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evildrganymede
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### Solar System dynamics discussion thread

These are really great articles, thanks! Question for you - You use "particles" to test orbits in the L4 and L5 points, do they have any mass? If not have we got any way to put anything with mass there to see what happens? There's a discussion about having planets in 'trojan orbits' (at the L4/L5) over on the Elite Dangerous forums and I'm wondering whether the mass of the "particle" would affect the stability of the L4/L5 orbits.

Watsisname
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### Solar System dynamics discussion thread

evildrganymede, good question!  In all my calculations and figures I treat the particle as massless (a good approximation if it is say, a spacecraft or asteroid as a Trojan of a planetary body).

If it is massless, then stability about the L4 and L5 points occurs if the primary mass is at least 24.96 times the mass of the secondary.  But what if the particle is not massless?  In generality, the condition for stability of orbits about L4 and L5 is given by

where you may call m1 the star's mass, m2 the mass of the big planet, and m3 the smaller (Trojan) body.

With this formula we can quickly confirm that if we set m3=0, then m1 must be at least 24.96 times m2.  But what if we let m1 be 1 solar mass and m2 be the mass of Jupiter?  How big can m3 be and still be stable as a Trojan body?

Even if both planets had the mass of Jupiter, then they could be stable in one another's orbital space, as Trojans of one another.  But they could not both have a mass of over 20 Jupiters, or else the stability would fail since together they'd be too massive relative to the star, and the chaos of the 3-body problem would appear.

In other words, for almost any reasonable choice of masses of star and planet, another planet could exist in a stable Trojan orbit.

Why then do we not find many Trojan planets in nature?

The analysis here has been for an idealized situation, where we considered only two masses (plus a co-orbital body) and for them to be in circular orbits.  Eccentric orbits will reduce the stability, as will the presence of additional planets in the system.  Many of the Trojan asteroids of Jupiter are not in stable orbits for example, and those asteroid groups have slowly been eroded over time.  Others may be newly caught there, and remain only temporarily.

But maybe an even bigger reason is that planetary systems are very dynamic, especially in their youth.  The planets form by accreting dust and gas out of the disk, and then their orbits migrate as they scatter nearby planetessimals away.  The planets also influence one another, potentially even leading them to swap places or get ejected from the system.  So even if a Trojan planet were to exist in a system's early history, it probably wouldn't remain there for very long while all this chaos is happening.

evildrganymede
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### Solar System dynamics discussion thread

Thanks - so 40 MJ... that's actually very close to 1/24.96 of a solar mass (that would be 41.96 Jupiter masses exactly)? I'm assuming that's not a coincidence? So are you saying that generally speaking *the total mass of everything in the orbit* (the main planet and any trojans) has to be at least 1/24.96 of the star's mass for the configuration to be stable? So Jupiter could say have an earth-mass planet in its L4 and L5 points and that would theoretically work? Heck, it could have a saturn-mass planet in both those points and that'd still be stable?

Watsisname
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### Solar System dynamics discussion thread

evildrganymede wrote:
Source of the post 40 MJ... that's actually very close to 1/24.96 of a solar mass (that would be 41.96 Jupiter masses exactly)? I'm assuming that's not a coincidence?

Good observation!  It is very close and not by coincidence.  It's not quite correct that the rule is "total mass of everything in the orbit must be less than 1/24.96 of the star, but it's a decent rule of thumb for a wide range of masses.  For example if we set m2 = m3, and m1 = 1, then m1/(m2+m3) must be 25.23.  Slightly more than 24.96.  However, this is splitting very fine hairs, as anything close to these boundary conditions for stability will only barely be stable anyway.

evildrganymede wrote:
Source of the post So Jupiter could say have an earth-mass planet in its L4 and L5 points and that would theoretically work?

This I do not know.  It would be stable for Jupiter and an Earth in L4 or L5, but I'm not sure about an Earth in both simultaneously.  That would turn this from the unrestricted 3-body problem to an unrestricted 4-body problem, for which I haven't seen the criteria for stability of equilibrium points.  I'm sure there must be a paper about this somewhere, though.

Watsisname
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### Solar System dynamics discussion thread

• Common misconceptions about tides. What are tides really?
• "The devil is in the detail" of tidal theory (coastlines, resonances, delays, tide-predicting machines, tides caused by planets, etc...).

This subject came up in the Astronomy Q&A thread, and I've made an attempt to explain the workings of the tidal interaction and its effects on both Earth and Moon there.  As the above implies, the details are actually incredibly complex, and I only give the very basic "schematic" idea.  Hopefully this is a good starting point.  There is much, much more that can be said about it!

FastFourierTransform
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### Solar System dynamics discussion thread

Watsisname wrote:
Source of the post This subject came up in the Astronomy Q&A thread, and I've made an attempt to explain the workings of the tidal interaction and its effects on both Earth and Moon there.  As the above implies, the details are actually incredibly complex, and I only give the very basic "schematic" idea.  Hopefully this is a good starting point.  There is much, much more that can be said about it!

Yeah! I saw it. Great and direct way to introduce the topic by the way. I was thinking about merging those posts into this thread but I prefer if you can make that decission since maybe is to messy for everyone to move 5+ posts here all of a sudden.
I'm still trying to figure out when I can contribute again and with what quality. Life... is complicated. But I have been gaining some confidence (that's what a Dunning-Kruger would say) with small aspects of tidal interactions so I will give it a try sooner or later probably.

Cantra
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### Solar System dynamics discussion thread

Would it be stable for a small planet to exist in the L4 or L5 points of the Earth, or possibly other planets too?

FastFourierTransform
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### Solar System dynamics discussion thread

Cantra wrote:
Source of the post Would it be stable for a small planet to exist in the L4 or L5 points of the Earth, or possibly other planets too?

In his spectacular explanation about Lagrange Points and their stability Watsisname mentioned and linked an article just about this. The last equation of that article yields the mass constraints for L4 and L5 trojans orbital stability. From the mechanics of the system it turns out that the mass of the main object must never exceed 25 times the mass of the trojan object. Well not exactly 25 but almost, the real constraint is

So, it turns out that even a planet like Earth could be a trojan of planet like Jupiter and still be a stable system since

In fact trojan planets is an important concept that is currently been investigated by planetary scientists. In 2002, Michael Nauenberg theorized that these trojan exoplanets should exist and be detectable with our current infrastructure (Kepler telescope). In 2004, Rudolf Dvorak et al. made numerical simulations and showed the feasibility of habitable zone trojan exoplanets. In 2007, Bálint Érdi et al. showed that even Jupiter sized trojans could exist in stable L4/L5 points. In 2008, Paul Cresswell et al. presented a formation and evolution model for trojan planets showing that they are even not so unfrequent (at least in theory). Gas and dust in the protoplanetary disk could get trapped in the L4 and L5 points of a protoplanet and accrete until a trojan planet is formed. In 2013, Anthony Dobrovolskis found a method to measure the mass of the trojan independently of the mass of the main planet in exoplanetary observations.

No trojan exoplanet has been conclusively found yet but there are even attempts of finding exotrojan asteroids. A good review of the current state of the search is given in John Dolan's 2018 excellent thesis (I reccoment to read it if you are interested in the topic).

A trojan planet is even part of the current models of the history of the Solar System and in particular it is the protganist of the Giant Impact Hypothesis, a scenario about the formation of the Moon by means of a planetary collision between Earth and this hypothetical object, we usually call Theia. The mass and orbit of the Moon highly constraint the impact parameters of Theia. A hard collision would have shatter Earth and Theia thus the relative speeds between both planets had to be quite slow and the collision had to be more like a glancing blow. This suggests that Theia and Earth where in a very similar orbit, so much that probably Theia was located either on the L4 or L5 points, where it would have grown with enought time until it got such a huge mass that the system became unstable and a collisional trajectory with Earth was possible. In fact the chemical composition of Theia (which is mixed with current Earth and Moon material) can be estimated and corresponds to the chemical composition of this region of the Solar System (almost identical to Earth in fact even if there is a debate about some methodological disagreements). Another impact parameter that has been constrained is the mass of Theia. For it to be able to form our Moon it had to be almost as large as Mars. It turns out that this makes a lot of sense if you think of it as a trojan that went unstable after growing to much, since for Earth the unstable limit corresponds to a mass of the same order of magnitude as Mars's. This was first proposed by Edward Belbruno et al. in 2005.

Cantra
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### Solar System dynamics discussion thread

FastFourierTransform wrote:
Cantra wrote:
Source of the post Would it be stable for a small planet to exist in the L4 or L5 points of the Earth, or possibly other planets too?

In his spectacular explanation about Lagrange Points and their stability Watsisname mentioned and linked an article just about this. The last equation of that article yields the mass constraints for L4 and L5 trojans orbital stability. From the mechanics of the system it turns out that the mass of the main object must never exceed 25 times the mass of the trojan object. Well not exactly 25 but almost, the real constraint is

Sounds about right to me, very interesting. Our Solar System is quite strange compared to the other solar systems we have seen, terrestrial planets first, and then gas giants with no super-earths. I wonder if Lagrangian point planetary bodies are common in the universe or if they are rare, I would imagine that they would be able to be disturbed by other planets unless they are held in place by a gas giant or something of that sort. I would argue that habitable gas giants could have Earth like worlds in their L4 or L5 points, which is certainly something to look into.

Watsisname
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### Solar System dynamics discussion thread

FastFourierTransform wrote:
Source of the post The last equation of that article yields the mass constraints for L4 and L5 trojans orbital stability. From the mechanics of the system it turns out that the mass of the main object must never exceed 25 times the mass of the trojan object.

It must be greater than, not less than!    I think the confusion is that in that relation, M1 and M2 are not the masses of the planet and trojan, but of the star and the planet.  The trojan is assumed to be massless, which is good enough approximation for asteroids.

A good rule of thumb is that the Sun must be at least about 25 times the combined mass of the planet and trojan body together.  If that is satisfied, then the relative masses of the planet and trojan body can be essentially anything.  (The actual relationship accounting for the trojan not being massless is more complicated, but doesn't change the number very much from 25.)

So yeah, we can imagine all sorts of planets existing as trojans of each other, even a planet as a trojan of Earth.  Since the Sun is much more than 25x the mass of Earth, or even Jupiter, it's actually hard to think of a realistic system where planet and star masses would not satisfy the criteria for stability.  Instead it seems (but based only on the limited data so far) that it's just unlikely for trojan planets to happen, probably because of interactions with other planets in the system and the migrations that happen early on, which could upset their stability.

FastFourierTransform
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### Solar System dynamics discussion thread

Watsisname wrote:
Source of the post It must be greater than, not less than!   I think the confusion is that in that relation, M1 and M2 are not the masses of the planet and trojan, but of the star and the planet.  The trojan is assumed to be massless, which is good enough approximation for asteroids.

Ups! Sorry for messing up that. Indeed one of the papers I was citing shows the correct relationship and the one you refered indeed says is the mass of the entire system. An example of how not to answer quickly and without enought care. Sorry for that.

Cantra
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### Solar System dynamics discussion thread

Would a counter-Earth be stable? I would imagine it wouldn't be, it would gradually move so we could see it from the other side of the sun. If it was a smaller planet, it'd be moved around by other planets more easily, but if it was a larger planet, it would be us that would be moved around more easily.

Salvo
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### Solar System dynamics discussion thread

Theoretically it's stable since it is in a lagrange point, and in reality it is. For example, there are a lot of asteroids orbiting in L3 lagrange point of sun-jupiter barycenter (Hilda family). The L3 point is the point of the orbit in which a counter-earth is supposed to lie. A small perturbation would indeed change this object's orbit, so it just depends on what you mean with "stable".

There are other parts of the orbit (they're called L4 and L5 lagrange points and they are "right before" and "right after" the orbiting mass, they make a 45° angle if you connect them with the center of the orbit) which are "more stable" than the point of counter-earth, but of course a perturbation would change the orbit, but this also works for planets.

In conclusion I think is impossible to this counter-earth to exists since a lot of probes went in the interplanetary space and they never found a counter-earth. It would be as bright as earth so it would be impossible to be ignored. Still there are a lot of people convinced of it

P.S.: Surely there are a few small asteroids in that orbit, if they count as "counter-earth" well it exits.
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Watsisname
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### Solar System dynamics discussion thread

Salvo wrote:
Source of the post Theoretically it's stable since it is in a lagrange point, and in reality it is.

L3 is not stable even in theory.  It is an equilibrium point, which means that in the corotating frame the forces all balance there.  However, the shape of the effective potential there is a saddle, like this:

I think you can easily see that this is not a stable place to be putting something.  A ball placed exactly at this point will stay there, and a small offset directly uphill would make it roll back and forth along a line.  But a tiny offset in the other direction would send it rolling down the hill, faster and faster.  So in general, even arbitrarily small offsets in a random direction will grow larger with time.  This is the definition of instability!

For a particle near L3 of Earth, a small deviation will be exponentially amplified, by a factor of e (2.718) every 150 years.  Combined with the fact that there are constant perturbations by the other planets, there is no chance that a planet could exist there naturally over geologic timescales.

The 150-year timescale does however make L3 a great place to put spacecraft.

Salvo wrote:
Source of the post For example, there are a lot of asteroids orbiting in L3 lagrange point of sun-jupiter barycenter (Hilda family).

The Hilda asteroids are not at L3.  They briefly pass through L3, but also L4 and L5 on eccentric orbits about the Sun, and share a 3:2 resonance with Jupiter.

Salvo wrote:
Source of the post There are other parts of the orbit (they're called L4 and L5 lagrange points and they are "right before" and "right after" the orbiting mass, they make a 45° angle if you connect them with the center of the orbit) which are "more stable" than the point of counter-earth

As I show earlier, L4 and L5 are stable (for large enough ratio of masses).  This is actually surprising if you think about it.  A particle very close to L4 or L5 will not stay there, but slide away just like a ball rolling down a hill.  There is no restoring force to keep a particle tied there.  Instead, the Coriolis force keeps the particle whirling around in the vicinity of those points.  Small arbitrary perturbations to their orbits do not amplify over time, and this is the definition of stability.  Orbits of L4 and L5 are truly stable!  Also the angle they form is 60°, not 45°.  Imagine them as making equilateral triangles:

Cantra
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### Solar System dynamics discussion thread

Very interesting, so the L4 and L5 points are possible for planets to lie, the L3 is not a stable point for planets to lie, though asteroids can lie there. How about the L1 and L2 points? Could an asteroid or a small planet stay in those areas over long periods of time?
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