I How can an asteroid get caught at a Lagrange point without a "brake"?

[Jonathan212]

Talking about the Jupiter Lagrange points at 60 degrees only. Hard to imagine a scenario where an asteroid comes from outside or inside the orbit of Jupiter and stops at a Lagrange point. That's like tossing a cone on a table and trying to make it end up standing on its nose. Or make the nose flat, it is still very hard. With asteroids it seems impossible without some sort of "brake" applied at the right time. Maybe the influence from a neighbouring planet passing by at the right time or something? How can it happen with a neighbouring planet and how without a neighbouring planet?

 

[DaveC426913]

Well, sure other planets or asteroids, but remember, it is Jupiter that is doing most of the work. Asteroids that cross Jupiter's path when it is in the right position will be slowed down or sped up, altering their orbits. The Trojan points collect bodies because Jupiter is accelerating/decelerating them till they rest there.

I think it very likely that a quick Google will get hits that show the specific mechanisms that cause it. Have you done any research?

 

[Jonathan212]

Give an example of an approach angle from outside the orbit that makes the asteroid end up at the Lagrange point. I think Jupiter would shoot it towards the inside or the outside, not towards the 60 degree Lagrange points.

 

[DaveC426913]

Give an example of an approach angle from outside the orbit that makes the asteroid end up at the Lagrange point. I think Jupiter would shoot it towards the inside or the outside, not towards the 60 degree Lagrange points.

Giving an example kind of sounds like a 'you' thing, not an 'us' thing.

What research have you done so far?

 

[Jonathan212]

How would you phrase it then? It's really what I want. Not to put you on any spot, just how can it happen.

 

[Jonathan212]

Maybe a gravitational simulator can be run in reverse order and we start with the asteroid at various points near a Lagrange point and with velocities near that point's velocity and see what happens, where the asteroid comes from. It has probably been oscillating around the point.

 

[DaveC426913]

BTW, this is a dynamic animation of how asteroids in the trojans behave.

The green asteroids - while remaining within the Lagrange area, still perform some acrobatics.

That doesn't really answer your question; it's just fascinating to watch.

 

[Runesmith]

Talking about the Jupiter Lagrange points at 60 degrees only. Hard to imagine a scenario where an asteroid comes from outside or inside the orbit of Jupiter and stops at a Lagrange point. That's like tossing a cone on a table and trying to make it end up standing on its nose. Or make the nose flat, it is still very hard. With asteroids it seems impossible without some sort of "brake" applied at the right time. Maybe the influence from a neighbouring planet passing by at the right time or something? How can it happen with a neighbouring planet and how without a neighbouring planet?

The asteroids at the Lagrange points don't "stop", nor do they orbit Jupiter. Those asteroids orbit the sun, and do so with the same periodicity as Jupiter. The trojan asteroids aren't actually "captured" (as in, they are flying somewhere, happen to cross the Lagrange point and suddenly turn into a trojan, kind of "capture").

What happens is that, some asteroids that orbit the sun with specific periodicities are pertubed by the gravitational pull of Jupiter. Over time, this pertubation adjusts their orbits. Some asteroids get fling out of the solar system altogether. Some get flung towards the sun, or end up in highly elliptical orbits. A (lucky) few get pertubed into the Lagrange points, where the interaction of gravitational effects of the Sun and Jupiter make that orbit very stable.

Such orbital pertubations happen over hundreds, if not thousands of orbits (and are only rarely successful in getting into the Lagrange points). The animation posted by DaveC426913 shows (in an exaggerated manner) how this pertubation works.

In summary, asteroids don't stop or suddenly change their trajectories. Therefore, the example that you ask for doesn't exist.

 

[Jonathan212]

Fascinating. So the Lagrange point opposite Jupiter cannot keep any asteroids. Maybe collisions with other asteroids are the only way they can get caught.

 

[Orodruin]

It should be fairly self-evident that an object in a non-stable orbit can enter a stable orbit without external perturbations such as interactions with objects other than the two massive ones creating the Lagrange points. In general, it is a well known fact that only some of the Lagrange points are stable in the sense of allowing stable orbits, i.e., the Lagrange points themselves are just equilibria, not necessarily stable equilibria. Only L4 and L5 have stable orbits around them, as shown in the animation of post #7.

 

[Jonathan212]

I think our issue is not whether a Lagrange point is stable, this is necessary but not sufficient for catching new asteroids from outside the orbit of Jupiter.

 

[Orodruin]

I think our issue is not whether a Lagrange point is stable, this is necessary but not sufficient for catching new asteroids from outside the orbit of Jupiter.

It was clearly an issue for you here:

Fascinating. So the Lagrange point opposite Jupiter cannot keep any asteroids. Maybe collisions with other asteroids are the only way they can get caught.

The reason L1-3 cannot keep any asteroids is precisely that they are unstable.

Apart from that, see the first sentence of #9. Being in a stable orbit around it is the definition of having been ”caught by” a Lagrange point.

 

[Jonathan212]

Read the topic title please. This is the issue. Anything can be mentioned if it is found interesting but that does not make it the issue in people's conversations. The definition of being caught is when you throw a ball and someone catches it. Being in a stable orbit is being kept in place after getting caught.

 

[Orodruin]

Read the topic title please. This is the issue. Anything can be mentioned if it is found interesting but that does not make it the issue in people's conversations. The definition of being caught is when you throw a ball and someone catches it. Being in a stable orbit is being kept in place after getting caught.

I read the topic. Did you read the answer? It does not seem so.

 

[Runesmith]

Read the topic title please. This is the issue. Anything can be mentioned if it is found interesting but that does not make it the issue in people's conversations. The definition of being caught is when you throw a ball and someone catches it. Being in a stable orbit is being kept in place after getting caught.

Please see my post (#8 in this thread). The definition of "caught" that you are using is wrong when applied to Lagrange asteroids. Therein lies the problem.

The asteroids don't get caught into Lagrange points. They get slowly nudged and herded into those positions over hundreds of orbits. This is similar to the mechanism that creates the gaps in Saturn's rings, for example.

 

[sophiecentaur]

We only observe Trojans and Lagrange points for a short time. I suspect that any mechanism that doesn't actually involve Energy loss would have a number of asteroids leaving these groupings, going off round the Sun and, later, returning. In the absence of any loss mechanism, the total number of asteroids could remain pretty constant; they just spend most of their lives in lagrange points and the remaining time in a bigger orbit.
Can we be sure that this is not actually the case? Are there significant losses to stabilise the situation? I know there are tidal effects on the larger moons but can that happen with small lumps of rock?
If we had the time, we could 'ring' some asteroids (as is done with migratory birds) and identify their 'migrations'.

 

[Jonathan212]

I thought all asteroids started their life somewhere else, not jupiter's orbit. So then we should be asking how did they get captured to jupiter's orbit before getting nudged and herded to the Lagrange points?

I propose that without collisions, and without neighbouring planets, capture to a Lagrange point is not possible and neither is convergence to it, for an asteroid coming from elsewhere. Any ideas/scenarios to refute this?

 

[Runesmith]

I thought all asteroids started their life somewhere else, not jupiter's orbit. So then we should be asking how did they get captured to jupiter's orbit before getting nudged and herded to the Lagrange points?

Majority of them start their life in the asteroid belt, between Mars and Jupiter. They don't stay there their whole life. The asteroids at the outer edge are pertubed by Jupiter, and those on the inner edge are pertubed by Mars. There are also collisions among the asteroids. These pertubations change the orbital mechanics of the asteroids. The lower energy orbits will move them closer to Jupiter and the higher energy orbits will move them closer to the Sun. In most cases, they will be forced into highly elliptical orbits, or get flung out somewhere.

I propose that without collisions, and without neighbouring planets, capture to a Lagrange point is not possible and neither is convergence to it, for an asteroid coming from elsewhere. Any ideas/scenarios to refute this?

As I said before, "capture" as you define it, doesn't happen - not with Lagrange points, not with anything else. So yes, what you posit is true. I don't think anyone who understands basic orbital mechanics would propose anything different.

 

[Orodruin]

I propose that without collisions, and without neighbouring planets, capture to a Lagrange point is not possible and neither is convergence to it, for an asteroid coming from elsewhere.

As I said, this much is obvious. Let me ask you a question: How do you think the Sun "captured" the planets? Regarding the Trojans, there are some theories regarding their origin already on the Wikipedia page that you took the image in the OP from.

 

[Jonathan212]

"capture" as you define it, doesn't happen

You forgot convergence, I also proposed convergence to jupiter's orbit or Lagrange points is impossible for objects coming from elsewhere etc.

If collisions are an option, capture as dictionaries define it, is possible. As in, a ball can stop after hitting another or just move a little.

As I said, this much is obvious.

You were not talking about the same thing (the impossibility of capture/convergence of objects that are NOT in the formations of asteroids near the orbit of jupiter).

How do you think the Sun "captured" the planets?

Smaller pieces were attracted to each other, hit each other and coalesced.

 

[Orodruin]

You were not talking about the same thing (the impossibility of capture/convergence of objects that are NOT in the formations of asteroids near the orbit of jupiter).

Yes I was, read it again. Where the objects are is irrelevant. Either an orbit is stable or it is not.

I also proposed convergence to jupiter's orbit or Lagrange points is impossible for objects coming from elsewhere etc.

Again, obvious.

 

[Runesmith]

You forgot convergence, I also proposed convergence to jupiter's orbit or Lagrange points is impossible for objects coming from elsewhere etc.

I would like to see your reasoning behind that, because you seem to be laboring under some misconception (assuming your term "coming from somewhere" means "originating in the asteroid belt").

Any asteroid crossing Jupiter's orbit while on a highly elliptical orbit won't be "captured" in Lagrange points. There is too much kinetic energy that needs to be released in such a case. If your "coming from elsewhere" refers to such objects - then, it's obvious.

If collisions are an option, capture as dictionaries define it, is possible. As in, a ball can stop after hitting another or just move a little.

Stop in relation to what? The entire solar system is in the Sun's gravity well, and objects near Jupiter are within its gravity well in addition. How can an object remain stationary in a gravity well?

Orbital mechanics are different from billiards. Unlike a billiard table, space has no friction. Capture, as you define it, is impossible. Nothing "stops" in space.

Smaller pieces were attracted to each other, hit each other and coalesced.

You didn't answer Orodruin's question. He asked how the Sun "captured" the planets. If you think about it, without being compelled to defend an untenable position, you will see the answer to the questions you have been asking.

 

[sophiecentaur]

Nothing "stops" in space.

That's a bit sweeping. If two objects are traveling towards each other with equal and opposite velocities in the Earth's frame (equal masses too) then they can coalesc and become stationary (Earth's frame) and be pulled directly to the Earth. An approximation to this could produce a stable Earth orbit. But this would need to involve loss of Energy in an inelastic interaction. A near miss through Jupiter's atmosphere could achieve something like that.

 

[Runesmith]

That's a bit sweeping. If two objects are traveling towards each other with equal and opposite velocities in the Earth's frame (equal masses too) then they can coalesc and become stationary (Earth's frame) and be pulled directly to the Earth. An approximation to this could produce a stable Earth orbit. But this would need to involve loss of Energy in an inelastic interaction. A near miss through Jupiter's atmosphere could achieve something like that.

True. But the combined object still doesn't stand still (ie. "stop"). This was the premise of the OP, who assumes that an asteroid actually stops at the Lagrange points and gets pulled/pushed along by Jupiter. It doesn't. If it did, it will crash into Jupiter and burn up. The asteroids at Jupiter's Lagrange points orbit the Sun - which the OP doesn't seem to understand.

 

[Orodruin]

The asteroids at Jupiter's Lagrange points orbit the Sun - which the OP doesn't seem to understand.

This is a coordinate system dependent statement. In the co-rotating Sun-Jupiter system they orbit the Lagrange point (assuming a stable orbit and no external perturbations), much like the moon orbits the Earth. This is the point of L4 and L5 being called stable Lagrange points. Saying that they orbit the Sun is like saying that the Moon orbits the Sun, which is true because it orbits the Earth which in turn orbits the Sun - just like the Lagrange points do.

 

[DaveC426913]

The asteroids at Jupiter's Lagrange points orbit the Sun - which the OP doesn't seem to understand.

The OP seems to have a better grasp of orbital mechanics than that.
I'd have said the he meant stop wrt Jupiter / Jupiter's orbit.

He is, after all, referring to bodies that cross Jupiter's orbit.

 

[Runesmith]

In the co-rotating Sun-Jupiter system they orbit the Lagrange point (assuming a stable orbit and no external perturbations), much like the moon orbits the Earth.

That's incorrect. Objects at a Lagrange point orbit the larger body - in this case, the Sun. They remain in the same orbit as the smaller body (in this case, Jupiter), orbiting the larger body with the same periodicity as the smaller one does.

 

[Orodruin]

That's incorrect. Objects at a Lagrange point orbit the larger body - in this case, the Sun. They remain in the same orbit as the smaller body (in this case, Jupiter), orbiting the larger body with the same periodicity as the smaller one does.

Objects at the Lagrange point, yes. Not objects orbiting the Lagrange point. Objects actually in the Lagrange point (with zero velocity) are stationary in the corotating frame and therefore obviously orbiting the larger body (or, more accurately, the barycenter of the two bodies) with the same period as the period of the two-body system. For objects that are not situated with zero velocity at the stable Lagrange points, but have stable orbits, they do orbit the Lagrange points in a fashion very similar to how the Moon orbits the Earth in the corotating frame. This is also evident from the animation shown in #7 (which is depicting the corotating frame). Saying anything else is misleading and wrong. Of course, in the inertial frame, an additional rotational motion about the barycenter of the two-body system is added.

It is also misleading to say that they have the same orbit as the smaller body if they are in the Lagrange point. This only holds in the limit where the ratio of the masses goes to zero.

 

[HopDavid]

Talking about the Jupiter Lagrange points at 60 degrees only. Hard to imagine a scenario where an asteroid comes from outside or inside the orbit of Jupiter and stops at a Lagrange point. That's like tossing a cone on a table and trying to make it end up standing on its nose. Or make the nose flat, it is still very hard. With asteroids it seems impossible without some sort of "brake" applied at the right time. Maybe the influence from a neighbouring planet passing by at the right time or something? How can it happen with a neighbouring planet and how without a neighbouring planet?

In an ideal 3 body system an object would not be able to drift in from the outside and end up at L4 and L5. These regions are actually called the Forbidden Realm. See this online text on 3 body systems. The forbidden realm is discussed on page 37 among other places.

However a Sun-Jupiter-Trojan isn't an ideal 3 body system. Outside perturbations can gradually dislodge a Trojan from the L-4 or L-5 location. And since orbits are time reversible, perturbations can gradually embed an object. I think of it as the solar system's Sargasso sea.

 

[James Demers]

This conversation seems to be descending into an argument about semantics: what's an "orbit" and what does it mean to be "captured"?

The Trojan asteroids appear to orbit their Lagrange points, but it is the sun (strictly, the Sun-Jupiter barycenter) that they actually orbit - that's where the gravitational force controlling their orbits is centered. The vast majority (those not smack on the Lagrange point) accelerate, decelerate, and move inward and outward in a metastable orbit that, in Jupiter's refrence frame, appears to be an orbit "around" the Lagrange point ... an illusion caused by their oscillating orbital velocity. These orbits may last for millions of years, but they're not truly stable.

The animation is informative because you can see how a "green" asteroid might, with slight perturbation, end up on a "red" trajectory and depart the Lagrange point. The reverse process, of course, is the answer the OP's question. An excellent overview here: https://www.lpi.usra.edu/books/AsteroidsIII/pdf/3007.pdf

Some orbits get perturbed to the point that the asteroid is ejected completely from the Sun-Jupiter system; most end up in solar orbits out in the "comet zone". It's likely that very few of the primoridal Trojan asteroids (those present at the formation of Jupiter) remain in their original orbits today. https://arxiv.org/pdf/1811.00352

 

[Orodruin]

The Trojan asteroids appear to orbit their Lagrange points, but it is the sun (strictly, the Sun-Jupiter barycenter) that they actually orbit - that's where the gravitational force controlling their orbits is centered.

Sorry, but this is misleading. The ”actually” is a matter of choice of coordinate system and as such not actually an ”actually”. You might as well say that they actually orbit the galactic center or that the Moon orbits the Sun. Also, the entire issue arises because the Jupiter-Sun system is not idealisable to a gravitational field due to the barycenter. What is being orbited is a matter of choice of reference frame - there is no ”actual” here because no reference frame is any more or less valid than any other as long as all appropriate effects are taken into account. In the co-rotating frame (not ”Jupiter’s frame”), L4 and L5 are stable Lagrange points that objects can orbit. Barring external perturbations (to which we may choose to include the ellipticity of the orbit), the stable orbits around these Lagrange points are stable.

 

[Phil Lawless]

I thought all asteroids started their life somewhere else, not jupiter's orbit. So then we should be asking how did they get captured to jupiter's orbit before getting nudged and herded to the Lagrange points?

I propose that without collisions, and without neighbouring planets, capture to a Lagrange point is not possible and neither is convergence to it, for an asteroid coming from elsewhere. Any ideas/scenarios to refute this?

Gravity is a central force and as such cannot change a satellie's angular momentum around the central body. For a simple system, some type of dissipation is necessary for an asteroid to be captured by a planet. Collisons with other asteriods come to mind. Tidal forces might break an asteroid, leaving one part with less energy and the other part with more.

The Lagrange points are points where the gravitational attraction of the sun is equal in magnitude to the gravitational attract of the planet. L1, 2, and 3 have the solar and planetry forces in roughly the same direction, There is no stabilty for an orbit to exist. At L4 and L5, the solar and planetary forces oppose generally and there is a local minimum of gravitational potential energy that will confine a small body with the kinetic energy corresponding to the orbital velocity of the planet.

The depth of the gravitational minimum depends on the masses of the sun and the planet. Small planets have small minima. Jupiter's is huge. Therefore, the region of stability is much larger for Jupiter than for any other planet. This allows a wide range of asteroid kinetic energies to remain stable once they are caught.

This discussion does not include the gravitational interactions with all the nearby asteroids. Those are generally weak, but they are numerous and over long periods of time

 

[Orodruin]

The Lagrange points are points where the gravitational attraction of the sun is equal in magnitude to the gravitational attract of the planet.

This in incorrect. The Lagrange points are the stationary points in the corotating frame. Not anything else.

L1, 2, and 3 have the solar and planetry forces in roughly the same direction, There is no stabilty for an orbit to exist. At L4 and L5, the solar and planetary forces oppose generally and there is a local minimum of gravitational potential energy that will confine a small body with the kinetic energy corresponding to the orbital velocity of the planet.

This is certainly not accurate. L4 and L5 are actually saddle points of the effective potential in the corotating frame (which also includes a centrifugal term). However, dynamics that govern the stability of the Lagrange points also include other effects of an accelerated frame, such as the Coriolis force. This is covered, for example, in the excellent free lecture noted by Tong.

 

[James Demers]

The ”actually” is a matter of choice of coordinate system and as such not actually an ”actually”.

Like I said - an argument over semantics. I choose to go with the gravitational center of the orbit as the thing being "actually orbited", precisely because it's independent of reference frame.

 

[Orodruin]

Like I said - an argument over semantics. I choose to go with the gravitational center of the orbit as the thing being "actually orbited", precisely because it's independent of reference frame.

Sorry, but this is self-inconsistent. Your "actually orbited" implies a preference for reference frame.

 

[James Demers]

Your "actually orbited" implies a preference for reference frame.

Semantics, again: reading what you want into "actually".
(In the reference frame of the asteroid, are you "actually" not orbiting anything at all?)

 

[Orodruin]

Semantics, again: reading what you want into "actually".
(In the reference frame of the asteroid, are you "actually" not orbiting anything at all?)

Which is why I am not using the word "actually" - you, on the other hand, are using it.

 

[DaveC426913]

Semantics, again: reading what you want into "actually".

Yes. We are wasting electrons clarifying (for all potential readers) the intended meaning.
Just specify a reference frame, and we can move on.

 

[Mister Perfessor]

The Tumbling Bananas do not stop for lunch:

 

[Runesmith]

Objects at the Lagrange point, yes. Not objects orbiting the Lagrange point.

As far as I know, objects don't "orbit" Lagrange points. They orbit the Sun, at the Lagrange points.

Lagrange point asteroids don't orbit the Lagrange points - they oscillate in their orbits around the Sun as they are pertubed by the other planets, and due to the slight elliptical nature of Jupiter's orbit, which alters the point of balance at different parts of Jupiter's orbit. When the frame of reference is made stationary, that gives the illusion of an orbit. There is no gravitational well at Lagrange point to orbit.

It is also misleading to say that they have the same orbit as the smaller body if they are in the Lagrange point. This only holds in the limit where the ratio of the masses goes to zero.

Agreed. A typical asteroid, however is negligible in mass compared to Jupiter or the sun. For all practical purposes we can assume the ratio is zero. We can turn a simple three body problem into a 10^27 body problem, and discover that a Lagrange point Trojan has an orbit that is 3mm closer to the Sun than Jupiter's, but for practical purposes, there's no reason to do so, is there?

As far as the answer to the original question goes, the answer still remains the same - asteroids don't have to have "brakes" to get nudged into Lagrange points.

 

[Orodruin]

As far as I know, objects don't "orbit" Lagrange points. They orbit the Sun, at the Lagrange points.

Again, this is a matter of your reference frame. Would you also say that the Moon does not orbit the Earth? It is the same distinction.

Lagrange point asteroids don't orbit the Lagrange points - they oscillate in their orbits around the Sun as they are pertubed by the other planets

This is incorrect. The perturbations from other planets is not the main reason for the motion of the asteroids around the Lagrange points. This motion would be present also for test particles in the ideal two-body system. Again, the stability of orbits around the Lagrange points is discussed in Tong’s lecture notes.

There is no gravitational well at Lagrange point to orbit.

The Lagrange points are, by definition, the stationary points of the effective potential in the comoving frame (which includes the gravitational potentials and the centrifugal potential). L4 and L5 are saddle points of this potential. However, this is insufficient to determine the stability of orbits around the Lagrange points as we are dealing with a rotating frame where there are also other effects (Coriolis). You therefore need to do the full analysis (as presented in Tong) to determine the Lagrange point stability.

Agreed. A typical asteroid, however is negligible in mass compared to Jupiter or the sun. For all practical purposes we can assume the ratio is zero. We can turn a simple three body problem into a 10^27 body problem, and discover that a Lagrange point Trojan has an orbit that is 3mm closer to the Sun than Jupiter's, but for practical purposes, there's no reason to do so, is there?

I am talking about the Jupiter-Sun mass ratio. The Lagrange points L4 and L5 are on the Jupiter orbit only in the limit if Jupiter’s mass going to zero.

 

[DaveC426913]

testing...

 

[Ophiolite]

Hmm. There seems to be a general agreement that by some means, chance variation in orbits generated by various gravitational interactions, some asteroids are "captured"into the Trojan points. There is also the suggestion that, again through chance variation etc. some of them may periodically leave these points.

I thought it might be worthwhile looking at what's in the recent literature. Nesvorny et al (2013) Capture of Trojans by Jumping Jupiter comment that ". . we tested a possibility that the Trojans were captured during the early dynamical instability among the outer planets (aka the Nice model), when the semimajor axis of Jupiter was changing as a result of scattering encounters with an ice giant. The capture occurs in this model when Jupiter’s orbit and its Lagrange points become radially displaced in a scattering event and fall into a region populated by planetesimals (that previously evolved from their natal transplanetary disk to ∼5 AU during the instability). Our numerical simulations of the new capture model, hereafter jump capture, satisfactorily reproduce the orbital distribution of the Trojans and their total mass."

In an earlier paper Morbidelli et al (2005) Chaotic capture of Jupiter’s Trojan asteroids in the early Solar System noted that "the Trojans could have formed in more distant regions and been subsequently captured into co-orbital motion with Jupiter during the time when the giant planets migrated by removing neighbouring planetesimals9–12. The capture was possible during a short period of time, just after Jupiter and Saturn crossed their mutual 1:2 resonance, when the dynamics of the Trojan region were completely chaotic. "

These two papers seem to reflect a consensus view that the Trojans originated early in the solar system and that giant planet migration was implicated in their capture. Caveat: I have not done an exhaustive literature search by any means and the two quoted papers have Morbidelli as a co-author, so this may be reflecting a particular take on the matter and other views may exist.

 

[Bandersnatch]

Again, this is a matter of your reference frame. Would you also say that the Moon does not orbit the Earth? It is the same distinction.

I'm thinking it isn't the same distinction. With the Earth-Moon system, one can model the motion of the two bodies around their barycentre using Newton's laws. I don't see how one could do the same with a system of an asteroid-Lagrange point, since the latter is not a physical entity and all mass is in the asteroid. In the latter case, the asteroid is orbiting the Sun-Earth barycentre.
The oscillating motion of the Trojans around the (nomen est omen) libration points would be in the same category as Lunar libration.

 

[Orodruin]

I'm thinking it isn't the same distinction. With the Earth-Moon system, one can model the motion of the two bodies around their barycentre using Newton's laws. I don't see how one could do the same with a system of an asteroid-Lagrange point, since the latter is not a physical entity and all mass is in the asteroid. In the latter case, the asteroid is orbiting the Sun-Earth barycentre.

I am not saying it is a question of looking at a barycenter. I am saying it is a question of having a periodic orbit around a particular point in the corotating system. This is also using Newton's laws, albeit in a non-inertial frame. Are you saying that the Moon is not orbiting the Earth? The case is exactly the same. In the corotating frame of the Earth and the Sun, the Moon orbits the Earth-Moon barycenter in a periodic fashion. In the corotating Sun-Jupiter system, the Trojans orbit the Lagrange points L4 and L5. Everything is described by Newton's laws of motion. The Lagrange point is just as much of a "physical entity" as the barycenter is, both are particular points in space that can be computed through a mathematical abstraction.