Do trojan satellites follow migrating planets?

[harrylentil]

Some or all of the planets are thought to have migrated long ago under the gravitational influence of Jupiter. Would the trojan matter at their L4 and L5 points have followed during their migration to new orbits? In other words, while the L4 and L5 points are approaching or receding from the sun, does the matter collected there remain at these moving libration points?

 

[jim mcnamara]

Partial answer: there are Kordylewski dust clouds at Earth & moon L4, L5. I do not know what you can derive in terms of persistence times or "migration".

http://www.sci-news.com/astronomy/kordylewski-dust-clouds-06547.html

 

[Vanadium 50]

There are Trojans for Uranus and Neptune. If you are asking if these are the same ones they had 4BY ago, how would you tell? I am also unsure that even without migration, these orbits are stable for billions of years.

 

[harrylentil]

There are Trojans for Uranus and Neptune. If you are asking if these are the same ones they had 4BY ago, how would you tell? I am also unsure that even without migration, these orbits are stable for billions of years.

If the points are stable today they were stable yesterday and they will be stable tomorrow. Gravitational perturbation can evaporate their contents, and one cannot know how long any object has been in an L4 or L5 point. I don't want to know the history of particular objects. I am interested in whether or not the contents will remain in the L4 and L5 points in theory.

 

[Vanadium 50]

If the points are stable today they were stable yesterday and they will be stable tomorrow.

Not even close.

There are no permanently stable solutions. The best you can do is say "remains in this window for this long".

 

[Ophiolite]

Migration is a phenomenom that occurs during the formation of a planetary system. My supplementary question would be, do significant bodies even exist in Trojan locations at that point in system history? This 2009 paper, Origin and dynamical evolution of Neptune Trojans – I. Formation and planetary migration provides a partial answer.
From the abstract: ". . . we find that scenarios involving the slow migration of Neptune over a large distance (50 Myr to migrate from 18.1 au to its current location, using an exponential-folding time of τ= 10 Myr) provide the best match to the properties of the known Trojans. . . . Scenarios which avoid disruptive perturbation events between Uranus and Neptune fail to yield any significant excitation of pre-formed Trojans . . . . Conversely, scenarios with periods of strong Uranus–Neptune perturbation lead to the almost complete loss of such pre-formed objects."

 

[trurle]

Migration is a phenomenom that occurs during the formation of a planetary system. My supplementary question would be, do significant bodies even exist in Trojan locations at that point in system history? This 2009 paper, Origin and dynamical evolution of Neptune Trojans – I. Formation and planetary migration provides a partial answer.
From the abstract: ". . . we find that scenarios involving the slow migration of Neptune over a large distance (50 Myr to migrate from 18.1 au to its current location, using an exponential-folding time of τ= 10 Myr) provide the best match to the properties of the known Trojans. . . . Scenarios which avoid disruptive perturbation events between Uranus and Neptune fail to yield any significant excitation of pre-formed Trojans . . . . Conversely, scenarios with periods of strong Uranus–Neptune perturbation lead to the almost complete loss of such pre-formed objects."

Looking on extrasolar systems, we can expect early (soon after gas giants formation) Solar System to have 10-100 times more asteroid debris compared to modern Solar System. Of course, many bodies would exist early in Trojan Clouds too, although the density contrast with the background will be smaller compared to today. It would be largely bodies still in current Trojan population. From dynamical perspective, L4 and L5 are as stable as planetary radiation belts (because both are kept populated by non-potential forces - the Coriolis and Lorentz forces respectively) - always do exist escape trajectories from Trojan cloud with zero energy expenditure. Planetary migrations do alter "escape path" direction, exposing more previously "safe" bodies and resulting in faster depletion of Trojan clouds. On the other way, re-populating Trojan cloud do require expenditure of energy (typically coming from asteroids collision) in modern nearly-empty solar system, therefore we can see many of previously depleted L4 and L5 points are nearly empty now.

 

[alantheastronomer]

Since the Trojans exist at stable locations ( the L4 and L5 points) in the orbit of a sufficiently massive object, the disruption of that orbit would result in those points no longer being stable. With the orbit being disrupted, the debris would no longer be stabilized at L4 and L5, and with the gravitational potential of the massive body "relocating" elsewhere, would have no reason by any physical principle to "tag along" with the migrating body.

 

[DaveC426913]

Since the Trojans exist at stable locations ( the L4 and L5 points) in the orbit of a sufficiently massive object, the disruption of that orbit would result in those points no longer being stable.

This is a hasty conclusion.

There is no good reason to think that if the planet migrated at a sufficiently slow rate, the Trojans wouldn't simply adjust. The L-points are an area, meaning the Trojans could be slightly displaced and yet still be within the stable zone.

There mere fact that there are multiple (nay, innumerable) Trojans indicates that there is a large volume wherein they are able to maintain a relationship wrt the planet.

If Trojan A found itself inexplicably between Tb and Tc, it's not like it's suddenly out of the stable zone.

 

[alantheastronomer]

This is a hasty conclusion.

There is no good reason to think that if the planet migrated at a sufficiently slow rate, the Trojans wouldn't simply adjust. The L-points are an area, meaning the Trojans could be slightly displaced and yet still be within the stable zone.

There mere fact that there are multiple (nay, innumerable) Trojans indicates that there is a large volume wherein they are able to maintain a relationship wrt the planet.

If Trojan A found itself inexplicably between Tb and Tc, it's not like it's suddenly out of the stable zone.

Sure, the Trojans would be stable amongst themselves, but with the parent body migrated out of it's orbit, the debris would no longer be at a stable location in the orbit. With the parent body migrating, even ever so slowly, it's gravitational attraction to the debris is going to weaken and there is no principle in celestial mechanics that would cause them to migrate along with it. They have to maintain constant angular momentum, they can't just pick up on a whim and leave. Since the parent body is migrating due to perturbations, unless they experience the same perturbations, they're not going to undergo the same migration pattern.

 

[DaveC426913]

...there is no principle in celestial mechanics that would cause them to migrate along with it...

Of course there is. Gravity.

The L points are stable because they shepherd objects. It's a restoring force. A rock that drifts a little this way or that way gets shepherded back. Whether it's the rock that's drifted or the planet that's drifted is immaterial.

 

[alantheastronomer]

Of course there is. Gravity.

The L points are stable because they shepherd objects. It's a restoring force. A rock that drifts a little this way or that way gets shepherded back. Whether it's the rock that's drifted or the planet that's drifted is immaterial.

Not if the gravitational force has weakened! The L points are stable only as long as the orbit is stable. Once the parent body has migrated, they're no longer stable!

 

[DaveC426913]

Once the parent body has migrated, they're no longer stable!

It's not a binary switch.
The Trojans didn't go to sleep and wake up to discover Jupiter migrated in their absence. Jupiter's presence is always there, every second.

Look, Jupiter's L points are billions of cubic miles in volume (As witnessed by the estimated almost quarter million bodies larger than 2km in each).

The migration of Jupiter by 350 million miles or so over the course of - what? A few million years ... is maybe a thousand miles per year. That's dwarfed by the volume of the Lagrange points - which range a from half billion miles from Jupiter to a billion and a half miles along its orbit.

In a volume with dimensions on the order of millions to billions of miles,.Jupiter moving 1000 miles over its orbit is nothing.Here's a (very) rough diagram of Jupiter's Trojans.

See if you can draw Jupiter moving 1000 miles away over the course of its year.
(Hint: 1000 miles is on the order of a millionth of a pixel.
or: you would have to blow this picture up until it was about 63 miles in diameter before you would see a single pixel change over the course of a Jovian year.)

Bonus marks:
What is the difference in gravitational potential between something at 500,000,000 miles and something at 1,500,000,000 miles? (the extent of the stable zone)

What is the difference in gravitational potential between something at 1,000,000,000 miles and something at 1,000,001,000 miles? Now divide that into a Jovian year.

*caveat: these are very rough numbers quickly pulled from Wiki. They are meant to show orders of magnitude, not accurate calculations. That should be enough to make the point - and then some.

 

[Ophiolite]

That should be enough to make the point - and then some.

You had me at "This is a hasty conclusion."

 

[alantheastronomer]

Here's a (very) rough diagram of Jupiter's Trojans.

This diagram is of Jupiter's present day stable orbit, after it's finished migrating...But let's take a look at it and see if we notice something of significance. As you say -

the Lagrangian points - which range from … a billion and a half miles along it's orbit.

Yet in the diagram you can see that the Lagrangian points are very well defined geometrically as being sixty degrees ahead of and behind Jupiter in it's orbit, making equilateral triangles with Jupiter and the Sun. Only we see the asteroids spread out over a long distance in the orbit. Does this mean that the region of stability extends throughout this region? No! So then what is happening? It turns out that Jupiter, with it's Trojan asteroids, at a distance of 5 A.U. from the sun and an orbital period of ten years, pass by Saturn, at a distance of 10 A.U. and period of thirty years. Every time the leading edge of the Trojans approach Saturn, Saturn's gravity tugs on them causing them to advance a little more along their orbit, and every time the trailing edge passes by Saturn it's motion is retarded a little bit, and the cumulative effect of this is to cause the Trojans to spread out in their orbit. It's a type of tidal force called dynamical friction. By the way,

(As witnessed by the estimated almost quarter million bodies larger than 2 km. each).

But according to the same Wikipedia entry on Jupiter's Trojan asteroids that the diagram came from: quote - "According to the new estimates, the total number of Jupiter Trojans with a diameter larger than 2 km. is 6,300 and 3,400 in the L4 and L5 swarms, respectively."

The migration of Jupiter...is maybe a thousand miles per year.

See if you can draw Jupiter moving 1000 miles away over the course of it's year.

A Jovian year is ten Earth years, so over the course of a Jovian year, it should have moved ten thousand miles! By choosing an arbitrarily small timescale, you're minimizing the effect that is occurring, similar to Zeno's paradox. What you're really doing amounts to selection bias. According to the migration hypothesis, Jupiter started out forming at a distance of 350 million miles from the sun, then migrated inward to 150 million miles, then, under the influence of Saturn, out to it's present distance of 500 million miles. If the Trojans migrated with Jupiter from it's former position to it's present one that would also mean that they keep the same angular relationship in order to maintain stability at the L points.

The Trojans didn't go to sleep and wake up to discover Jupiter migrated in their absence.

Let's not anthropomorphize; their motion is constrained by the laws of gravity, and as Jupiter moves away, two things happen - one, the gravitational attraction weakens so their trajectory is influenced less and two, the geometric relationship between them is distorted so they're no longer in a region of gravitational stability. Also, there's a third effect, and that is as their orbit separates from Jupiter's their orbital periods are no longer in sync, which will reduce the stability even more. Finally, if I haven't convinced you, let me quote from the Wikipedia entry on the formation and evolution of the solar system referencing the migration theory: "...when Jupiter and Saturn were near their 1:2 resonance, the orbits of pre-existing Jupiter Trojans became unstable. This process was also reversible allowing...numerous objects...to enter this region and be captured. These new Trojans..." So evidently, the migration theory model finds that the original Trojan's orbits become unstable and new Trojans come in and replace them at the Jupiter's new orbit.

 

[Ophiolite]

TA Jovian year is ten Earth years, so over the course of a Jovian year, it should have moved ten thousand miles!

Dave very clearly stipulated that he was referring to a movement of 1,000 miles over the course of a Jovian year. His words:

"See if you can draw Jupiter moving 1000 miles away over the course of its year." (Emphasis added.)

That's "its year", not "a year", or "a terrestrial year", but "its year". i.e. a Jovian year. Consequently your subsequent remarks seeking to reject this point are invalid.

 

[alantheastronomer]

Dave very clearly stipulated that he was referring to a movement of 1,000 miles over the course of a Jovian year. His words:

"See if you can draw Jupiter moving 1000 miles away over the course of its year." (Emphasis added.)

That's "its year", not "a year", or "a terrestrial year", but "its year". i.e. a Jovian year. Consequently your subsequent remarks seeking to reject this point are invalid.

Yes he did! BUT before that he claimed quote - "The migration of Jupiter by 350 million miles or so over the course of - what? a few million years...is maybe a thousand miles per year." Now maybe you might want to argue that he meant a Jovian year, but if Jupiter was migrating then a Jovian year didn't yet exist... By changing to a Jovian year he's minimizing the impact Jupiter's migration has on it's gravitational influence over the Trojans. Also, according to the migration theory model, migration occurred over a period of about half a million years, not a few million; also tending to minimize the impact...the other points I made in my argument are still valid and still do hold!