would be captured at those points. But if we deliberately placed satellites there, any debris from them would stay there for a lot longer than with L1 and L2 points (and perhaps L3, as mentioned before).


Edit: Apparently, I've misread the question initially and was answering for the Sun-Earth lagrange points instead of the Earth-Moon ones. OK, no problem, most of the problems remain the same in theory, only the L1-L3 saddle points are even more unstable. Moon's orbital eccentricity is ~ 0.055, so L1-L3 points move even more along the Earth-Moon axis. On average, EML1 is 326,380 km away from Earth and 58,019 km away from the Moon, EML2 448,914 km and 64,515 km respectively, and their velocities would be (again, average) ~ 0.87 and 1.2 times average orbital velocity of the Moon (1.022 km/s). They are even more perturbed, especially by the Sun itself and the complexity of the Moon's orbit relative to the Sun (it never curves on itself in loops though, contrary to popular beliefs).

Here's a nice image depicting Earth-moon lagrange points:

    enter image description here

    The Lagrange points for the Earth-moon system. Credit: David A. Kring, LPI-JSC Center for Lunar Science and Exploration

And this is how Lissajous orbits of the ARTEMIS (Acceleration, Reconnection, Turbulence and Electrodynamics of the Moon’s Interaction with the Sun) mission's P1 spacecraft's EML1 and EML2 orbits looked like:

    enter image description here

     The view from above of the ARTEMIS orbits as they make the transition from the kidney-shaped Lissajous orbits on either side of
     the moon to orbiting around the moon. Credit: NASA/Goddard Space Flight Center

    enter image description here

    Illustration of Artemis-P1 liberations orbits, side or ecliptic view. Credit: NASA/Goddard

Sources:

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What about L1-L2 points of Earth-Moon System? Wikipedia link –  osgx Mar 21 at 0:12
    
@osgx Oww dang I misread the question, didn't I? Oh well, I'll see what I can do when I'll have the time for it... not much changes tho, it's mostly just the numbers that are wrong, and perturbation is of course even stronger and faster (Sun). –  Noordung Mar 21 at 0:17
    
@osgx I've managed to write a fast update. Please let me know if I've not managed to clarify something or again made some errors. And thanks for pointing out I was answering for Sun-Earth L-points, not the Earth-Moon ones! –  Noordung Mar 21 at 1:03
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gravity in these multi-body systems is so chaotic. pretty amazing. –  Stu Mar 21 at 13:13
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The video, from the ARTEMIS pages you linked, showing their transfer to lunar orbit is truly amazing. –  Chris Mueller Mar 23 at 17:56

The Lagrange points L1, L2 and L3 are stable in prograde- and retrograde direction, but unstable on the radial axis. That means any object on these points will drift away in radial direction unless it uses small amounts of thrust to balance on these points, so any concentration of natural mass or debris at these points is impossible.

Only the points L4 and L5 are stable, and objects, so-called "Trojans", tend to orbit these points. There is also one which orbits the Earth/Sun L4: 2010 TK7. It's rare for trojans to orbit closely to the lagrange points. The reason is that these points would only be 100% stable when the Sun and Earth would be the only objects in the solar system. But due to the gravitational influence of the other planets, an object parked exactly at L4 or L5 would be slowly but steadily dragged away from its point and end up in an orbit around it.

The L4 and L5 points of Moon/Earth appear to be clean, except for some faint clouds of dust, and even the existence of those is disputed.

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Just to be clear though. The L4 and L5 points are stable in the sense that an object there can orbit with the same orbital speed as the planet regardless of its mass (this is true of the other points too). They are not stable in the sense that a perturbation of their orbit is restored, i.e. a spring like force. The gradient is much weaker at the L4 and L5 points though so perhaps objects can live there longer. –  Chris Mueller Mar 21 at 2:15
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After reading a little more, I realized that my comment is not generically true. To quote the Wikipedia article on Lagrange points: ", the triangular points (L4 and L5) are stable equilibria, provided that the ratio of M1/M2 is greater than 24.96." This applies to the Sun-Earth and Earth-Moon systems. Thanks for you answer. –  Chris Mueller Mar 21 at 2:27

There's a number of missions that have headed to various of the Lagrange points, see Wikipedia for a good list. Let's look at what it takes to stay there:

  • James Webb- Proposed for L1 point, can only take enough fuel for a 10 year mission.
  • Hershel Space Telescope: Was deliberately moved from L2 to a heliocentric because " the spacecraft would be in a slow tumble, receding from its stable L2 orbit, subjected to solar radiation pressure. And as ESA’s ground stations were no longer communicating with it, so we wanted to basically check the orbits and make sure that for future science, it was not mistakenly detected as an asteroid."

There has never been an L3 mission, but the same sort of logic follows, fuel usage is required to maintain that orbit, and as a result, there shouldn't be a large amount of debris left behind there.

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Both Webb and Hershel are near Lagrangian points of system Sun-Earth, not points of the system Earth-Moon. –  osgx Mar 21 at 0:10
    
I didn't read the title carefully I suppose... The same thing applies for Earth/Moon as Earth/Sun. –  PearsonArtPhoto Mar 21 at 0:16

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