Objects as L3 and L4 at once - possible?

[Shraa]

Specifically, Earth-sized objects. I know that another Earth-sized object could exist at either L3 or L4 in the Sun-Earth system. However, could Earth-sized objects exist at L4 and L5 at the same time? I'm unsure because, at that configuration, the other two objects (Earth4 and Earth5) would be at 120 degrees to one another - which, in their own systems, would not be Lagrangian points. Could they exist at the same time?

If that solution is impossible, would having six Earths, each at 60 degree intervals, remain stable?

And finally, would the presence of a single moon (same as ours) orbiting Earth4 and Earth5 affect the balance?

[URL]http://upload.wikimedia.org/wikipedia/commons/b/b8/Lagrange_very_massive.svg[/URL]

EDIT: Sorry, typo in the title - it should read 'objects at L3 and L4 at once'.

DOUBLE EDIT: Referring to wrong Langrangian points in title. Really not doing good tonight.

 

[IsometricPion]

However, could Earth-sized objects exist at L4 and L5 at the same time? I'm unsure because, at that configuration, the other two objects (Earth4 and Earth5) would be at 120 degrees to one another - which, in their own systems, would not be Lagrangian points.

Your intuition is correct, since there are bodies that are not at the Lagrange points as they move they will tend to cause the other bodies to move out of the configuration in which they were initially "placed" (outside of these special configurations, orbits do not usually exactly repeat cf. n-body http://en.wikipedia.org/wiki/N-body_problem" on Scholarpedia interesting.

And finally, would the presence of a single moon (same as ours) orbiting Earth4 and Earth5 affect the balance?

If there were no other objects in the universe, yes. In the world as it exists, no, though I'm sure the moon-like object would remain nearby for a while.

 

[tony873004]

Specifically, Earth-sized objects. I know that another Earth-sized object could exist at either L3 or L4 in the Sun-Earth system. However, could Earth-sized objects exist at L4 and L5 at the same time? I'm unsure because, at that configuration, the other two objects (Earth4 and Earth5) would be at 120 degrees to one another - which, in their own systems, would not be Lagrangian points. Could they exist at the same time?

If that solution is impossible, would having six Earths, each at 60 degree intervals, remain stable?

And finally, would the presence of a single moon (same as ours) orbiting Earth4 and Earth5 affect the balance?

[URL]http://upload.wikimedia.org/wikipedia/commons/b/b8/Lagrange_very_massive.svg[/URL]

EDIT: Sorry, typo in the title - it should read 'objects at L3 and L4 at once'.

DOUBLE EDIT: Referring to wrong Langrangian points in title. Really not doing good tonight.

Yes, yes, and yes.
You can have triple planets, provided that the ratio of mass of (planet1 + planet2 + planet3) / sun < ~0.04

Yes, 6 planets spaced 60 degrees apart would be stable.

Yes, they can all have a moon.

 

[D H]

Yes, yes, and yes.
You can have triple planets, provided that the ratio of mass of (planet1 + planet2 + planet3) / sun < ~0.04

Yes, 6 planets spaced 60 degrees apart would be stable.

Yes, they can all have a moon.

No, no, and no!

That 4% figure pertains only to a central object such as a star, a smaller object such as a planet whose mass is less than about 4% of the mass of the central object, and a third object of negligible mass. This is the restricted circular three body problem. Make that third object have a non-negligible mass and you have entered the realm of the unrestricted three body problem. In particular, that 4% figure is now invalid.

With three planets you now have entered the realm of the N body problem. A system of six planets of equal mass and equally spaced around the central Sun is kinda stable, kinda not. Chop off half of that weirdly stable system and you get an unstable system. Add in Moons and you get a chaotic mess. What is stable is a system of N+N planets orbiting in a common orbit (but different phases) about a massive star, with planets alternating between heavy and light along the common orbit.

 

[Shraa]

D_H, could you clarify what 'N+N' is? And thanks for the answers so far. :)

 

[tony873004]

I guess I need to make one clarification. There is a difference between being stable and being a solution to the 3-body problem. Because nothing in the real universe is unperturbed, there can't be any real systems that represent a solution to the 3-body problem. A real system with a star, a planet, and a particle of insignificant mass 60 degrees from the planet would be stable. But perturbations would cause the particle to orbit the L4 or L5 point, rather than sit exactly on the spot as would be required for a solution. So stable doesn't mean that they'll sit there exactly 60 degrees apart. Rather, they'll oscillate around this configuration, without colliding or ejecting each other out of the system.

I'm assuming that the OP is wondering about real systems. There are many trojan configurations that are stable: after long periods of time, they're still in a trojan configuration.

No, no, and no!
That 4% figure pertains only to a central object such as a star, a smaller object such as a planet whose mass is less than about 4% of the mass of the central object, and a third object of negligible mass...

From Eric Ford's paper, Using Transit Timing Observations to Search for Trojans of Transiting Extrasolar Planets
http://www.citebase.org/fulltext?format=application%2Fpdf&identifier=oai%3AarXiv.org%3A0705.0356
"While the mass ratios of the Trojan systems in our solar system are rather extreme, it is possible that extrasolar planets may have much more massive Trojans. Indeed, theorists have already outlined several possible mechanisms to form Trojans with mass ratios potentially including unity." (unity meaning the trojan masses are equal)

According to Christian Marchal in chapter 8 of his book "Simple Solutions of the Three Body Problem", a very massive body, such as the Sun, and two much less massive bodies, such as Earth-mass bodies, is a simple solution to the three-body problem, provided that the ratio of their combined mass and the mass of the primary is
0 <= R < 0.02860...or 0.02860... < R <= 0.03852...
I don't have access to this book at the moment. I'm quoting someone else quoting the book. Interestingly, this leaves an unstable zone at 0.02860 sandwiched inbetween two stable zones. I tried simulating this, using the Sun, two Earth-mass objects separated by 60 degrees and found the following:
The first column is mass (planet1 + planet2) / mass Sun, the second is how many years this configuration remained stable. I stopped at 10000 years.

0.05	    21
0.041	   430
0.04	10000+
0.0388	10000+
0.0386	10000+
0.0385	10000+
0.031	10000+
0.03	   800
0.299	   658
0.0295	   400
0.0294	   450
0.0293	   400
0.029	   400
0.0289	   440
0.0288	   490
0.0287	   536
0.0286	   990
0.0285	10000+
0.028	10000+
0.02	10000+

This simulation supports Marchal's claim that three bodies can have mass and still remain stable, and that a small zone of instability exists within the otherwise stable zone.

As for three bodies, spaced 60 degrees apart, with the two outer bodies being 120 apart, we have one example of this in our own solar system. Saturn's moons Tethys, Telesto & Calypso are in this configuration, although Telesto and Calypso oscillate around the +- 60 degree points, rather than sit directly on Tethys' L4 & L5 points. Here's a page I made showing a simulation of Tethys, Telesto & Calypso
http://www.orbitsimulator.com/gravity/articles/tethys.html


I also tried simulating 6 bodies, spaced equally around an orbit. They are all Earth-mass at 1 AU. I gave them all a single moon at 0.0123 Earth masses with an SMA of 384000 km. I also added Jupiter as a perturber at 5.2 AU. These 6 bodies didn't keep their perfect spacing. Rather, they act like 6 cars on a one-lane circular track. Sometimes they bunch up, and sometimes they spread out, but they keep their original order. No planet ever overtakes or falls behind another planet. No planets are ejected from the system, and no planets collide. This sim was run for 23000 years.

Here's a paper on a 3-body solution where all three bodies have mass:
http://arxiv.org/PS_cache/math/pdf/0011/0011268v1.pdf
One of the users on my website made simulations based upon this author's work. They can be found here:
http://www.orbitsimulator.com/cgi-bin/yabb/YaBB.pl?num=1181200864/0 and
http://www.orbitsimulator.com/cgi-bin/yabb/YaBB.pl?num=1250625868

 

[Shraa]

So it is possible for both posited systems to work?

I'm grateful for the active investigation, by the way!

 

[FtlIsAwesome]

I was under the impression that objects at the Lagrange points must be of insignificant mass.
tony873004, are you saying that they can be massive? Interesting.