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The L1 and L2 points are thought to be unstable "saddle" points, meaning that there is stability in two directions of movement, but not in the other. That raises an obvious question - when a spacecraft stationed there diverges toward either side of the saddle, and station-keeping is no longer maintained, where will it go?

There are several projects planned for these points. The JWST won't be at the Sun-Earth L2 point forever. Likewise, we're looking at a plan for moving an asteroid to the Earth-Moon L2 point, and a lot of people raise questions about what would happen if it goes astray (although I'm not worried). Unless we keep sending supply missions in perpetuity, it will drift away eventually.

I'm most interested in the Earth-Moon system. We have 2 points, and 2 directions it can fly off into. Does that result in 4 distinct, deterministic, trajectories? Which of these would crash into the moon and the Earth? Would it happen with one orbit? Or would any of them be chaotic?

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There's an online app that relates to your question: astro.u-strasbg.fr/~koppen/body/LagrangeHelp.html –  Jerard Puckett Feb 19 at 14:51
    
@JerardPuckett That particular applet seems to only map the potential. But the site has several available, and it looks like the 3-body simulation might do it. It has the Earth & Moon and let's you simulate a test particle. Only problem is, I can't figure out how to set its initial velocity, so I can't get it to start at the true L1 point. But maybe someone can figure that out. astro.u-strasbg.fr/~koppen/body/ThreeBody.html –  AlanSE Feb 19 at 15:56
    
That's the very reason Lagrange points are the beginning of a multitude of chaotic trajectories. A small perturbation may lead you anywhere in the Solar System, eventually. –  Deer Hunter Feb 19 at 20:20

1 Answer 1

Thanks to a very helpful suggestion in the comment, I was able to do an initial test with an online orbital simulator. It can be found here:

http://astro.u-strasbg.fr/~koppen/body/ThreeBody.html

This starts off (helpfully) with the Earth moon parameters, for the mass ratio and the spacing. All we need to do is tell it where to put the initial particle. Here is the screen where you can do that:

parameters

This is "details of initial situation". As a side note, I believe they marked both the system CM and Earth, but they partially obscure each other.

You can see that I've changed the parameters to what we need for simulation starting at L1. It is around 84% of the way from Earth to the moon. Make sure the angle is set correctly too. Then we have two parameters for initial velocity - set both to zero because we are in the co-rotating reference frame.

Here's what I get doing the simulation:

plot

You can see that it initially moves tangentially to the Earth-moon line, and then crashes into Earth.

I can't tell if this is the answer I'm looking for. Simplistically, I don't see how it can move in that direction. Given the shape of the saddle point, I believe it should move either to the right or the left in the above plot. But the mechanics in the co-rotating frame are notoriously spooky. It still seems possible that this is right, but I can't tell.


The problem was that I misunderstood what the input parameters were. The initial tangential velocity is still given in reference to a non-rotating frame. In order to be "stationary" relative to the Earth-moon system, this parameter needs to be equal to the radius parameter. See help:

http://astro.u-strasbg.fr/~koppen/body/ThreeBodyHelp.html#BUTTON

init.radial velocity with respect to the Centre of Mass, given in units of the oribital speed of the Moon

init.tangent.velocity ditto, positive velocities point in the same sense as the lunar motion

So in the above L1 example, it the initial tangential velocity needs to be 0.84. If you do that, you can get this:

L1 moon

This is much more reasonable, and more likely correct. In this example, it does crash into the moon, but only after about a month. The scenarios appear to fit:

  • L1 moon-side drop: orbits for 1 month, crashes into moon
  • L1 earth-side drop: goes into high Earth orbit, looks stable
  • L2 moon side drop: goes into moon orbit, can transition into high Earth orbit
  • L2 far side drop: orbits both Earth and moon at very far distance
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So it falls toward one of the "stirrup" sides of the saddle. –  Jerard Puckett Feb 19 at 16:29
    
@JerardPuckett No, the input was wrong. I'll write an update. –  AlanSE Feb 19 at 16:44

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