UPDATE: I found another reference! While I always enjoy a good video starring Jimmy and Linda Carter, this one has Dennis Wingo describing ISEE-3's original Halo orbit. He describes Sun-Earth $L_1$ as "a point about 1.5 million miles from earth where a spacecraft can safely orbit without using any fuel."

The site http://spacecraftforall.com/a-new-orbit is an interactive thing - if you leave it alone it will usually start the video in about 10 or 15 seconds.

enter image description here

While finding some good patent examples to add to the question Spacecraft Maneuvers as Intellectual Property? Wow!, I ran across this patent:

US7744036B2 Method for designing an orbit of a spacecraft (J. Kawaguchi and K. Tarao, JAXA)

You can click on the PDF icon to download it.

My question comes from reading the following parts of these two sections:


…By the way, it has been revealed by previous studies that if a Lissajous orbit is enlarged to a much more large scale one to enter into a non-colinear area, the non-linear effect makes the spacecraft trajectory closed. And the orbit of a spacecraft, if the spacecraft is positioned close to point L2 can avoid shadow by the earth and if the spacecraft is positioned close to point L1, can avoid passage in front of the Sun. When viewed from the Earth it looks like a halo around the sun, and thus is called a halo orbit….

and then:


As stated above, the natural halo orbit is adventageous on the fact that it is stable and draws a closed locus without any special artificial correction operation maintaining the motion. However, the natural halo orbit disadvantageously leads to an extraordinarily large orbit which must depart from a co-linear line connecting two celestial bodies. The distance may be as large as, for example, about one million km from the Lagrange point referred to here. The resulted natural Halo orbits around point L1 or L2 are too far and very much apart from the aimed points originally. Because of this, the merit associated with the halo orbits, that is, no maintenance of the position of a spacecraft is needed, is inevitably lost, if the trajectory is controlled and maintained artificially...

I take this to mean that when a halo orbit around an $L_1$ or $L_2$ point (let's say Sun-Earth) is sufficiently large, it is (at least within the CR3BP model) actually stable to small perturbations, i.e no station keeping is required for sufficiently large halo orbits about $L_1$ or $L_2$.

Question: Is this true? Are there other stable orbits associate with unstable Lagrange points?

I had always thought that all orbits around $L_1$ or $L_2$ were also similarly exponentially unstable just like the points themselves, (again, within the CR3BP model).

note: I am not asking a question about this patent. I am referencing it only because of the clear statement it makes about Halo orbits - this is where I read it. The patent discusses an artificially maintained smaller halo orbit. Halo orbits are a special class of Lissajous orbits where the in-plane and out-of-plane periods are equal so that it stays "open" and away from the central exclusion zone.

Figure 3 from the Patent US7744036B2:

Fig 3 from US7744036B2

  • $\begingroup$ I'm thinking about that for two days now, and I doubt what is the question actually. Without note all clear, I can run model and check if it's stable, without note - just asking about another strange and interesting orbits? (connected to L-points) $\endgroup$
    – MolbOrg
    May 14 '16 at 19:35
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    $\begingroup$ @MolbOrg thanks for your interest. Within CR3BP $L_1$ and $L_2$ (and $L_3$) are always unstable. While a test mass with zero velocity placed exactly at those points will remain fixed, even the smallest mathematical deviation in position or velocity will result in an exponential growth of the deviation and within a few periods it leaves the area. My question is - are there some orbits around $L_1$ or $L_2$ that are stable - where small deviation will result in only a slightly different orbit? This is purely a mathematical question needing a yes/no answer with math or reference backup. $\endgroup$
    – uhoh
    May 15 '16 at 0:59
  • $\begingroup$ @MolbOrg I'm asking if such an orbit exists, not "is orbit X stable?" A stable orbit within CR3BP would then be a starting point for a search for a sufficiently stable orbit in the real world where the CR3BP restrictions no longer hold. In "space exploration" language, that would manifest itself as a spacecraft in a halo orbit that would not require frequent state vector measurements and stationkeeping thrust maneuvers to remain in the orbit for, say, a half-dozen periods (e.g. >3 years for sun-earth). The existence of such an orbit might be quite helpful to know about! $\endgroup$
    – uhoh
    May 15 '16 at 1:12
  • $\begingroup$ So your definition of stable orbits are - more then 3y without maneuvering. note CR3BP solutions will not be a start for searching more stable orbit, because of venus, mars, barycenter movements and solution is unstable by it's nature, so then possible set of solutions in CR3BP most likely will not be same as more general situation. Answering this will be not an easy task, as example this $\endgroup$
    – MolbOrg
    May 15 '16 at 14:21
  • $\begingroup$ @MolbOrg in the case of orbits around sun-earth $L_1$ or $L_2$, a half-dozen orbits is long enough to rule out exponential instability, and short enough that those perturbations might not kick it completely out yet, and potentially long enough for an artificial satellite to actually get something useful done. The word "stable" is tricky - it is given different definitions in different contexts. If I were trying to find a useful sun-earth $L_1$ or $L_2$ in the real world, I'd start by looking near a CR3BP solution rather than just start shooing with a random number generator but that's just me. $\endgroup$
    – uhoh
    May 15 '16 at 14:43

I take this to mean that when a halo orbit around an $L_1$ or $L_2$ point (let's say Sun-Earth) is sufficiently large, it is (at least within the CR3BP model) actually stable to small perturbations, i.e no station keeping is required for sufficiently large halo orbits about $L_1$ or $L_2$.

Yes, this is true but the devil is in what "large" means. The stable orbits are by any means not orbit about the $L_1$ or the $L_2$ points. They wander far off the ecliptic plane, and a body on those stable orbits spend little time on the axis Sun-planet. I happen to have given a detail answer to this very issue in another question a couple of hours ago.

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    $\begingroup$ This is the mathematical stability of an orbit in the CR3BP model, not the real solar system, so no need to apologize for its shape ;) I'll take/need some time to read your great answer there and get an idea of the size and shape of the stable region as it would translate into the Sun-Earth system, then see if ISEE-3 was actually in this region or not. Thanks! $\endgroup$
    – uhoh
    Jun 11 '17 at 10:04
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    $\begingroup$ Note the orbits presented by Howell are for $\mu=0.04$ whereas $\mu=3\times 10^{-6}$ for Sun-Earth. Conclusions do not change qualitatively iiuc but the size of orbits and especially minimum $x$ and maximum $z$ do change in a systematic manner. I can add more about that later if you wish. $\endgroup$
    – user20022
    Jun 11 '17 at 13:37
  • $\begingroup$ Why don't I write up a separate question for it tomorrow? I'll leave a message for you when it's posted. I'll probably refer to the math here as well. Thanks!! (fyi the bounty function is asking me to wait 24 hours before I can award it to you.) $\endgroup$
    – uhoh
    Jun 11 '17 at 13:51
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    $\begingroup$ The practical implication of the orbit being large should be that that the orbiting body will at some point wonder to a point where it will be perturbed by some other body in the Solar system. $\endgroup$ Jun 11 '17 at 23:18

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