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I've recently said

you see one co-linear libration point, you seen 'em all. That's what I always say

and while halo and other orbits associated with Sun-Earth L1 and L2 work similarly, L3 is 2 AU away from Earth compared to 0.01 AU for the first two.

So I'd like to ask if there can be halo or Lissajous orbits associated with L3, and if not, are there any orbits associated with that point?

note: I'm asking about the general circular three body problem (CR3BP) so the two major bodies can have any ratio of masses. Answers don't need to be necessarily constrained to the Sun-Earth system.

I can has cheezeburger cat meme

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  • $\begingroup$ Why shouldn't be? Sorry to answer with a question but a quick glance through the literature show there are $\endgroup$ – Julio Jan 14 at 15:41
  • $\begingroup$ @Julio haven't said "should" or "shouldn't" but did point out dramatic difference in distances, in the case of the Sun-Earth system it's a factor of 200 which is a factor of 40,000 in force so I feel that it's not a given. There are plenty of distinct orbits in CR3BP (though I don't know exactly how many) but I've asked specifically if there are "halo" or "Lissajous" orbits. If you think this is answerable as asked, please consider posting an answer demonstrating said orbit(s) exist, thanks! $\endgroup$ – uhoh Jan 14 at 19:08
  • $\begingroup$ @Julio if the question is problematic, perhaps because of the term "halo" orbit, that might be a useful answer as well, or perhaps the question needs adjustment. $\endgroup$ – uhoh Jan 14 at 19:08
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    $\begingroup$ Given two equal masses it seems to me L2 and L3 would be indistinguishable from one another. $\endgroup$ – HopDavid Mar 11 at 22:06
  • $\begingroup$ @HopDavid yep, to me to! :-) $\endgroup$ – uhoh Mar 12 at 0:45
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Halo orbits definitely exist around any of the collinear Lagrange point and in most regards your comment:

you see one co-linear libration point, you seen 'em all. That's what I always say

is correct. Halo orbits are just special cases of Lissajous orbits. You can generate these orbits in the same way you would a halo orbit around L1 or L2 (usually some differential correction method).

Now the reason why halo orbits are called halo orbits is because in the Earth-Moon system, a halo orbit in EM-L2 looks like a halo around the Moon when viewed from Earth. I have no numbers to back this up but I would be surprised if you could find a halo orbit in SE-L3 with a large enough amplitude to be able to see it as a halo around the Sun. This however doesn't mean that they're not halo orbits.

There is some literature available on L3 halos but since for now they seem pretty much useless except for sci-fi supervillains, they are not treated as much as L1 or L2 halos. The principles remain the same however. I added to articles below for reference.

Transfers from Earth to EM-L3 halos

Three-Dimensional Periodic Halo Orbits

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