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

[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.