Stability of L4 and L5... where to stick an extra mars?

[KerikBalm]

So I've read that L4 and L5 are stable, I've also read that the best hypothesis for the formation of the Moon involves a protoplanet Theia forming at Earth's L4 or L5 point, and then getting destability once its mass reached 10% of Earth's.

Something to do with Jupiter perterbations.... I'm not sure.

How possible would it be for 2 roughly evenly sized planets to exist in each others L4/L5 points? Would one need to be significantly smaller than another?

I ask because I'd like to add such a planet to the Kerbin system as a mod, but I'm not sure how realistic it would be.

I'm thinking either a duna sized planet at kerbin's L4/L5 (at the edge of this 10% stability "rule"), or a duna "sister" of a very similar mass.

I'm not super happy with either option... Kerbin's moon suggests a similar protoplanet already smacked into it... *two* duna sized planets formed at both L4 and L5, but only one smacked into it... seems rather... unlikely...

A duna-sister would also seem unlikely because of both the similarity in size, and that duna already has a moon, which may also suggest a lagrange impactor.

Where would be the most realistic place in our solar system to put an extra mars sized body? by extension/analogy... where in the kerbin system could we best accomodate another duna sized object....

in both cases, the distance from the sun should be such that polar water ice caps could exist, and liquid water could exist if the pressure was high enough?

Could we stick another mars sized object between trhe orbits of Earth and mars and have it be stable in the long term, perhaps with some sort of resonance?

How would I calculate such a resonance?

[Bill Phil]

L4 and L5 are only stable when the two bodies are below a certain threshold of relative mass. But that's between the primary and the secondary, in this case sun and planet. 

L4 and L5 aren't enormously stable. They're more stable than the other L-points, but they're still not very stable.

[pincushionman]

Like Bill said, none of the Lagrange Points work at all unless the planet is small compared to the star. Likewise, the third body has to be similarly small compared to the planet. That is, negligible.

[kerbiloid]

2 hours ago, pincushionman said:

Likewise, the third body has to be similarly small compared to the planet

Otherwise you call it "the main planet", while the original one - it's trojan companion...

P.S.
Probably, not the mass itself plays role, but the ratio of their masses, The closer are their masses - the greater would be their oscillations around their orbit, the more unstable this system.
While when one of co-planets has much greater mass (Jupiter vs Trojans, Earth vs clouds of trash), it forces the lesser companion to keep the discipline,
L4 and L5 are equal, the only difference is aesthetic.

 

 

[KerikBalm]

So then, the trojan planet must be smaller than the main planet... ie 10% as I originally speculated?

Duna is less than 10% the mass of kerbin, so if I add a Duna-mass planet to kerbin's pseudo L4/L5, it would plausibly be stable?

[Bill Phil]

35 minutes ago, KerikBalm said:

So then, the trojan planet must be smaller than the main planet... ie 10% as I originally speculated?

Duna is less than 10% the mass of kerbin, so if I add a Duna-mass planet to kerbin's pseudo L4/L5, it would plausibly be stable?

What percentage is Kerbin of Kerbol? That percentage or less.

[KerikBalm]

Well, Theia got a lot bigger than Earth's percentage of the sun's mass, so I don't think that's valid

"Two-dimensional computer models suggest that the stability of Theia's proposed trojan orbit would have been affected when its growing mass exceeded a threshold of approximately 10% of the Earth's mass (the mass of Mars).In this scenario, gravitational perturbations by planetesimals caused Theia to depart from its stable Lagrangian location, and subsequent interactions with proto-Earth led to a collision between the two bodies."

http://iopscience.iop.org/article/10.1086/427539/fulltext/

"The configuration is stableprovided the mass ofEarth and the massof the giant impactorare both below 0.0385times the mass ofthe Sun, which isthe case. But wenumerically demonstrate that gravitationalperturbations from other growingplanetesimals can eventually kickthe giant impactor intoa horseshoe orbit andfinally into an orbitthat is chaotically unstablein nature, allowing escapefrom L4."

"Thus, wepropose the following scenario:Debris remains at L4(as the Trojan asteroidsprove). From this debrisa giant impactor startsto grow like Earththrough accretion as describedabove. As the forminggiant impactor reaches asufficient mass (0.1m_E), itgradually moves away fromL4 through gravitational encounterswith other remaining planetesimalsand it randomly walksin peculiar velocity. Itgradually moves farther andfarther from L4 approximatelyon Earth's orbit ina horseshoe orbit at1 AU, until itacquires a peculiar velocityof approximately 180 ms^-1. The giant impactorthen undergoes breakout motionin which it performsa number of cyclesabout the Sun, repeatedlypassing near Earth. Ina time span roughlyon the order of100 years, it collideswith Earth on anear-parabolic orbit."

 

[Bill Phil]

6 hours ago, KerikBalm said:

Well, Theia got a lot bigger than Earth's percentage of the sun's mass, so I don't think that's valid

"Two-dimensional computer models suggest that the stability of Theia's proposed trojan orbit would have been affected when its growing mass exceeded a threshold of approximately 10% of the Earth's mass (the mass of Mars).In this scenario, gravitational perturbations by planetesimals caused Theia to depart from its stable Lagrangian location, and subsequent interactions with proto-Earth led to a collision between the two bodies."

http://iopscience.iop.org/article/10.1086/427539/fulltext/

"The configuration is stableprovided the mass ofEarth and the massof the giant impactorare both below 0.0385times the mass ofthe Sun, which isthe case. But wenumerically demonstrate that gravitationalperturbations from other growingplanetesimals can eventually kickthe giant impactor intoa horseshoe orbit andfinally into an orbitthat is chaotically unstablein nature, allowing escapefrom L4."

"Thus, wepropose the following scenario:Debris remains at L4(as the Trojan asteroidsprove). From this debrisa giant impactor startsto grow like Earththrough accretion as describedabove. As the forminggiant impactor reaches asufficient mass (0.1m_E), itgradually moves away fromL4 through gravitational encounterswith other remaining planetesimalsand it randomly walksin peculiar velocity. Itgradually moves farther andfarther from L4 approximatelyon Earth's orbit ina horseshoe orbit at1 AU, until itacquires a peculiar velocityof approximately 180 ms^-1. The giant impactorthen undergoes breakout motionin which it performsa number of cyclesabout the Sun, repeatedlypassing near Earth. Ina time span roughlyon the order of100 years, it collideswith Earth on anear-parabolic orbit."

 

It would potentially take tens of millions or hundreds of millions of years for an unstable system to do its thing. Uranus' inner moons, for example.

[KerikBalm]

indeed, stability is relative.

in Ksp, two moons of Jool are definitely unstable. Someone did N body simulations, and almost everything in Ksp was "stable" over the time period modeled... but Val was ejected very fast, and sometime later, so was Bop... so bop is comparatively stable, but in geological/astronomical timescales... very unstable.

I looked it up again... seems Theia hit Earth 20-100 million years after earth's formation... so that didn't last too long. I wonder what it would take to have something be "stable" on the order of 1-2 billion years.

Surely these things aren't directly proportional... how long could we expect a planetoid to last at L4/5 if it was 5% of the mass of Earth(M_e) rather than 10%? I suspect much longer than 2x the 20-100 million that Theia did (at 10% M_e).

This whole idea is fascinating to me... what would it have taken for Theia to capture instead of impact... imagine living in a double planet system (some argue the moon is big enough that we already do)... or imagine a "planet of damocles" at... say 3% M)e sitting at the L4/5... growing ever more unstable. What could be done to buy more time? some asteroid redirects? what would knowledge of such an inevitable apocalypse do to human civilization, given that it could be 100's of thousands, or millions of years before it happened (lets assume its hard to predict if it will still be orbiting L4/5 in 10 thousand year, or 10 million, or a billion years).

 

Anyway, for KSP, I'm sticking it at kerbin's L4, and its roughly 9% of Kerbin's mass... because I've got no other place to put it unless someone can tell me an orbit between Kerbin and Duna's orbit that would be stable over billions of years... Is there such an orbit (replace Kerbin and Duna with Earth and Mars) in our solar system?

[PB666]

A stable orbit for second mars is in orbit around Venus. Pretty much same fate but Venus might fair batter off if its most volatile gases had be retained. 

[KerikBalm]

"Pretty much same fate but Venus might fair batter off if its most volatile gases had be retained."

Huh? You're proposing it as a moon?

Better off how?

[PB666]

Orbit would be more stable around venus compared to any Earth la grange point. Venus is the only safe place in the inner solar system. 

[magnemoe]

13 hours ago, KerikBalm said:

indeed, stability is relative.

in Ksp, two moons of Jool are definitely unstable. Someone did N body simulations, and almost everything in Ksp was "stable" over the time period modeled... but Val was ejected very fast, and sometime later, so was Bop... so bop is comparatively stable, but in geological/astronomical timescales... very unstable.

I looked it up again... seems Theia hit Earth 20-100 million years after earth's formation... so that didn't last too long. I wonder what it would take to have something be "stable" on the order of 1-2 billion years.

Surely these things aren't directly proportional... how long could we expect a planetoid to last at L4/5 if it was 5% of the mass of Earth(M_e) rather than 10%? I suspect much longer than 2x the 20-100 million that Theia did (at 10% M_e).

This whole idea is fascinating to me... what would it have taken for Theia to capture instead of impact... imagine living in a double planet system (some argue the moon is big enough that we already do)... or imagine a "planet of damocles" at... say 3% M)e sitting at the L4/5... growing ever more unstable. What could be done to buy more time? some asteroid redirects? what would knowledge of such an inevitable apocalypse do to human civilization, given that it could be 100's of thousands, or millions of years before it happened (lets assume its hard to predict if it will still be orbiting L4/5 in 10 thousand year, or 10 million, or a billion years).

 

Anyway, for KSP, I'm sticking it at kerbin's L4, and its roughly 9% of Kerbin's mass... because I've got no other place to put it unless someone can tell me an orbit between Kerbin and Duna's orbit that would be stable over billions of years... Is there such an orbit (replace Kerbin and Duna with Earth and Mars) in our solar system?

Yes, Val is unstable as its too close to Tylo and Laythe. Bop is also to close to Tylo. 
It might well be more stability problems if you simulated millions of years. This would not be realistic either but would show stability problems. 
We are not able to predict Harley's comet for more than a few orbits with any accuracy, small changes will easy dramatically change Ap and therefore then it reenter. 

The early solar system was not stable, its belive it was 50 planets or dwarf planets in the inner solar system in the beginning, Theia was probably one of the last unstable ones, also likely something hit Mercury hard. 

[KerikBalm]

Yes, I know it wasn't stable early on, things took time to reach relative stability.

Realistically, how big of an object can I stick at kerbin L4 while having it be relatively stable on the order of a billion years or so?