# Are some Halo Orbits actually Stable? (stable orbits about unstable Lagrange points)

update: some more sources; the broken site spacecraftforall.com/a-new-orbit used to have an interactive simulation, here's an old screen shot:

Hat tip to @NgPh for finding this Space College page linking to the video ISEE_3 Original

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)

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

## BACKGROUND OF THE INVENTION

…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:

## SUMMARY OF THE INVENTION

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:

• 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) May 14 '16 at 19:35
• @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.
– uhoh
May 15 '16 at 0:59
• @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!
– uhoh
May 15 '16 at 1:12
• If you want an animation for illustration, you should replace the broken link by this initial ISEE-3 one Nov 29 '21 at 16:37
• @NgPh thanks and great find! I've updated and also included a link to the video itself in case that page also evaporates.
– uhoh
Nov 29 '21 at 23:34

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.

• 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!
– uhoh
Jun 11 '17 at 10:04
• 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.
– user20022
Jun 11 '17 at 13:37
• 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.)
– uhoh
Jun 11 '17 at 13:51
• 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. Jun 11 '17 at 23:18

Periodic orbits about the 3 collinear points in the circular-restricted 3-body problem are Liapunov and orbitally unstable. These collinear points are labelled L1, L2, L3. Each is an unstable equilibrium point in the CR3BP. A spacecraft inserted into any periodic orbit about any of the collinear points will require some real stationkeeping to maintain its periodic orbit. There's NO discussion of value about this to the contrary.

Halo orbits are very special cases of Lissajous orbits and occur when there exists a certain specific nonlinear relationship between in-plane (ecliptic or Earth-Moon) amplitudes and out-of-plane amplitudes. This causes the nonlinear in-plane and out-of-plane frequencies to be equal. Hence a 3-D periodic orbit. No such relationship exists with general Lissajous orbits which if given a long enough evolution will be ergodic in collinear-point 3-space.

The first references that should be researched and understood deal with the ISEE-3 mission launched in 1978. The name of Bob/Robert Farquhar will pop up, as this mission was his idea and his "baby". He is the "father" of halo orbit theory and use. His original work extends back to the mid 1960's.

The pioneering paper on the analytical development of periodic halo orbits was published in 1973 by Farquhar and Kamel: "Quasi-Periodic Orbits about the Translunar Libration Point", Celestial Mech, Vol 7, pp 458-473. Application strictly to the CR3BP was published in 1980 by D.L. Richardson: "Analytic construction of periodic orbits about the collinear points", Celestial Mech, Vol 22, pp 241 - 253.

The discovery of stable, large-amplitude halo orbits was published in 1979 by Breakwell and Brown: Celestial Mech, Vol 20, pp 389 - 404. Stability of these orbits was found by applying FLOQUET stability theory using numerical integrations. That paper can be found in many places on the internet. Here is one:

https://www.eng.buffalo.edu/~psingla/Teaching/CelestialMechanics/HaloOrbitPaper.pdf

This reprint is low resolution for some reason.

Perhaps the info above will take care of the guess work, personal opinions, and inaccuracies expressed earlier on this subject.

• There's a well-received answer here that links to a longer answer in Physics SE; there are some stable halo orbits about L1 and L2 in the CR3BP. If you'd like to say there are not, you'll have to refute both of those answers and the sources they cite.
– uhoh
Nov 26 '21 at 10:06
• Uhoh: The ISEE-3 mission which I worked on for 4 years (doing trajectory - mission analysis) orbited the interior collinear point of the Earth-Sun system. Asking if the ISEE-3 mission orbit was in a "stable region" of the collinear point makes no sense. The entire region of near-by collinear 3-apace is liapunov unstable. There is no region of stability of PRACTICAL value. At amplitudes of around 900,000 km, collinear orbits become FLOQUET stable. See Breakwell and Brown, Celestial Mech, 1979. ISEE-3 required less than 2% of its hydrazine propellant allocated for stationkeeping. Nov 26 '21 at 11:55
• Using FLOQUET stability theory with numerical integrations of the CR3BP, it has been found that stable, large-amplitude orbits about all 3 collinear points do exist. These orbits are of no practical value because perturbations from either the moon (or sun) and the planets as well as eccentricity effects render these orbits unsuitable for most any mission unless you have robust on-board stationkeeping. This hasnt been extensively studied as no one can think of a mission that would benefit from orbits like these. Nov 26 '21 at 12:03
• "link-only" answers are discouraged in Stack Exchange. "Just read the paper then you'll understand" is not considered a proper Stack Exchange answer. I posted this question to facilitate discussion and hopefully a consensus on a boolean "yes" or "no" first, and then the nuances can follow. If in the CR3BP with a $\mu$ representative of the Sun-Earth system there is a halo orbit that is somehow stable against small displacements; it forms a new closed periodic orbit rather than drifting off, then the answer to my question is "yes". If there isn't, then it's "no".
– uhoh
Nov 26 '21 at 13:10
• For whoever interested, there is a downloadable (better) copy of Breakwell&Brown. Nov 26 '21 at 18:10

Most of the questions about collinear orbit stability and stationkeeping and etc can be answered by several papers written by Robert Farquhar on his method of stationkeeping for the 1978 ISEE-3 mission. Search for this topic with his name. His papers are not overly technical, and most anyone with a decent math background can read them for content.

ISEE-3 stationkeeping control was extremely simple. A 40 km radius torus was imagined about the nominal (CR3BP) mission trajectory, and stationkeeping was only applied when the spacecraft wandered near the boundaries of this imaginary torus. Wandering of significance only happened when the spacecraft trajectory brought it near to the end points of the orbit's long axis as viewed in the Sun-Earth plane (ecliptic). This scheme used less than 2% of the total stationkeeping hydrazine allocated for the entire mission.

If left unattended, the spacecraft would have "missed" the long-axis turn (think perigee/apogee) of its trajectory, and after about half an orbital period later, it would head straight for the sun. This behavior was verified in simulations many times by numerical integrations of a very detailed force model. Orbital instability in the real world was always present in the ISEE-3 mission trajectory, but those real-world perturbations were surprisingly very easy to handle.

I am taking the risk to submit this answer. Not because I have working experience or attended graduate courses on astrodynamics, or the like. It is rather because the question arouse my curiosity and I have discovered beautiful concepts/technologies, by reading various bibliographic resources recommended by the answerers.

It is perhaps also because, to my eventual understanding, the two answerers concur among themselves. Moreover, they arrived at this agreement by citing more or less the same source! I can't understand why there was a dispute between the OP and the second answerer (@Ange Purs) about the correctness of the first answer (@user20022), that the OP accepted.

• First, let's review the main part of the question (emphasis added):

... when a halo orbit around an L1 or L2 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 L1 or L2. Q: Is this true? ... I had always thought that all orbits around L1 or L2 were also similarly exponentially unstable just like the points themselves, (again, within the CR3BP model).

Now, just for those who are not yet familiar with the terminology used (like me before), the OP was referring to orbits around (2 of) the equilibrium points named after the French mathematician Joseph-Louis Lagrange who discovered them (1772). They are usually called "Lagrangian points", also "libration points".

The class of trajectories that would make a spacecraft stay in the vicinity of these points, without orbit corrections or with limited corrections, is of obvious interest to Space Exploration. This is because the Lagrangian points remain, mathematically, at fixed distances from the 2 celestial bodies defining them. Think of it as an extension of the Geosynchronous concept, although the analogy is wrong (as all analogies are!). So while the bodies are moving in space, a spacecraft that "hugs" these equilibrium points will be also at bounded distances from the 2 celestial bodies, instead of moving away at "exponential" speed, after some time.

A "halo" orbit is a subclass of these trajectories (more on this later). First investigated by Farquhar in his Phd thesis ("The Control and Use of Libration-Point Satellites", 1968), they have since seen multiple practical uses, the International Sun-Earth Explorer-3 (or ISEE-3) being the first spacecraft to exploit this possibility. NASA tribute to Farquhar after his death wrote:

One of the more famous of Farquhar's pre-APL missions was the International Sun Earth Explorer-3, or ISEE-3, launched in August 1978 to study space weather. ISEE-3 was the first mission to exploit Farquhar's development of "halo orbits" around libration points, where the gravitational pull from two celestial bodies is balanced. After ISEE-3's initial mission was accomplished, Farquhar and long-time collaborator David Dunham designed an intricate series of orbits and engine firings that sent the spacecraft away from Earth to perform the first encounter with a comet. Renamed the International Cometary Explorer (ICE), the spacecraft made a textbook pass through the tail of comet Giacobini-Zinner on Sept. 11, 1985.

But the OP has chosen to explicitly frame the question in the realm of a mathematical model, the Circular Restricted 3-Body Problem (CR3BP). In essence, the OP's question can be interpreted as: Ignoring real-world imperfections (as the CR3B model indeed does), is there a class of orbits that, gravity alone can keep a spacecraft indefinitely in the vicinity of the libration point, this despite any small, natural or artificial, instantaneous force acting on the spacecraft? The real-world imperfections are for example: the presence of other celestial bodies, the non-circular orbits of the 2 celestial bodies of interest, ...

Unfortunately, to justify the question, the OP took an extract from a patent text (granted to JAXA in 2010. JAXA is the equivalent of NASA in Japan). This blurred the goal of limiting the discussions to solutions in the theoretical domain (and somewhat irritated the second answerer). Recall that patents are granted based on demonstrations of practical use (among other things). While their claims must be constructed with a certain legal formalism, they are seldomly judged by reviewers on their technical/scientific correctness. So, don't expect terminology or any statements in a patent to be 100% scientifically correct.

• The first answer (@user 20022, June 2017)

This answer is a "Yes, but". It provided a link to a more extended discussion in Physics SE.

I happen to have given a detail answer to this very issue in another question a couple of hours ago.

The detailed answer is a discussion of the results published in 1984 by Kathleen Howell, who is undisputably a reknown academic figure in this field. The paper is titled ""Three-dimensional, periodic, halo orbits".

• The second answer (@Ange Purs Nov 26, 9:09).

recommended that we look-up publications by Farquhar and a paper by John Breakwell.

.... The discovery of stable, large-amplitude halo orbits was published in 1979 by Breakwell and Brown: Celestial Mech, Vol 20, pp 389 - 404. Stability of these orbits was found by applying FLOQUET stability theory using numerical integrations.

Now, John Breakwell happens to be the Phd supervisor to both Howell and Farquhar. In fact, Howell and Farquhar have co-authored a paper in recognition of Breakwell's scientific contributions ("John Breakwell, the Restricted Problem and Halo Orbits", Acta Astronautica Vol. 29, no 6, pp 485-488, 1993).

Here are relevant extracts of the Abstracts of the publications cited by the two answerers:

• Breakwell& Brown 1979 (emphasis added):

The halo orbits, originating in the vicinities of both L1 and L2, grow larger, but shorter in period, as they shift towards the Moon. There is in each case a narrow band of stable orbits, roughly half-way to the Moon.