# Talk:Lagrangian point

Active discussions

## How about an explanation for the common man?

I think that at least the first sentence should be changed. "The Lagrangian points are the five positions in an orbital configuration where a small object affected only by gravity can theoretically be part of a constant-shape pattern with two larger objects." is horrible. The problem is the "be part of a constant-shape pattern with two larger objects" bit. This might be the best general description of the phenomenon but it's difficult to understand. It should either be replaced with something that doesn't include the term constant-shape pattern, or immediately afterword there should be a paragraph explaining it. I would suggest something like: "The speed at which an object orbits around another is related to how close it is. The closer the two objects get, the faster the orbiting object must be. This means that distance between two objects orbiting the same celestial body at different altitudes can not remain the same with the exception of five points provided one of the orbiting objects is of negligible mass. At these five Lagrange points the gravitational attraction of the larger orbiting object on the object of negligible mass either reduces or increases the speed required to orbit the celestial body at a certain altitude by exactly the amount needed so the smaller object can have the same orbital period as the larger orbiting object at a different altitude." Now I realize that of course all three objects are orbiting a common center but to include that would make it unclear. Also I might be a bit wrong because I'm not a physicist. Especially suspect is the use of equidistance as a goal, which is probably only valid for circular orbits (I don't know if that is the case). If it isn't valid for elliptical orbits, perhaps relative angles should be used or circular orbits should be specified. I came here hoping to understand the concept and then clicked back and went to the ESA webpage on the subject because this article starts too abstract and then just gets very technical. I haven't made the modifications directly, because I could be partially or totally wrong. I would kindly request that someone with more knowledge of orbital mechanics check my paragraph and include it or a modified version of it. Kaanatakan (talk) 16:13, 22 October 2013 (UTC)

## Add M1 M2 to diagram

The article discussed M1 M2 etc., but these symbols do not appear in the diagram. One would have to read the whole article to discover the definition of M1 and M2. This is quite frustrating. —Preceding unsigned comment added by 75.53.54.121 (talk) 18:32, 28 February 2011 (UTC)

## Exact position of L3, plus minor amendments

#### Beginning

The L-points are not necessarily in interplanetary space.

Corrected to "in orbital configuration".
> OK ("in an orbital ..." ??)

#### History and concepts

His name was hyphenated : Joseph-Louis Lagrange.

Good catch.

It has "It took hundreds of years before his mathematical theory was observed". His theory was published around 1772; Trojans were observed around 1905. Thet's not "hundreds of years" later.

"Over a hundred years"?
> OK

#### Diagrams

Recreated contour plot with n=25

The first, "... contour plot ...", diagram shows Earth, L3, L4 & L5 on a Sun-centred circle, and L1 & L2 reasonably close to Earth. That's satisfactory.

Actually, it shows L3 just outside the circle. It may not be all that clear.
> Agreed, agreed.
The problem is that the contour plot clearly shows a system where the ratio of masses primary:secondary is of the order of 10:1-50:1. In that case L3, L4 and L5 will be visibly off the secondary's orbit. I recreated the diagram here. –EdC 00:36, 5 February 2007 (UTC)

The second, "... far more massive ...", diagram shows L3 outside the circle. But if Earth, L4 & L5 all lie (as far as can be seen) on a primary-centred circle, then L3 should be similarly on that circle; not outside it.

The circle should be the orbital path of the secondary (centred on the barycentre), in which case L3 lies outside it. The diagrams should be fixed by moving the primary away from the barycentre, and L4 and L5 outside the circle.
> Doubt. Could be better to have "very much more massive" with Moon L3 L4 L5 on a circle centred on Earth, and L1 L2 very near Moon, AND also "considerably more massive" with everything properly shown. If the latter is a bit bigger, it will serve also for the L4 L5 geometrical srgument.
Yes, that could work. In that case the "very much more massive" diagram should have L1 and L2 pulled in as far as practicable. –EdC 00:36, 5 February 2007 (UTC)
Done, now we need to decide how to fix the contour plot. –EdC 02:16, 5 February 2007 (UTC)

The blue triangles (showing the gradient to be downhill going away from the points) indicate that L4 and L5 are unstable equilibria, whereas they are actually stable equilibria. —Preceding unsigned comment added by 208.71.200.91 (talk) 05:26, 24 February 2010 (UTC)

Yes L4 and L5 are the stable equilibria. But they really are the maxima of the pseudopotential; it takes the Coriolis force to keep objects from falling away from those points.
—WWoods (talk) 09:21, 24 February 2010 (UTC)
The triangles are equilateral and do not act effectively as arrows. It is therefore difficult if not impossible to be certain which direction they are intended to be pointing.
—thereaverofdarkness (talk) 20:47, 10 August 2015 (UTC)

#### Section "L3"

The page says : "L3 in the Sun-Earth system exists on the opposite side of the Sun, a little farther away from the Sun than the Earth is" - my italics. That wording will naturally be taken as saying that L3 is further from the centre of the Sun than the centre of the Earth is.

The better calculations measure distances from the barycentre, and show that L3 is a little further from the barycentre than the centre of the Earth is. But it seems that L3 is a little nearer to the centre of the Sun than the centre of the Earth is.

Hm. Yes, it is, isn't it?
> Not a lot of people know that, though.
Fixed - I hope. –EdC 01:05, 5 February 2007 (UTC)

New point: the article says "Example: L3 in the Sun–Earth system exists on the opposite side of the Sun, a little outside the Earth's orbit but slightly closer to the Sun than the Earth is." But how can it be OUTSIDE the Earth's orbit but CLOSER to the sun??

Because the Sun also orbits the barycenter – hence the Sun is closer to the far side of the Earth's orbit (if we ignore eccentricity, perturbation from other planets, and possibly a bunch of other things I forgot). :) — the Sidhekin (talk) 21:48, 24 May 2008 (UTC)

The issue of whether L4 and L5 are stable has been raised several times above. Could this be sorted out please. At the moment there's a straight contradiction between the contour plot, showing blue arrows leading "downhill" from L4 and L5, and the statement "the triangular points (L4 and L5) are stable equilibria ...". I can't fix it myself because I don't know which one is right. Occultations (talk) 11:37, 24 February 2008 (UTC)

I came to this Talk page for the same reason. L4 and L5 would have to be low points in the potential in order for them to be stable (according to this NASA page) and have objects orbit those points. The fact that it's stable only if conditions regarding the M1/M2 ratio should come later as, presumably, the "islands" of stability just get smaller as the ratio decreases until they eventually disappear. bcwhite (talk) 12:09, 31 May 2012 (UTC)
The points are dynamically unstable (an object perturbed from L4 will continue to move away) but form stable equilibria (Coriolis force will curve an object's path back to L4). EdC (talk) 15:46, 24 February 2008 (UTC)
The plot's caption used to say, "Counterintuitively, the L4 and L5 points are the high points of the potential." Maybe something about the stability of L4 and L5 should be added to the introduction?
—WWoods (talk) 17:23, 24 February 2008 (UTC)
Stability of L4 & L5 depends on the mass ratio of the primary and secondary bodies. For Earth/Moon they have been analytically proved to be stable (ca ~1980 I think; in the sense that a small test mass near either L4 or L5, moving at low speed with respect to the Lagrange point, will remain in its vicinity), and I believe this result also applies to Sun/Earth & Sun/Jupiter, as ratio m1/m2 is even larger (~80 for Earth/Moon). However, the stability depends on the Coriolis force, as an object a small distance from the L4 (say) point will at first move away, but then move back towards it, looping I think in a rosette sort of path. Keith Symon's old textbook Mechanics (2nd edition, 1960, Addison Wesley) discusses this problem fairly extensively at an advanced undergraduate level, although in 1960 the long-term question was still open. I am not certain that the Earth/Moon proof actually applies to the real physical situation, with a somewhat (~0.05) eccentric orbit, perturbed by the Sun, but I believe long accurate numerical integrations suggest they are stable. Wwheaton (talk) 16:11, 6 May 2010 (UTC)

I'm still confused. This article doesn't mention "Coriolis" anywhere, yet that appears to be critical to stability. Obviously, L4 and L5 look unstable in terms of effective potential. Understanding that earth is orbiting the sun and therefore that L4 and L5 are orbiting the sun, I can see the potential for orbiting-like behavior around those points, but I haven't worked out the details. Also, I've never seen "effective potential" before. It looks like effective potential drops off at a distance as centrifugal force takes over. In a two-body problem, I think the effective potential for a given angular momentum would be an annular trough, the bottom of which would indicate the the radius of the stable circular orbit (and the trough shape indicating stability).
I think this article needs a description of the effects of perturbation of a mass near all Lagrange points, particularly L4 and L5. I'd like to see something like this: "Suppose a small object is placed at L4 with the same angular velocity as earth. Were earth not there, it would be in orbit around the sun, but with earth there, it slows down (falls back toward the earth), causing it to drop to a lower orbit, speed up angularly, move forward (away from the earth), etc., orbiting L4." Is that even right? —Ben FrantzDale (talk) 14:49, 4 March 2015 (UTC)

All we need is a simple analogy. L4 and L5 are stable. Like a marble in the centre of a valley. If it is moved away from the centre a tiny bit, it rolls back to the centre. The other points are unstable. Like a marble on the tiny flat spot on the top of a hill. If it is moved away, even a tiny bit, it will then roll away from the hill. — Preceding unsigned comment added by 121.212.53.120 (talk) 03:47, 1 February 2018 (UTC)

I added the following clarification to the stability section, which I'm hoping resolves this issue: "Although the L4 and L5 points are found at the top of a "hill", as in the effective potential contour plot above, they are nonetheless stable, as such a diagram ignores Coriolis acceleration (which depends on the velocity of an orbiting object and cannot be modeled as a contour map)." expensivehat (talk) 22:11, 5 September 2018 (UTC)

## Changes made to the Intuitive Explanation

The Intuitive Explanation as presented was simply wrong. The outward force sensed by the hand twirling the string is a real physical force, namely the tension. It is not the centrifugal force. Further, the fact that when released the revolving mass travels on a straight tangential trajectory has nothing to do with the centrifugal force being 'fictitious'. When the string is cut, there is no longer any centrifugal nor centripetal force, so we are in an entirely new dynamical situation which is unrelated to the previous state. In addition to fixing these problems I have made the section less verbose and replaced the word 'weight' (which actually means 'gravity force') by a specific item, the stone. PlantTrees (talk) 20:26, 12 March 2008 (UTC)

### Possibly related question

Is there a stable orbit which runs through L4 L5, but into the 3rd dimension in the diagram? Or any pair of L1 L2 L3 ? Or any other pair? Just a mental rambling on my part, L4 L5 looks fairly obvious, but I am not able to do the math. —Preceding unsigned comment added by 78.32.144.39 (talk) 18:26, 10 April 2009 (UTC)

If you are still here 9 years later, then Yes and Yes. There are a number of excellent YouTube videos that show actual orbits of the Trojan (at the L4 & L5 points) and Hilda (at the L3 point) asteroids around Jupiter. It shows them in a sort of a "perspective" projection, which means you get some idea of "vertical" motion. You can see all combinations. Some asteroids stay at L4 and L5, some move between L4 and L5, some move through all three points. Be aware that the "easiest to understand" visualisations are shown as rotating with Jupiter, so Jupiter itself appears stationary. Search YouTube for "Trojan asteroids". As part of "background", these usually show the normal asteroid belt in green. — Preceding unsigned comment added by 58.166.224.167 (talk) 00:19, 3 March 2018 (UTC)

Here it is: https://www.youtube.com/watch?v=yt1qPCiOq-8 — Preceding unsigned comment added by 121.212.147.109 (talk) 03:26, 3 March 2018 (UTC)

## History and concepts

In the final paragraph of the section, the word "area" appears inappropriate. It seems to me that, in rotating co-ordinates, the system of two bodies and five points maintains a constant shape, varying in size. Then, at any instant, each L-point has a point location, which moves in and out in rotating co-ordinates and on an elliptical track in fixed co-ordinates. 82.163.24.100 (talk) 16:40, 23 June 2008 (UTC)

http://www.merlyn.demon.co.uk/gravity5.htm can now show the Lagrange points for elliptical orbits as seen by a co-rotating observer, who sees that each of the five points moves in a straight line. The observer's angular velocity is of course not constant. The path traced out by the points, in these coordinates, does not enclose an area. 94.30.84.71 (talk) 22:37, 12 June 2011 (UTC)

## Proposal to add section "Proposed objects"

It might worth noting somewhere the proposals (Space sunshade and Solar shade) to place a large "sunshade" at the L1 point to counteract the effects of global warming. (Give these proposals some though before dismissing them out-of-hand.) There are sections for existing and fictional objects at the Lx points, but there is no section for proposed objects. 220.76.15.253 (talk) 18:12, 27 October 2008 (UTC)

## Earth-Moon L2 Point

The article doesn't mention go into detail on the Lagrangian point on the other side of the moon; for example this orbit is an option for Constellation Program Moon relay satellites. MithrasPriest (talk) 18:46, 5 November 2008 (UTC)

## Rubber sheet analogy

If the contour map were replaced with a rubber-sheet visualization (i.e., the kind that are popular for depicting gravitational distortions of space-time), would it be accurate to say that the L-points are the highest points on the ridges between the astronomical bodies? If that's the case, it would be fairly obvious that these locations are not stable, and that if an obejct at such a point is nudged away from that point, it will "fall into" the gravity well of one of the bodies, right? | Loadmaster (talk) 17:43, 3 December 2008 (UTC)

L-4 and L-5 are the highest points of the pseudopotential, yes. Picture a volcano, with a deep central crater, and a smaller crater breaking its rim.
If an object is released near one of those points, it falls away from them, but the Coriolis force pushes it sideways, driving it along the contour lines which are closed around the points, so it never leaves the vicinity.
By contrast, L-1, -2, and -3 are saddle points. An object released there can wander about the system
—WWoods (talk) 20:24, 3 December 2008 (UTC)

## Kidney bean-shaped orbit round L4 and L5?

The paper by Cornish cited in the article shows that orbits around L4 and L5 are characterised by two frequencies. Since these are not in general harmonically related, it seems unlikely that these orbits will have a simple closed form.

Suppose that we rotate the coordinate axes in Cornish's paper so that the x-axis is parallel to the 'ridge' in the potential 'hill'. The effect of this on the evolution matrix is to diagonalise the 2x2 submatrix forming its bottom left-hand corner. It then becomes

${\displaystyle {\begin{bmatrix}0&0&1&0\\0&0&0&1\\{\frac {3}{4}}\left(2-{\sqrt {3\kappa ^{2}+1}}\right)\Omega ^{2}&0&0&2\Omega \\0&{\frac {3}{4}}\left(2+{\sqrt {3\kappa ^{2}+1}}\right)\Omega ^{2}&-2\Omega &0\\\end{bmatrix}}.}$

The eigenvalues (which indicate the characteristic frequencies of the system when the orbit is stable) are of course unaffected by this rotation. If ${\displaystyle \epsilon }$  is an eigenvalue, it can be shown that

${\displaystyle {\frac {\delta y}{\delta x}}={\frac {2\Omega \epsilon }{{\frac {3}{4}}\left(2+{\sqrt {3\kappa ^{2}+1}}\right)\Omega ^{2}-\epsilon ^{2}}}}$

Substituting the eigenvalues in turn into this, we obtain

${\displaystyle {\frac {\delta y}{\delta x}}=\pm i{\frac {\sqrt {2-{\sqrt {27\kappa ^{2}-23}}}}{2-{\frac {1}{4}}{\sqrt {27\kappa ^{2}-23}}+{\frac {3}{4}}{\sqrt {3\kappa ^{2}+1}}}}}$

and

${\displaystyle {\frac {\delta y}{\delta x}}=\pm i{\frac {\sqrt {2+{\sqrt {27\kappa ^{2}-23}}}}{2+{\frac {1}{4}}{\sqrt {27\kappa ^{2}-23}}+{\frac {3}{4}}{\sqrt {3\kappa ^{2}+1}}}}}$

That these are pure imaginary is a consequence of having diagonalised part of the evolution matrix. It follows that the orbit has the general form

${\displaystyle \delta x(t)=A_{1}\sin \left({{\frac {1}{2}}{\sqrt {2-{\sqrt {27\kappa ^{2}-23}}}}\Omega t+\phi _{1}}\right)+A_{2}\sin \left({{\frac {1}{2}}{\sqrt {2+{\sqrt {27\kappa ^{2}-23}}}}\Omega t+\phi _{2}}\right)}$
${\displaystyle \delta y(t)=A_{1}{\frac {\sqrt {2-{\sqrt {27\kappa ^{2}-23}}}}{2-{\frac {1}{4}}{\sqrt {27\kappa ^{2}-23}}+{\frac {3}{4}}{\sqrt {3\kappa ^{2}+1}}}}\cos \left({{\frac {1}{2}}{\sqrt {2-{\sqrt {27\kappa ^{2}-23}}}}\Omega t+\phi _{1}}\right)}$
${\displaystyle +A_{2}{\frac {\sqrt {2+{\sqrt {27\kappa ^{2}-23}}}}{2+{\frac {1}{4}}{\sqrt {27\kappa ^{2}-23}}+{\frac {3}{4}}{\sqrt {3\kappa ^{2}+1}}}}\cos \left({{\frac {1}{2}}{\sqrt {2+{\sqrt {27\kappa ^{2}-23}}}}\Omega t+\phi _{2}}\right)}$

where ${\displaystyle A_{1}}$ , ${\displaystyle A_{2}}$ , ${\displaystyle \phi _{1}}$  and ${\displaystyle \phi _{2}}$  are constants chosen to match a particular set of initial conditions. The orbit is thus a kind of epicycle in which the circular motions have been replaced by elliptical ones; it is not, however, a squashed epicycle since the eccentricities of the ellipses differ.

The relationships between the shapes of these elliptical components and that of an equipotential are shown in the above. The innermost (solid) ellipse is an equipotential while the middle (dashed) ellipse shows the shape of the slower of the two elliptical components and the outer (dotted) one that of the faster.

The above shows part of an orbit for a system with an earth-moon mass ratio. The periods of the elliptical components are about 3.35 and 1.05 times that of the two-mass system. It is fairly apparent from the picture how, as the body falls down the potential hill, the Coriolis force changes its direction of motion to bring it back up again.

Can the orbit ever be kidney bean-shaped? Cornish's paper - and thus the above - deal with approximations to orbits which lie 'close' to a Lagrangian point. Presumably, if (i) the size of the orbit became such that the curvature of the potential 'ridge' were large enough to 'bend' the ellipses and (ii) if we had a situation analogous to that in which either ${\displaystyle A_{1}}$  or ${\displaystyle A_{2}}$  were 0, we would obtain a kidney bean-shaped orbit rather than an elliptical one but the probability of the second condition occurring naturally would seem to be 0.

--IanHH (talk) 17:27, 16 March 2009 (UTC)

My impression is that when they talk about a "kidney bean" shaped orbit, they usually mean with respect to a frame co-rotating with the smaller mass. So if you revolve your Spirograph, then as you said, depending on your A1/A2/A3, you may well be able to get a "mutant" kidney shape.

Finally, a number of simulations shown on YouTube of specifically the Trojan asteroids, show several that visit around L3, L4 and L5. It's a bit hard to tell, though whether they are "kidney beaning". I would have made the asteroids different colours :D — Preceding unsigned comment added by 121.212.147.109 (talk) 03:31, 3 March 2018 (UTC)

## References

Reference 6 may be dead. 82.163.24.100 (talk) 14:02, 13 April 2009 (UTC) Now http://wmap.gsfc.nasa.gov/media/ContentMedia/lagrange.pdf 82.163.24.100 (talk) 15:18, 13 April 2009 (UTC)

Current Reference 3 is at present a 404. Also (1867–92) are highly unlikely dates for Lagrange, which is what they look like there. They are probably the dates of the 'Oeuvres'. The Reference should be removed; the link to ULg serves. 94.30.84.71 (talk) 20:14, 15 August 2011 (UTC)

## limiting mass

I assume that there is a limiting mass for the body placed at the L point to have a stable orbit. That would be relevant for the destabilization of Theia. Anyone know? kwami (talk) 13:34, 3 May 2009 (UTC)

See page 2 top of http://www.lesia.obspm.fr/perso/bruno-sicardy/biblio/biblio/Sicardy_Gascheau_CelMec10.pdf - Gascheau was the first to study the stability of the L4 and L5 points (1843), showing that the motion of three non-zero masses m1,m2 and m3 in a rotating equilateral configuration becomes linearly unstable if (m1 + m2 + m3)2 / (m1 · m2 + m1 · m3 + m2 · m3) ≤ 27. That condition is approximately equal to (m2 + m3)/m1 < 0.04, as illustrated in http://www.merlyn.demon.co.uk/gravity4.htm . 94.30.84.71 (talk) 16:26, 13 May 2012 (UTC)
Thanks! (Should that be a ">"? Negligible m2 and m3 should be stable.)
Since (m2 + m3)/m1 could never be > 0.04 (as m2 = 0.003 and m3 is less than that), it would seem that this cannot explain the destabilization of Theia.
The solution may be that it wasn't a 3-body problem. (See Giant impact hypothesis#Possible origin of Theia.)
kwami (talk) 12:36, 17 October 2012 (UTC)

## Rocheworld

Surely the Rocheworld waterfall is at L1, not L4?

Comsats were placed at L4 and L5.

82.163.24.100 (talk) 20:09, 4 May 2009 (UTC)

Indeed. —Tamfang (talk) 06:34, 11 May 2009 (UTC)

## L2 is relatively close to the Earth

The first diagram claims to be "A contour plot of the effective potential of a two-body system (the Sun and Earth here) ..." Isn't that misleading? The radius of the Earth's orbit is 150 million km, and the distance from the Earth to L2 is 1.5 million km, so L2 is really snugged up close to the Earth. I see that a contour plot on that scale would be too difficult to read, but I think the caption should be changed to make the relationships clear, or at least not misleading. Maybe making the plot for the Earth and the Moon would help, since their masses are more nearly comparable. --Art Carlson (talk) 08:34, 19 May 2009 (UTC)

...likewise L1. —Tamfang (talk) 19:48, 19 May 2009 (UTC)
Right. I'm fixating on the Planck satellite. I'm pretty sure the islands around L4 and L5 will shrink down to specks as well. --Art Carlson (talk) 20:40, 19 May 2009 (UTC)

## ESA explanation

I found this article today on L points at the ESA website: http://www.esa.int/esaSC/SEMM17XJD1E_index_0.html. The animations are very easy to understand, as are the explanations. Maybe it's just me, but I found this much easier to follow than the "stone at the end of a string" examples in the 'Intuitive explanation' section. I'm wondering if ESA material is PD and could be incorporated into the article somehow. —divus 05:55, 6 July 2009 (UTC)

As of yesterday, the ESA material, while generally good, did not appear completely accurate. 94.30.84.71 (talk) 10:33, 24 July 2011 (UTC)

External link "Astronomy cast - Ep. 76: Lagrange Points Fraser Cain and Dr. Pamela Gay" seems unworthy of mention. 82.163.24.100 (talk) 18:40, 9 August 2009 (UTC)

Also, the fourth link in the References, to Gallica, does not now work. 82.163.24.100 (talk) 19:31, 9 August 2009 (UTC)

Should it be http://gallica.bnf.fr/ark:/12148/bpt6k229225j.image.r=Lagrange%2C+Joseph-Louis%2C.langEN.f274.pagination ? 82.163.24.100 (talk) 19:39, 9 August 2009 (UTC)

The present Reference 3 should probably now link to http://gallica.bnf.fr/ark:/12148/bpt6k229225j/f819.image.r=lagrange.langFR . 94.30.84.71 (talk) 19:20, 12 June 2011 (UTC)

In the examples of L4 and L5, the link to "NASA's animated clip" doesn't go to the correct clip. — Preceding unsigned comment added by 69.70.154.110 (talk) 15:20, 18 July 2013 (UTC)

## L2 and Umbra

For circular orbits and a relatively light secondary, it is sufficiently easy to show that L2 is within the umbra if the secondary is less than three times denser than the primary. The density ratio for L2 being at the umbra tip is quite sensitive to the mass ratio. See in http://www.merlyn.demon.co.uk/gravity4.htm. At present, the Article says little about that. 82.163.24.100 (talk) 15:21, 16 September 2009 (UTC)

## Missing words

Sorry if I'm wrong on this, but are there words missing from the caption in the rotating diagram? It just ends "in relation to..." Thanks InspectorSands (talk) 01:27, 25 October 2009 (UTC)

Done -- I'm not seeing a problem now. TNKS, Charles Edwin Shipp (talk) 00:26, 4 April 2015 (UTC)

## Execrable Prose Style!

Regardless of technical content, the prose style of this article is uniquely bad among WP science articles, which as a group are the worst content on WP. This is a real shame and a true mystery. Something should be done to organize the WP science pages and ruthlessly weed out this awful writing. Antimatter33 (talk) 15:19, 9 December 2009 (UTC)

## L1

The distance of L1 from the smaller body should be added. I suppose it is 1/C^2 the distance between the two large bodies, where C is the quotient of their masses? --Roentgenium111 (talk) 13:23, 10 December 2009 (UTC)

I rather doubt that 1/C2 can be accurate, though it may be a first approximation. http://www.merlyn.demon.co.uk/gravity4.htm may help. 82.163.24.100 (talk) 20:01, 2 February 2010 (UTC)
For Sun/Earth the distance ratio is near 100, and the mass ratio is near 300,000. The distances for L1 & L2 are almost equal for the Sun/Earth case, both ~1.5e6 km from Earth. The animation figure (Lagrangianpointsanimated.gif) shows the distance to L2 is much larger than to L1, which is wrong and needs to be fixed. Both of those distances are shown as several times too large, >>0.01 AU; which would be hard to see if rendered correctly, but this could be fixed by dropping the caption that refers to the Sun and Earth. Wwheaton (talk) 17:16, 6 May 2010 (UTC)

Can't possiby be 1/C^2. Jupiter is about 1/1000 the mass of the Sun, and the table shows 1-L1/SMA to be 6% . — Preceding unsigned comment added by 121.212.147.109 (talk) 03:37, 3 March 2018 (UTC)

## The High Frontier

The first reference to "The High Frontier" by Dr. Gerard K. O'Neill should not be within a section headed "In fiction". 82.163.24.100 (talk) 20:25, 2 February 2010 (UTC)

## sonde at L4 and L5

The Japanese Sonde Hiten launched 1990 made swingbys at moon and visited Lagrangian points L4 and L5 to collect dust there. NASA. Retrieved 3.2.2010. —Preceding unsigned comment added by Helium4 (talkcontribs) 13:05, 3 February 2010 (UTC)

...shouldn't this be added under the missions section? 128.210.206.145 (talk) 08:38, 18 March 2013 (UTC)
Added--agr (talk) 17:16, 18 March 2013 (UTC)

## Do we need a lengthy "in fiction" list of trivia here?

It is a leading question, I know. My opinion is that lengthy lists of every science fiction book and cartoon series that mention a given topic are not very useful. Look at the size of this list in relation to the article as a whole. I am trying to imagine the reader that comes to this article that would find such a list helpful. I am tempted to delete the entire section unless somebody wants to talk me out of it. CosineKitty (talk) 00:58, 22 March 2010 (UTC)

I concur. But I don't see much, if any, of a trivia section left. Maybe someone already deleted it. N2e (talk) 04:10, 25 July 2010 (UTC)
Personally, I liked it and am sorry to see it gone. There should at least be some sort of link to such a list, at Wikipedia or elaewhere. 94.30.84.71 (talk) 21:18, 8 February 2011 (UTC)
There is a list of references, fiction and non-fiction, mainly early ones, at http://www.merlyn.demon.co.uk/gravity4.htm#R3 . 94.30.84.71 (talk) 17:42, 4 July 2012 (UTC)

## How stable is "stable"?

I don't really understand this article. Would something placed at a Lagrange point really stay there, or would it need to fire thrusters every now and then to really stay there? Is it more like a gravitational hole where things placed there are pulled towards the center of a lagrange point, or is it just that you wouldn't feel any gravity there, but since you can't set your inertia precisely you would inevitably drift away from that point? 89.12.78.225 (talk) 19:09, 24 July 2010 (UTC)

L1, L2, L3 are unstable equilibria: there is no net force (in a rotating frame) at those precise points, but a body will fall away from them along the line between the major masses (but back toward that line) – like a ball balanced on a saddle. L4, L5 are stable equilibria, effectively attractors: a body perturbed from either will continue to wander in the neighborhood, like a ball in a bowl. Or so I (mis)understand. —Tamfang (talk) 22:13, 25 July 2010 (UTC)

Not exactly. Yes, the initial force will be along that line, but because the radius of the orbit changes, then due to conservation of angular momentum, the velocity will change, and the object will move perpendicular to that line. So if it moves "inward" the velocity will increase; conversely, if it moves "outward", the velocity will decrease. — Preceding unsigned comment added by 58.167.55.219 (talk) 07:43, 7 March 2018 (UTC)

With active and sufficiently accurate control, the amount of propellant needed to station-keep at the equilibrium point can be made arbitrarily small. The acceleration (read thrust) needed can also be very small. A small ion drive unit, as are now used on spacecraft at GSO, should be entirely sufficient, I think. Small perturbations by the Sun (for Earth/Moon), Moon (for Sun/Earth), planets, solar radiation pressure, etc, may be the limiting factors. Wwheaton (talk) 19:41, 8 August 2010 (UTC)
L4 and L5 are completely stable, provided that the primary/secondary mass ratio exceeds about 25 and external perturbations are minor. Otherwise, only L1 L2 L3 require station-keeping. 94.30.84.71 (talk) 11:18, 9 February 2011 (UTC)

## That diagram

The five Lagrangian points (marked in green) at two objects orbiting each other. (Here a yellow sun and blue earth)

This diagram is seriously flawed. The distances from the Earth to Sun/Earth L1 & L2 are nearly equal, at 1.5 million km, ~1% of the Earth Sun distance. The L3 point is nearly the same distance from the Sun as L2, I believe, a little over 1 AU. Can someone who knows how to fix the .gif modify it? If not, I think it adds little and needs to be removed. Thanks. Wwheaton (talk) 19:50, 8 August 2010 (UTC)

L3 is more nearly the same distance from the primary as the secondary is. 94.30.84.71 (talk) 11:20, 9 February 2011 (UTC)
It is easy to take a copy of the GIF and edit it in Windows Paint. Uploading is another matter. 94.30.84.71 (talk) 11:58, 9 February 2011 (UTC)
Best replacement would be a new SVG modeled on the PDF at caltech.edu: "Dynamical Systems," etc. by Koon, Lo, Marsden and Ross, page 9. Rursus dixit. (mbork3!) 08:35, 30 July 2011 (UTC)

In the Article, the second diagram (shown here; http://en.wikipedia.org/wiki/File:Lagrangianpointsanimated.gif) should be replaced by the third diagram, http://en.wikipedia.org/wiki/File:Lagrange_very_massive.svg. It would be well is someone who knows how would enlarge the text within the latter. 94.30.84.71 (talk) 17:51, 4 July 2012 (UTC)

Text enlarged... cmɢʟee୯ ͡° ̮د ͡° ੭ 19:26, 14 February 2013 (UTC)

## Introduction

It seems to me that the sentence "The Lagrange points mark positions where the combined gravitational pull of the two large masses provides precisely the centripetal force required to rotate with them." should be improved by inserting "for the object" between "required" and "to". If you agree - do it! And "pull" should be "pulls"? 82.163.24.100 (talk) 13:00, 28 October 2010 (UTC)

http://www.merlyn.demon.co.uk/gravity5.htm shows Lagrange points moving under various conditions; it might be worth adding, but it is linked from gravity4.htm (the old g5 is now g6). 94.30.84.71 (talk) 21:23, 8 February 2011 (UTC)

## The L2 distance

I think this says that the distance of Sun-Earth L2 from Earth is such that, if there were no Sun, a body at that distance would orbit Earth in 1/√3 year. Is that right? —Tamfang (talk) 19:04, 7 March 2011 (UTC)

Linguistically it says (subclause interpretation):

or (ellipsis interpretation):

I cannot make any sense of it. Rursus dixit. (mbork3!) 08:55, 30 July 2011 (UTC)

A proof of this follows the first part of http://www.merlyn.demon.co.uk/gravity4.htm#Ap1. 94.30.84.71 (talk) 12:28, 11 May 2012 (UTC)

"At the L1 point, the orbital period of the object becomes exactly equal to Earth's orbital period. L1 is about 1.5 million kilometers from Earth.[4]"
" ... at the L2 point that orbital period becomes equal to Earth's."


Since the distance of L1 from Earth is given, the distance of L2 should also be given. L2 is just as important as L1. — Preceding unsigned comment added by Martin Coles (talkcontribs) 00:12, 25 April 2015 (UTC)

## Trojans

In a general article on Lagrangian points, the string "Trojan" ought to appear at least once. 94.30.84.71 (talk) 22:03, 15 June 2011 (UTC)

I count sixteen. Is that excessive? —Tamfang (talk) 03:40, 16 June 2011 (UTC)
Oops. No. Perhaps I used a case-sensitive search. I understand that there is one Greek in the Trojan camp, and vice versa. 94.30.84.71 (talk) 22:25, 26 June 2011 (UTC)
Yes, named before the convention caught on. —Tamfang (talk) 00:12, 29 June 2011 (UTC)

## Image is wrong.

I'm sorry. I don't speaking english. TT I Want you to understand it.

Arrow around L4 and L5 in Vector Image (http://en.wikipedia.org/wiki/File:Lagrange_points2.svg) is wrong. The surrounding arrow is to make the opposite direction because of L4 and L5 is a stable equilibrium point. --Goldenbug (talk) 19:58, 12 July 2011 (UTC)

Yes it is wrong and generally chaotic. It needs (at the very least) the following fixes:
• the L4 and L5 surrounding arrows each needs a 180° turnaround, and being colored red,
• the gravity well indicating curves should all be removed, they make a visual chaos — instead I propose indicating the gravity well with gray shades, f.ex. gray deep in the well and white in gravity-zero-space,
• the direction of earth's orbit movement needs a clearly visible arrow,
• earth is almost invisible against the background, when choosing an appropriate background color for the gravity well, earth needs a strongly contrasting color,
• the figure should be adapted to low sizes, the letters already are big – good! – earth and the lagrange points need to be big,
• the lagrange points themselves should need a large clearly visible symbol that is not round or any conventional astronomical symbol (stars, planetary nebulae or such), f.ex. haircrosses or squares (maybe)
First fixer gets all the glory! Rursus dixit. (mbork3!) 07:50, 30 July 2011 (UTC)
/harrumph/! Diging into the topic, I'm a little unsure what the "arrows" (red and green blue triangles) represent. It might be that the blue arrow pointing away from the L-point might instead be an uphill mark as used in geographic maps. Very misleading indeed. Maybe they should be replaced with arrows pointing "downhill" to indicate movement tendencies? Rursus dixit. (mbork3!) 08:11, 30 July 2011 (UTC)
No the arrows all point downhill. In the pseudopotential, L-1, -2, and -3 are saddle points, while L-4 and -5 are the highest points. I've tried to clarify this in the caption.
—WWoods (talk) 19:17, 30 July 2011 (UTC)

I have added to the Article two links to images of Lagrange's "Essai sur le Problème des Trois Corps". That is something which anyone considering contributing to the Article ought to have read. 94.30.84.71 (talk) 10:31, 24 July 2011 (UTC)

For the benefit of those unfortunates who cannot understand French as PDF images, I am about to add a link to a translation into English as HTML of the words of the "Essai". For the best experience, have JavaScript enabled, and a copy of the original also open to show the equations. 94.30.84.71 (talk) 19:17, 15 August 2011 (UTC)

Since Cain and Gay wrote that gravity cancels out at the L-points, I suggest that their link should be removed. 94.30.84.71 (talk) 17:30, 24 July 2011 (UTC)

## Geostationary analogy

"They are analogous to geostationary orbits in that they allow an object to be in a fixed position relative to one or two bodies rather than any more general orbit in which its relative position changes continuously."

This geostationary analogy is not at all compelling and will only confuse anyone who does not already understand what a Lagrange point is. It should be removed rather than re-worked but I suspect that if I just remove it someone will revert the edit.154.5.32.134 (talk) 04:52, 28 July 2011 (UTC)

Agreed. I commented it out. 94.30.84.71 (talk) 20:25, 15 August 2011 (UTC)

## General Review

### The Situation

There are two main problems with the Article as it stands today.

(1) Contributors have not read, or have not understood, Chapters I & II of Lagrange's Essai of 1772 (I'm assuming that to be his only relevant publication; I've read the rest of the Essai and all the Subject entries of the Oeuvres).

(2) Contributors, some of inadequate understanding, have shovelled in anything they can think of that is vaguely related. An encyclopaedia article should concentrate on its subject, and should link to other articles rather than include what should be their content.

94.30.84.71 (talk) 22:15, 15 August 2011 (UTC)

#### Introduction

It describes the common view. But Lagrange was trying to solve the general three-body problem, and, from the equations he derived, he proved that there are two types of exact solution in which the pattern of the bodies, without regard to scale, are preserved. One type was known to Euler (L1 L2 L3), the second was found by Lagrange (L4 L5). Lagrange paid no particular attention to circular orbits or even to whether the system was rotating - the paths of the bodies are, he found, conic sections.

Replace, above, "One type was known to Euler (L1 L2 L3)" with "A non-rotating collinear constant-pattern solution was found by Euler. 94.30.84.71 (talk) 11:21, 27 August 2011 (UTC)
I've now examined the Euler Archive site. Judging by the titles, only E.327, E.400, and E.626 can be relevant. All deal only with motion in a straight line, without any rotation. That cannot be said to amount to a prediction of L1 L2 L3 as commonly known. 94.30.84.71 (talk) 17:20, 21 August 2011 (UTC)

Using Lagrange's approach, but with virtually no equations, it is easy to show (easier than in the present Article) that the equilateral pattern, with the three massive bodies initially given a symmetrical pattern of velocities, is persistent. One can see that, for some spacing, any three bodies in a line must also be capable of persistence.

Previous sentence "One can see" - indeed, one can easily prove it, and solve for the spacing, by following Lagrange's approach. 94.30.84.71 (talk) 11:16, 27 August 2011 (UTC)

#### History and concepts

A reference to Euler's relevant work is needed (i.e. to Euler's own writing).

The present Reference 2, Koon et al, is unsatisfactory as it does not load, perhaps being too big. It needs to be replaced or supplemented by something else providing information on Euler's contribution. 94.30.84.71 (talk) 19:45, 16 August 2011 (UTC)
I have succeeded in fetching it. The relevant part is wrong, so there should be no Reference for Euler. 94.30.84.71 (talk) 11:25, 27 August 2011 (UTC)

I do not think that, in the Essai, Lagrange used 'action' as described.

I've looked again, and he did not. 94.30.84.71 (talk) 19:45, 16 August 2011 (UTC)

He did not use negligible mass or circular orbit.

The 'host bodies' cannot both be planets, though one of them usually is.

#### The Lagrangian points

The caption should include the word "circular". It would be better to use the Earth-Moon system, since for that L1 & L2 are more separated from the secondary and can be marked more clearly.

##### L1

"It is the only L-point which exists in non-rotating systems." Remove, as incorrect.

The idea is that if there are two stationary masses, there's a point between them where their gravitational forces are equal-and-opposite; this corresponds to L1. There's nothing like the other four, without the centrifugal force for balance.
—WWoods (talk) 20:38, 16 August 2011 (UTC)

"In a binary system," - phrase not needed. For a system of two massive bodies, whatever they may be, the Roche lobe apex is at L1.

##### L2

Unnecessary art; the previous diagram suffices.

The diagram may have been taken from ESA's Lagrange Point page - permission? 94.30.84.71 (talk) 19:27, 16 August 2011 (UTC)

"beyond the smaller of the two" - should be "beyond the lighter of the two". Size is not important.

##### L4 and L5

A better argument, valid for three massive objects independently of the shape of the orbit, exists.

#### Intuitive Explanation

"Lagrangian points L2 through L5 only exist in rotating systems," - incorrect. Well, a bit of a moot point! The entire raison d'etre of Lagrangian points is in a rotating 3-body system !

"Imagine a person spinning a stone ...". Inappropriate. The place for that, if any, is in the Wikipedia articles on centrifugal/centripetal force.

"Unlike the other Lagrangian points, L1 would exist even in a non-rotating (static or inertial) system." - Balderdash. In Lagrange systems, the only true forces are the gravitational ones. If there is no rotation, the bodies/particles all fall into the barycentre, maintaining their pattern (what happens when they get there is not germane). Of there is infinitesimal rotation, the bodies will orbit perpetually in ellipses of infinitesimal but proportionate widths, swooping round the barycentre in a manner worse than in Niven's "Neutron Star".

The rest of the section would, if correct, be superfluous. This : "or spirals in toward the barycenter" illustrates the shallownwss of its author's understanding

#### Lagrangian point missions

That material would be better placed elsewhere, in "List of objects at Lagrangian points" or in a new similar article.

#### Natural examples

Likewise. There are enough examples above; the Article should be about theory illustrated by sufficient examples, no more.

#### Other co-orbitals

2010 TK7 is "at" S-E L4. It should not be further described here; it has its own article.

Cruithne, Epimetheus, Janus are not Lagrangians. They deserve, if that, no more than a mention to plant a link.

94.30.84.71 (talk) 22:15, 15 August 2011 (UTC)

The article should, I think, say a little more about the size and shape of the useful volumes aroundd L1 L2 L3, and of the size and shape of the regions in which particles bound to L4/L5 may be found. If the primary/secondary separation is taken as 1, and the orbits as near-circular. the only relevant variable will be the mass ratio.

Stability : I assert that the triangular configuration is stable for two bodies and a particle in circular orbit if the mass ratio secondary/primary is smaller than a number about 0.04 which can be calculated exactly. I think that if the primary mass is 1, the system is stable if the sum of the other two masses is less than about 0.04, with very little dependence on the ratio of the smaller masses. That needs independent proof.

Article should include something about "Stable if   27(m1m2 + m2m3 + m3m1) < (m1 + m2 + m3)^2". That derives from Routh's Criterion, but I don't have any really good reference to offer for that expression. 94.30.84.71 (talk) 19:37, 16 August 2011 (UTC)

That's enough for tonight. 94.30.84.71 (talk) 22:15, 15 August 2011 (UTC)

## History and concepts

Paragraph 2, "in 1772, ..." : Is there evidence that Lagrange previously did significant work on the more-than-three body problem? If so, a link of some sort is needed; otherwise, there is no need for the second sentence, "Originally ... achieved.". 94.30.84.71 (talk) 17:19, 2 May 2012 (UTC)

Paragraph 3, "The logic ... trajectory.". seems superfluous. 94.30.84.71 (talk) 17:19, 2 May 2012 (UTC)

Paragraph 4, "Lagrange, however, ... in 1906." : Is it clear that, for this work, Lagrange used "action"? I have not recognised it in the Essai. In the Essai, I see nothing about "negligible mass" or "near-circular orbit" or rotating frames of reference. A body in a curved path CANNOT be experiencing zero ner force; centrifugal force is fictitions and superfluous. Only L4 L5 "follow" the orbit of another body, and they do not do that exactly. I think that the author of this section has regurgitated material from a second-rate textbook or lecture course. 94.30.84.71 (talk) 17:19, 2 May 2012 (UTC)

## New Section for Article : General Explanation

This is a draft new section, which will enable the removal of almost all of the existing Intuitive explanation section. If any of the styling needs fixing, please fix it in situ, if you can. After a few days, I'll move the new section over. 94.30.84.71 (talk) 18:42, 8 May 2012 (UTC)

### Bodies Moving in Constant Patterns

The "pattern" of a set of bodies depends only on the angles between them, and does not depend on position, orientation, and scale. The pattern of a set of massive bodies affected only by their mutual inverse-square gravitation may remain constant. Such a pattern may or may not be stable against small perturbations.

The relative acceleration of any two parts of such a constant pattern must be in constant proportion to the inverse square of the current size of the pattern. The barycentre ("centre of gravity"), which does not accelerate, can be considered as a part of the pattern. The total angular momentum about the barycentre is constant, and must be divided in a constant fashion among the bodies. So the angular momentum of each body around the barycentre is constant, and the net field at each body must be (bary-)central.

Newton has shown that the path of a body subject only to a central inverse-square net force is a conic section (circle, ellipse, parabola, hyperbola, straight line). Any conic section is a possible path for a body in a constant pattern, and laws similar to Kepler's will apply.

### The Lagrange Points

Three bodies can remain in a constant pattern in two ways, with no more than the position, the size and the absolute orientation of the pattern varying with time, as shown below.

#### Euler's Work

Euler and Lagrange shared a Prize for general Moon theory. Euler (E.304) worked in Sun-Moon-Earth theory, but got no nearer to the Lagrange Points than solving the case of three bodies moving in a fixed straight line (E.327).

P.S. The Euler Archive is at http://www.math.dartmouth.edu/~euler/ . 94.30.84.71 (talk) 18:28, 16 May 2012 (UTC)

#### Lagrange's Work

Lagrange, in the Essai sur le Probléme des Trois Corps, attempted the general three-body problem. He concentrated extensively on the behaviour of the distances between the three bodies, rather than on their positions. Using his general equations, he found that there are two types of constant-pattern solution, one with the bodies all collinear and one with them all equidistant. The first gives what are now known as L1, L2 and L3, the second L4 and L5. He did not consider the stability of the patterns.

### The Direct Approach

Dr J R Stockton, by following Lagrange in using the distances between the bodies, has proved the constant-pattern solutions directly, without considering the general three-body problem.

### A Direct Proof for L4 and L5

An equilateral triangle ABC with opposite sides a b c remains equilateral if the variable rates of change of the lengths of its sides (db/dt, etc.) are always equal. If those rates are initially equal, they remain equal if their own variable rates of change (d2b/dt2, etc.) are always equal.

For massive bodies A B C at the corners of an equilateral triangle currently of side s, the gravitational field component at A along side b is G(C+B/2)/s2. So at any instant d2b/dt2 is G(A+B+C)/s2, etc., therefore d2a/dt2 = d2b/dt2 = d2c/dt2, so the triangle remains equilateral.

Note : that applies for any three massive bodies, regardless of the shape of their paths, and does not use the inverse square law.

### A Direct Proof for L1, L2 and L3

Let massive bodies A B C be initially in a straight line. Motion of their barycentre can be disregarded. Let their initial velocities be mutually parallel and in signed proportion to their distances from the barycentre. The bodies will initially remain collinear, and the ratios of their mutual distances will initially remain the same. The pattern will initially be constant.

If AB is infinitesimal, and BC is not, the accelerations will be such that AB/BC will clearly reduce; and vice versa. There will be an intermediate initial ratio AB/BC for which AB/BC remains constant. All of the initial conditions then still hold, and the pattern is preserved for ever.

The traditional Lagrange Points L1, L2, L3 correspond to setting the masses of the bodies A B C to represent in any order a primary body, a secondary body, and a particle.

For the velocities to remain proportional to the distances from the special point the accelerations, and hence the fields, must be similarly proportional. For that, one must construct the necessary equations (which reduce to a single quintic equation) and solve them. That is done in detail in section "Collinear Pattern - L1 L2 L3" at Simplified Lagrange Point Theory, which includes the algebra for the quintic equation and a Form for the iterative solution of the equations.

Note : that applies for any three massive bodies, regardless of the shape of their paths.

### Further Work

For more detail and further work, see Simplified Lagrange Point Theory.

### Comment

Here. 94.30.84.71 (talk) 18:42, 8 May 2012 (UTC)

## Error in Introduction

The Introduction has "Lagrangian points are the stationary solutions of the circular restricted three-body problem.". That is an impractical definition, since no planet or satellite orbits in a circle. It would, though, be OK to have "The stationary solutions of the circular restricted three-body problem are called Lagrangian points." - although that does not say anything about real off-circular Lagrange Points. 94.30.84.71 (talk) 17:41, 9 May 2012 (UTC)

## Formal Definition

Is there a formal definition of Lagrange and/or Lagrangian Point - one with real authority, not a dictionary or a personal opinion? If so, the Article should use and cite it. If not, then there should be. Any definition needs to be entirely compatible with what Lagrange actually wrote (by the way, did he publish anything relevant other than the Essai?). It should also be fully compatible with current usage, but might be broader. 94.30.84.71 (talk) 22:02, 9 May 2012 (UTC)

The Lagrange Points are the relative positions of the lightest body, when of negligible mass, in constant-pattern solutions of the three-body problem. ?? 94.30.84.71 (talk) 18:54, 12 May 2012 (UTC)

## Lagrangian point missions

WMAP seems no longer operational. 94.30.84.71 (talk) 20:31, 20 May 2012 (UTC)

yes, fixed--agr (talk) 17:03, 18 March 2013 (UTC)

## History and concepts

I have rectified section "History and concepts" to agree with what Euler and Lagrange actually wrote, using the translations at the Merlyn site and directly reading other Euler papers. I have for the moment retained some parts which are apparently unnecessary but are not known by me to be actually false, adding HTML comment at the end of each. 94.30.84.71 (talk) 12:08, 4 July 2012 (UTC)

Query : Are there any other writings of Lagrange relevant to the topic - I have found none. 94.30.84.71 (talk) 12:08, 4 July 2012 (UTC)

## Introduction

I have removed the inappropriate presumption of circularity from the Introduction. There are in practice no truly circular orbits, and Lagrange's work applies (exactly) to constant-pattern paths of any conic section, as his readers will see. 94.30.84.71 (talk) 13:14, 4 July 2012 (UTC)

## Special Cases

If there is no rotation and the third body has negligible mass, both L4 and L5 become a circle around the mid-point of the line joining the other two bodies. 94.30.84.71 (talk) 13:21, 4 July 2012 (UTC)

If the two large masses are equal, then (L2 L3) and (L4 L5) are two indistinguishable pairs. 94.30.84.71 (talk) 13:21, 4 July 2012 (UTC) Uhh, that can't be right, surely? (L2 L3) are collineat with the two large masses, (L4 L5) are equilaterally triangular. How can they possibly be the same? Or have I misunderstood this sentence, and if so, does that mean the sentence is "bad" even if it is correct???

In External Links, I have changed the Euler link from the E.327 PDF at Dartmouth to the E.304 information page at MAA. The reasons will be obvious to those who have read the Euler Archives. 94.30.84.71 (talk) 22:38, 23 October 2012 (UTC)

Moreover, the present Reference 3 is to E.327, which discusses motion along a FIXED straight line, and so cannot be said to contain a discovery of the any Lagrange Points, since the Points are commonly considered to have near-circular paths. 94.30.84.71 (talk) 20:59, 24 October 2012 (UTC)

In E.304, now cited, Euler discovered L1 and L2 (by the rather drastic step of moving the Moon there), but did not discover L3 (a step too far? - but would have been compatible with his general reasoning). Nothing else by Euler appears to have revealed L3. The academics responsible for Reference 2 appear to have relied on rumour, without checking their facts, and should not be quoted on this matter. 94.30.84.71 (talk) 20:59, 24 October 2012 (UTC)

## Circular and Restricted

The Lagrange Points are sometimes described as solutions of the circular restricted three-body problem. That is bad practice. Lagrange found the two constant-pattern configurations for the general three-body problem, and the merlyn site shows how easily those solutions can be verified if one avoids any attempt to solve for arbitrary initial conditions. The forms of the paths are in neither case artificially constrained; the solutions allow ANY conic section. Moreover, it is generally agreed that Lagrange Points exist in the Solar System, but every planet and moon has an elliptical orbit. 94.30.84.71 (talk) 21:16, 24 October 2012 (UTC)

Although the calculations allow all conic sections as paths, one might choose to use "Lagrange Points" only for the near-circular solutions. That seems ungenerous to Lagrange. 94.30.84.71 (talk) 21:16, 24 October 2012 (UTC)

## External Link to be Removed

The External Link to "The Five Points of Lagrange" by Neil deGrasse Tyson needs to be removed, as it has historical and technical errors, and adds nothing that is needed. 94.30.84.71 (talk) 12:52, 26 October 2012 (UTC)

## L3 - further question

Under the L3 section it says - "the Sun–Jupiter system is unbalanced relative to Earth (that is, the Sun orbits the Sun–Jupiter center of mass...)". Should that not be "the Sun–Jupiter system is unbalanced relative to Earth (that is, the Earth orbits the Sun–Jupiter center of mass...)"? BigSteve (talk) 15:48, 28 February 2013 (UTC)

PLEASE someone explain HOW the Earth's gravity EQUALS the Sun's at L3...at a distance of 300 million kilometres...??? BigSteve (talk) 09:15, 23 March 2013 (UTC)
It doesn't, obviously. Where does the article say this? --JorisvS (talk) 11:42, 23 March 2013 (UTC)
Well, the fact that L3 is stable means that it is. For want of better scientific phraseology...how else do objects stay stuck there? BigSteve (talk) 13:57, 23 March 2013 (UTC)
No, it doesn't. The combined gravity of planet+Sun means that an object near the planet's L3, which is located slightly outside its orbit, can orbit there with the same (average) orbital period as that planet. --JorisvS (talk) 14:47, 23 March 2013 (UTC)
OK, but that still doesn't explain Why and How! :-) I get the complex mathematical formulas (well, actually, I don't, but I trust you that they are correct), but can someone please explain in the article for non-mathematicians how and why L3 exists... BigSteve (talk) 16:03, 23 March 2013 (UTC)
Well, it basically does. An object can orbit there with the same period as the planet. This is because the object at the L3 feels things like a slightly higher gravitational attraction coming from the Sun (which is because of the planet on the other side the Sun). This means that at a slightly greater semi-major axis it can orbit with the same orbital period as the planet. This is the reason that the L3 exists. The reason that it is stable is that when the object deviates from the L3, it effectively feels a small force directed towards the L3. --JorisvS (talk) 10:46, 24 March 2013 (UTC)
That's crazy... Could you add this explanation in the article? Perhaps adding a mention of what seems to me, looking at the contours, as if the Earth's gravity almost goes around the Sun...?! A bit like a magnetic field would. Which I guess also has something to do with it? BigSteve (talk) 11:01, 24 March 2013 (UTC)
Crazy? What do you mean? Gravity does not behave like magnetic fields. --JorisvS (talk) 11:58, 24 March 2013 (UTC)
The contours are that of the gravitational potential, not of the gravitational force. --JorisvS (talk) 12:01, 24 March 2013 (UTC)

"Crazy" in a positive way! As far as the gravity contours... File:Lagrange points2.svg – seems pretty magnetic-like to me, esp. the contours around L4 and L5, which seem to me to be definig the position and stability of L3...by going around the Sun. Unless I'm not understanding it? BigSteve (talk) 12:06, 24 March 2013 (UTC)

That's a gravitational-potential plot, which definitely does not behave like magnetic fields. Magnetic fields are vectors, whereas gravitational potentials are scalars. I don't know, however, what the blue and red triangles are supposed to mean. As the image description says, the lines in the picture are gravitational-potential contours and objects in free fall will trace out such a contour (or intermediate ones). --JorisvS (talk) 14:05, 24 March 2013 (UTC)
Hm... BigSteve (talk) 21:25, 24 March 2013 (UTC)

## Why 'Lagrangian point' instead of 'Lagrange point'?

I've studied physics for a long time, and I never heard the term 'Lagrangian point' until looking at this Wikipedia article. Everyone I know says 'Lagrange point'. 'Lagrangian' sounds stupid, because it means something else very different in classical mechanics. I get the feeling someone unfamiliar with this subject edited the title. I could be wrong, but.... John Baez (talk) 04:18, 11 March 2013 (UTC)

Lagrangian is simply the adjective (more famously used as a noun too of course), and some people prefer it. I think it could be good to pick one form and use it consistently throughout the article, while introducing both forms in the lede. Martijn Meijering (talk) 17:52, 11 March 2013 (UTC)

## L4 and L5 on File:Lagrange very massive.svg

The caption for this image in the The Lagrangian points section reads "In such a system, L3–L5 will appear to share the secondary's orbit, although in fact they are situated slightly outside it" (emphasis mine) with the "although" clause in agreement with the later math analysis. However, the image has L4 and L5 on the circle even when enlarged to 2000px, whereas L3 is indeed very slightly outside. Since I assume we're trying to illustrate the qualitative ideas here (rather than an exact solution for a specific situation), L4 and L5 should also be seen that way. DMacks (talk) 17:33, 2 April 2013 (UTC)

## More general than what?

A paragraph in "History and concepts" begins with "In the more general case of elliptical orbits, there are no longer stationary points ...."

More general than what? The previous paragraph discusses the general case and conic sections, and also constant pattern solutions. Perhaps the referent here is when one mass is negligible, where "positions" are discussed, not "points". Perhaps there was a mention of circular orbits which was deleted. In any case, it's not at all clear what changes the points into areas. This sentence needs to be revised to make sense.

Thanks Learjeff (talk) 19:12, 21 May 2013 (UTC)

## History and concepts

The section contains "The three collinear Lagrange points (L1, L2, L3) were discovered by Leonhard Euler a few years before Lagrange discovered the remaining two." That is doubly false.

The cited paper, E.327, deals with purely rectilinear motion and so cannot be said to have revealed the Lagrange Points. Euler did predict, in E.304, L1 and L2 - but not L3, though it is fairly obvious from what he wrote.

Lagrange did not discover any points, as he never wrote about setting one mass to near-zero, and moreover he wrote that he did not expect the situation to occur in nature (or, at least, involving the Earth). But the points are rightly named, since Lagrange's work provided the explanation of the objects later observed in the first decade of the twentieth century.

Before citing a paper, one should read and understand it in its original form, to ensure that it says what one wishes to assert that it says.

The article should give (for possible correction) the earliest use of "Lagrange" or "Lagrangian" in naming the points themselves.

94.30.84.71 (talk) 18:38, 25 January 2014 (UTC)

## Swap Mission Colour Coding?

Would it not be more natural to colour the successful missions green and the in-progress missions yellowy orange? Currently they are the other way around. 31.52.5.40 (talk) 14:58, 2 April 2014 (UTC)

## Graphs of effective potential

The article includes two three-d graphs of effective potential. The first is static, and (as far as I can tell) correct. The second differs from it in two important ways:

1. it is animated
2. it has the contour lines outside the orbit of the secondary body going the wrong way, so that instead of its main feature being the deep potential well of the primary, it is an unexplained ridge surrounding the primary.

I see little benefit from the animation. All it does is rotate. This is easy to envisage from a static diagram.

The second difference is seriously misleading. I suggest deleting the second diagram (and merging some of the content of its caption into that of the first), and will do so unless persuaded otherwise. Maproom (talk) 10:57, 25 December 2014 (UTC)

## Script for plotting the centrifugal/gravitational force

If anybody wishes to replot the data or just play with the problem, the following python script may be useful: https://gist.github.com/FilipDominec/4d5067717cecb4f699ad It shows the overall force in the Sun-Earth system. --FDominec (talk) 05:17, 25 January 2015 (UTC)

## Orbiting a Lagrangian point

I'd like to remind JorisvS of WP:BRD, which discourages reinsertion of edits that have been reverted without bringing it to talk first.

A body that orbits a Lagrange point still orbits the Sun as well (and the Earth too), so there is no inherent contradiction. How you describe it depends on your reference frame. Now clearly, a rotating reference frame is not an inertial one, but it's still an important conceptual tool. The point of the original wording is that in a rotating frame for the two dimensional problem, there are closed orbits that surround L4 and L5, which are equilibrium points in the rotating frame. Martijn Meijering (talk) 21:13, 20 May 2015 (UTC)

No, it doesn't. A Jupiter trojan does not orbit Jupiter. A body can only orbit a single other body (which can be composite, like in the case of a circumbinary planet, and that body can, in turn, orbit another body). Trojans, do not orbit the planet with which they are associated. It is a special case of a co-orbital configuration (which is, in turn, a special case of an orbital resonance), with gravitational interactions keeping the configuration intact. A trojan's trajectory with respect to its Lagrangian point is called libration, and can have all kinds of forms, as seen from a corotating reference frame, but are not ellipses (like orbits are). The description should be in a way that can be understood without making reference to a specific reference frame (unless describing the form of the trajectory in a specific reference frame). And Lagrangian points are equilibrium points regardless of reference frame: physics does not depend on one's reference frame. --JorisvS (talk) 08:45, 21 May 2015 (UTC)
Citation needed for the assertion that a body can only orbit a single other body. Clearly, if you are inside the sphere of influence of the moon, then the moon's gravity is the dominant force, so you might consider it more natural to say that you orbit the moon rather than the earth. Nevertheless, if you leave the moon out of the picture, it looks pretty darn close to an Earth orbit. On the other hand, if you are outside the moon's sphere of influence it is unreasonable to say that you are in lunar orbit, instead we say that the orbit is perturbed by the moon. If you are exactly at a stable Lagrange point, it is unreasonable to deny you are in a (perturbed) Earth orbit.
And for your information, the definition of an orbit doesn't require it to be elliptical (or even a conic section), it encompasses the phenomenon of libration too, hence the term libration *orbit*. Also, consider perturbed Earth orbits. The orbit of the ISS is an example of a lightly perturbed orbit, while low lunar orbits are strongly disturbed by the effect of mascons. It would be unreasonable to assert that these are anything else than orbits even though they are not elliptical.
Furthermore, a Lagrange point is most definitely *not* an equilibrium point for the equations of motion in an inertial frame, rather it corresponds to a circular (or elliptical) orbit. Once you switch to a rotating frame that rotates with the period of the circular orbit (i.e. roughly a month in the case of the Earth moon system), the periodic solution is transformed into an equilibrium solution. Both frames are used extensively in orbital dynamics.
If you have concerns about the wording, I'll be happy to work with you to accommodate them, but I don't appreciate your reinsertion of your edit over my objections, in violation of WP:BRD.Martijn Meijering (talk) 16:42, 21 May 2015 (UTC)
That's why I said that the body that is being orbited can itself orbit another body. And the trojans are actually far outside the SOI of the body they are associated with! And the orbit is actually a librating orbit: the orbit itself is a simple Keplerian that is gradually being perturbed by the larger body the trojan is associated with, causing gradual change (libration) of the orbit with the Lagrangian point as the average and a period much larger than their orbital periods. And if the equations of motion are integrated, the result has to be the same regardless of the reference frame one uses, so either they're equilibrium points, or they're not. Of course, in a non-rotating reference frame, the points themselves move along the orbit of larger body as it orbits the central one, and are hence time-dependent. I regularly employ an edit–revert–explain-revert–(rerevert–)discuss cycle for those cases where the edit is hardly "bold", but where the other editor has a hard time understanding the reason for the edit, because typically the explanation in the edit summary suffices (and hence then starting a new thread on the talk page is a waste of time and effort). --JorisvS (talk) 18:44, 21 May 2015 (UTC)
I never denied that a body that is being orbited can itself orbit another body, it is very obvious that it can. Also note that both 'librating orbit' and 'libration orbit' are correct, as a simple Google Scholar search will show. As for different frames having to give the same result: obviously they have to give *equivalent* results, but not *identical* results. A libration orbit is not an equilibrium solution in an inertial frame, rather it is a periodic solution. Consider L1 in the Earth moon system, seen from an inertial frame centered around the center of mass. Seen from above / below, the orbit of a spacecraft is almost circular, rotating with the same angular velocity as the moon, as opposed to the higher angular velocity that would be expected if it had moved only under the influence of the Earth at the same distance. In a corotating frame, the spacecraft appears to be stuck at a point, i.e. in an equilibrium orbit. See https://www.youtube.com/watch?v=cGm-nrgAIzM for an illustration. Bear in mind that in this video we are looking at Sun-Earth L1, not Earth-moon L1, but the principle is of course the same. Martijn Meijering (talk) 09:27, 23 May 2015 (UTC)
Perhaps you are confusing the term orbit, which is the actual trajectory a spacecraft is in with its osculating orbit, which is the instantaneous Keplerian approximation to that orbit. Martijn Meijering (talk) 09:37, 23 May 2015 (UTC)
Also check out this article for an explanation. http://www.wired.com/2012/12/does-the-moon-orbit-the-sun-or-the-earth/ Martijn Meijering (talk) 10:27, 23 May 2015 (UTC)
The video is misleading; it does not work like that. In fact, objects near the L1, L2, L3 even require periodic corrections to prevent them from drifting away. Objects near L4 and L5 do not, because a deviation from the point corrects itself. The Moon is unique among the known natural satellites in that it never appears to fall away from the Sun (its orbit is always convex), but it is still squarely within the SOI of Earth; trojans are not. Now, whether one uses the term "orbit" or not for trojans' association with Lagrangian points is, at the end of the day, just terminology. The fact remains that a trojan's trajectory/orbit/whatever with respect to its Lagrangian point is physically very different from that of a natural satellite, including the peculiar case of the Moon, and that's really my whole point. --JorisvS (talk) 13:56, 23 May 2015 (UTC)
I'm not sure what you're getting at. Periodic orbits of a *stable* Lagrange point (seen in a corotating frame) can be similar to the ones seen in a simple Keplerian system, though the details are different. No such orbits exist for the unstable ones, though there do exist three-dimensional halo orbits, which is worth pointing out. In any event, in the stable case it is entirely reasonable to describe the periodic orbits as orbiting the Lagrange point (in a rotating frame), and whether there is a physical object such as a spacecraft nearby is immaterial.
The video and the article I linked to correctly depict how an object can orbit one object (say the Earth) while at the same time orbiting another (say the sun), contrary to your earlier assertions. This is why I mentioned them. Yes, L4 and L5 are stable (in the idealised case, though not in the Earth moon system, due to the combined effect of the eccentricity of the moon's orbit and external perturbations), while the others are not. This is precisely why there are periodic solutions surrounding L4/L5 in a corotating frame, while no such solutions exist for the collinear points. This can be clearly seen in the plot of the effective potential (i.e. with the effect of the fictitious centrifugal pseudoforce due to the corotating frame accounted for) in the WP article.
Perhaps the article needs to do a better job of explaining the equivalence of equilibrium solutions in corotating frames and periodic solutions in inertial ones. Pointing out the relationship between general orbits and their instantaneous osculating Keplerian approximations would likely also be helpful. I'd also like an explanation that, depending on the reference frame, an object can reasonably be described as orbiting several different bodies.
Given that you've now made several demonstrably incorrect technical assertions it might perhaps be wise to be a bit more reluctant to reinsert changes you've made over the objections of others. Martijn Meijering (talk) 14:27, 23 May 2015 (UTC)
Whether or not there is a central body to orbit is hardly "immaterial", it makes the situation very different! If there is no object, there is no centripetal force acting on the body. "Perhaps the article needs to do a better job of explaining the equivalence of equilibrium solutions in corotating frames and periodic solutions in inertial ones." and "Periodic orbits of a *stable* Lagrange point (seen in a corotating frame) can be similar to the ones seen in a simple Keplerian system, though the details are different.": why don't you try me here!
"This is precisely why there are periodic solutions surrounding L4/L5 in a corotating frame, while no such solutions exist for the collinear points.". Collinear points have no movement associated with them, astronomical bodies and Lagrangian points move, which makes all the difference. The centrifugal force is a fictitious force to correct for the non-inertial reference frame. --JorisvS (talk) 16:39, 23 May 2015 (UTC)
"The video and the article I linked to correctly depict". The article, sure (I never said it didn't), but I can't fathom how you could say that about the video. Why don't you try me!
"Given that you've now made several demonstrably incorrect technical assertions": state them! (And don't talk about terminology, which really is immaterial, but the actual concepts and definitions involved.) --JorisvS (talk) 16:39, 23 May 2015 (UTC)

## Ln or Ln ?

I know of no other work in which L1 is written as L1, etc.; I suggest that the suffixing should go. 82.163.24.100 (talk) 13:45, 10 August 2009 (UTC)

I am relocating this section in an attempt to resurrect it. It seems like this fundamental question was never resolved, and the article remains inconsistent with respect to the format of the labels. I have no educated opinion on which is correct; however, I do suggest we should pick one format and stick to it throughout. It's a really simple point but it impacts the credibility of the article. Wdchk (talk) 04:39, 11 June 2015 (UTC)

I left a comment here: User_talk:Ohms_law#Lagrange_templates --S Philbrick(Talk) 14:40, 15 December 2015 (UTC)

## Problems with Equations for L3?

I think that for L3, the approximate value for r should be close to ~R, not R(7/12)(m2/m1). The exact equation given for solving for r diverges to infinity at r->R (the first term on the left: m1/(R-r)^2), so it's not going to work either. Synecdodave (talk) 12:30, 12 June 2015 (UTC)

Hey, SNAP! I just happen to have made a similar observation. Part of the trouble is that I cannot trust the wording of the article which says "r now indicates how much closer L3 is to the more massive object than the smaller object". If that means how much outside the Earth's orbit L3 is, OK, but if not, could someone who has worked it out please confirm that L3 is some 262500 metres outside (or inside) Earth's orbit? Or are we missing a cube root somewhere? Furthermore, I would have thought that if that were correct, then it should read something like "here r is how much further L3 is from the more massive object than the less massive object is from the more massive object; in other words how much greater the radius of the orbit of an object at L3 is than the radius of the orbit of the smaller mass"
(In my calculation I used masses of 2e30kg for Sol, 6e24 kg for Earth and 15e10 m for Earth's orbital radius, and assumed circular orbits.) That is such a tiny smidgin that since I was using approximate values and formulae, I could not confirm my values directly. That does not matter much but either the formulae given are wrong, or the wording is unclear, neither of which is tolerable in WP. If some kindly applied mathematician could confirm whether I have got it right, or if wrong, then how, then I would be grateful and would even would be happy to do some editing to make the article more intelligible. PLEASE? JonRichfield (talk) 16:06, 16 June 2015 (UTC)

## Footnotes four and five; same or different?

I noted that both footnote 4 and footnote 5 were dead links. I fixed footnote 4 with a link to the Internet archive source. I also looked at the source associated with footnote 5: Wayback link. At first glance, it appears to be the identical content as that in footnote 4 except for the header comment. Does anyone disagree if not we should simply replace footnote 5 with footnote 4. If it actually is different we should create a reference which picks up the Internet archive source.--S Philbrick(Talk) 14:43, 15 December 2015 (UTC)

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## Why aren't they simply called balance points?

The intentional confusion and obfuscation around Mathematics in English is tiresome and childish. — Preceding unsigned comment added by 2600:1700:EA10:E5D0:A0A7:570F:5F5E:8A (talk) 06:53, 12 February 2018 (UTC)

## L1/2 equations possibly wrong

In "mathematical details", in the starting equations, the last two terms come from omega*(R+r) and omega*(R-r) and then the R cancels one power of R in the denominator in the first of the two last terms - so why is the numerator only M1 and not M1+M2? Yes, later there is a discussion of the case when M2<<M1, but the starting equation are written as exact, so that's not the reason. 109.239.67.153 (talk) 20:15, 11 February 2020 (UTC)

## Removing clarification needed from the "L1 point" section

The unnecessary clarification was requested by Etymographer on 7 June 2020. The reason stated for clarification was an incorrect intuition that is already explained in the paragraph's text. The smaller large mass M2 orbits the larger large mass M1, and a relatively minute mass (say, a spacecraft) occupies L1, a point between M1 and M2. Thus, M1 pulls the craft one direction and M2 pulls it the opposite direction, but the pulls are unequal (M2's is considerably smaller). The net effect on the craft is that M1 cannot pull on the craft as strongly as it would in the absence of M2. Thus, the net gravitational force at L1 is less than that of M1 alone. The craft thus orbits M1 in a lower orbit which takes exactly as long as M2 takes to orbit M1. It is the reduced net gravity at point L1 that is responsible for changing the orbital mechanics at that point enough to reduce the orbit of the craft, which is exactly what makes L1 special. The sentences following the clarification notice say just the same thing in other words. The clarification needed flag is removed. Evensteven (talk) 07:29, 8 July 2020 (UTC)

Just a comment: this article takes fewer pains towards being understandable in favor of greater strains in trying to be technically accurate and mathematically complete. I have a Bachelor's degree in math, and like a good formula as well as any, but I see the use in the applications of things too, and this article could do a better job of communicating those things. For a more practical type of presentation, check out [1]. They manage to get a lot of basic ideas across without technical overkill. Wikipedia could learn from them. Unless, of course, the object here really is to train post-graduate students in orbital mechanics. Evensteven (talk) 22:41, 8 July 2020 (UTC)

## Mars terraforming

In the "future / proposed mission," I think mention should be made of the idea of stationing a magnet at the Mars-Sol L1 point to create an artificial magnetosphere on Mars. See also the Terraforming_of_Mars page.

Oops. I see that there is a mention of that in the article, but I still think it ought to be added to the table, as it has been "proposed."

Nsayer (talk) 22:39, 22 July 2020 (UTC)