Planets in Trojan points?

[ajtribick]

I've had a look at several sites on the internet about the stability of the Trojan points L4 and L5, and I've come across the constraint that the orbiting mass should be <4% the mass of the primary for L4 and L5 to be stable.

However I have not seen any constraints on the mass of the body located in the Trojan point, except that it must be lower than that of the first two objects.

Would it, for example, be possible to have an Earth-mass planet located in the Trojan point of a gas giant, or would this configuration be unstable?

[Guest]

WAG-
I don't think a planet would form in the trojan point of a gas giant; Jupiter, Neptune and Mars all have trojan asteroids but they have not accreted into single objects over billions of years.

Incidentally I need to put some orbiting habitats in the various Lagrange points around the Earth and Moon for illustration purposes in our Space Trader Sim;
what would the orbits be in Celestia for L1,2,3,4,5 in the Earth -Moon system? I have tried to make the orbits up myself but they end up all over the place (or inside the Moon):)

[eburacum45]

logs in to claim last post-

[Guest]

what would the orbits be in Celestia for L1,2,3,4,5 in the Earth -Moon system? I have tried to make the orbits up myself but they end up all over the place (or inside the Moon):)

Unfortunately, the Earth-Moon libration points are pretty much impossible to display accurately in Celestia. The plane of the Moon's orbit precesses relatively rapidly, requiring a CustomOrbit defined within Celestia in order to display it at all accurately. An EllipticalOrbit designed to place a libration point in the correct place on a given date will be noticeably inaccurate within the span of a few months.

Grant

[chris]

Would a new orbit type for specifying Lagrange points be useful? The orbit of Saturn's moon Calypso could then be given as:

LibrationPointOrbit
{
Point "L4"
Body "Tethys"
}

That's only approximately correct as Calypso probably doesn't stay exactly at the libration point . . . Grant and anyone else: is this a feature worth incorporating? It's pretty easy to add.

--Chris

[Evil Dr Ganymede]

Can you not just do L4 and L5 points by using the same orbit as the planet, but with a MeanAnomaly of +/- 60 degrees of the planet's?

[selden]

Oh Evil one,

It depends on how accurate one is trying to be. Bodies at L4 & L5 orbit around the minima rather than being fixed at exactly one location. But it's a good first approximation!

[Cormoran]

Hmm, we define L4 and L5 as invisible objects and put our Lagrange objects and stations in orbit around that.

Must give it a try. Back in a week

Cormoran

[granthutchison]

Can you not just do L4 and L5 points by using the same orbit as the planet, but with a MeanAnomaly of +/- 60 degrees of the planet's?

Oh yes. But if the object has a CustomOrbit that shifts its nodes and pericentre rapidly, you tend to end up with the libration points sitting high and dry some distance from their correct location - in the case of the Moon, there's a 5-degree tilt to its orbit and the nodes swing around significantly in the space of a year - so after a while you end up seeing (for instance) the Moon below the ecliptic while L4 and L5 perch mysteriously above the ecliptic.
So Chris is offering the option (I think) of having Trojan points that track CustomOrbits. Much better than trying to fake it with an EllipticalOrbit, Chris, but I guess best of all would be to have CustomOrbits for things like Calypso as well, so we could actually watch them wander around in the vicinity of the Trojan point.

Grant

[ajtribick]

WAG-
I don't think a planet would form in the trojan point of a gas giant; Jupiter, Neptune and Mars all have trojan asteroids but they have not accreted into single objects over billions of years.

Hmmm... but suppose an Earth-mass object migrated into the Trojan point or something, would the system be stable?

[wcomer]

What if you have more than one object at the L4? You need some additional parameters.

[marc]

Would a new orbit type for specifying Lagrange points be useful? The orbit of Saturn's moon Calypso could then be given as:

LibrationPointOrbit
{
Point "L4"
Body "Tethys"
}

That's only approximately correct as Calypso probably doesn't stay exactly at the libration point . . . Grant and anyone else: is this a feature worth incorporating? It's pretty easy to add.

--Chris

Yes please Chris, like Oroins Arm, I'd like to use it for space stations.
I imagine it would be a useful education feature as well.

[Falck]

To put in L4 or L5 as an orbit isnt really valid.

Objects orbit the Lagrange points, they dont sit directly on them.

[Spaceman Spiff]

I've had a look at several sites on the internet about the stability of the Trojan points L4 and L5, and I've come across the constraint that the orbiting mass should be <4% the mass of the primary for L4 and L5 to be stable.

However I have not seen any constraints on the mass of the body located in the Trojan point, except that it must be lower than that of the first two objects.

What was your source of the <4%? It may be an assumption made to allow working out these points, not a requirement for stability. I looked up a paper on the mathematics of the Lagrange points: http://www.physics.montana.edu/faculty/cornish/lagrange.pdf. The results are really for the special case of the 'restricted three body' problem (as it's called) but is applicable in all cases found in nature so far. It defined M1 as the central body mass, M2 as the other body mass. The Trojan is assumed massless. What I notice about this paper is that these solutions assume M2 is much less than M1, which might be the source of <4% - i.e., it's not a matter of stability, just approximations to let solutions be found?

I don't think a planet would form in the trojan point of a gas giant; Jupiter, Neptune and Mars all have trojan asteroids but they have not accreted into single objects over billions of years.

I agree it's unlikely a planet would form there. Isn't the thinking instead that Trojans are captured into those resonances? Given that objects wander about...

Hmmm... but suppose an Earth-mass object migrated into the Trojan point or something, would the system be stable?

... then since the theory is that planet formation becomes more 'stochastic' in its later processes (G W Wetherill), one could expect that ocassionally an Earth sized 'planetesimal' wanders into the L4/L5 resonances of a giant planet and gets trapped there. There's a suggestion that Uranus' rotation axis was tipped to 98° by a colliding Earth sized planetesimal. If that quasi-Earth had wandered into Jupiter's L4/L5 point it would have been stable. Letting Earth mass = 1, then Jupiter mass M2 = 318, and Solar mass M1 = 332,946, which fulfills the assumptions. In general, I suspect as long as both orbiting masses are much less than the mass of the orbited body, then one can say that one is the Trojan of the other. They could even be similar masses, and you might simply decide the 'Trojan' is the smaller of the two.

Would a new orbit type for specifying Lagrange points be useful? The orbit of Saturn's moon Calypso could then be given as:

LibrationPointOrbit
{
Point "L4"
Body "Tethys"
}

That's only approximately correct as Calypso probably doesn't stay exactly at the libration point . . .

Yes. The key is to transfer the Mean Anomaly of the 'Trojaner' to the 'Trojanee' (ahem...). However...

What if you have more than one object at the L4? You need some additional parameters.

...to allow the different orbits of Trojans to be kept, we need to be able to specify eccentricity, inclination, ascending node and argument of periwotsit as well as which body and L point is meant. Mean anomaly and semi-major axis should be defaulted to this:

L4 means +60° Mean Anomaly, same period, semi-major axis.
L5 means -60° Mean Anomaly, same period, semi-major axis.
L1 means same Mean Anomaly, same period, smaller semi-major axis = a(1-((M2/(M1+M2))/3)^(1/3)).
L2 means same Mean Anomaly, same period, larger semi-major axis = a(1+((M2/(M1+M2))/3)^(1/3)).
L3 means ±180° Mean Anomaly, same period, slightly larger semi-major axis = a(1+5(M2/(M1+M2))/12).

This comes from the paper I mentioned above, and is only accurate for the case it mentions.

Hmm, we define L4 and L5 as invisible objects and put our Lagrange objects and stations in orbit around that.

Well, it's a start, but not quite that simple - objects don't orbit L4 or L5 in circles or even anything like Keplerian orbits (offset ellipses). If you look at a real Trojan asteroid's orbit about its L5 point, it appears as a brazil nut shape streteched along Jupiter's orbit. This cannot be modelled well by Keplerian orbits referenced about an L4/L5 point. I think it's much better to model by a Keplerian orbit about the Sun, Moon, whatever - just use appropriate parameters.

Can you not just do L4 and L5 points by using the same orbit as the planet, but with a MeanAnomaly of +/- 60 degrees of the planet's?

Oh yes. But if the object has a CustomOrbit that shifts its nodes and pericentre rapidly, ---8< snip 8<---

Quite right Grant, but if we are lucky enough not to be forced into working with a CustomOrbit elsewhere then simple Keplerian orbits should suffice in the meantime. If we have a CustomOrbit, I think all we need is Chris's new keywords to help out with transferring that Mean Anomaly to that Trojanee. I don't know how CustomOrbit is done, so I don't know if what I wrote is sensible.

Yes please Chris, like Oroins Arm, I'd like to use it for space stations.

One thing to remember about these proposed artificial high-tech space stations, etc. We can assume they'll do wonderful 'station keeping' so one could pretend they will always stick faithfully to a decent L4/L5 orbit... Look at geostationary satellites, or even Soho as examples of station keeping. Remember, the L1, L2 and L3 points are unstable.

Finally, some background: The actual Trojan asteroids trail behind Jupiter near the L5 point, except for 624 Hektor, which was the first discovered (http://en.wikipedia.org/wiki/624_Hektor) near the L4 point. After further asteroids were discovered about L5 as well as L4, the convention became to name L4 asteroids after the Greeks of the Iliad, the L5 asteroids after the Trojans. Hektor was left on the wrong side. The name Trojan is then an over-generalisation, maybe even a misnomer (a bit like trojan virus from 'Trojan horse' - which was built by the Greeks ).

Still, if we do get 'Trojan' functionality, can we have 624 Hektor as the first one modelled in solarsys.scc? It's also a possible contact binary asteroid - might look nice.

Spiff.

[granthutchison]

What was your source of the <4%? It may be an assumption made to allow working out these points, not a requirement for stability.

No, it's a stability requirement. The equations of motion in the vicinity of a libration point are summarized by a biquadratic equation which includes the mass ratio as a constant. If all four roots of this equation are imaginary, then the motion is restricted to periodic oscillations; but if there are real roots, there is an exponential runaway. The value of the mass ratio determines whether there are real roots, and that occurs when the mass ratio (secondary to secondary+primary) falls below

(9-root[69])/18 = 0.038520896...

That corresponds to a secondary-to-primary mass ratio of 0.040064205..., so "<4%" is pretty tight.

Only Hal Clement could have used this fact as the plot of a science fiction story - he did it in Trojan Fall in 1944: "... Don't ask me why; I couldn't show you the math; but I know it's true - the stability function breaks, with surprising sharpness, right about the twenty-five-to-one mass ratio. ..."

Grant

[Spaceman Spiff]

What was your source of the <4%? It may be an assumption made to allow working out these points, not a requirement for stability.

No, it's a stability requirement.

Pfffff...

Spiff

[Spaceman Spiff]

(...recovers) Well Grant, good work! I think that sounds like one of the most certain figures Celestia users can use as a reality check. No trojans in typical binary star systems then, but could Earth really be stable as a trojan to Jupiter? Did that maths say anything about the upper mass limit of the trojan?

Spiff

[granthutchison]

Did that maths say anything about the upper mass limit of the trojan?

Now, that one is an assumption.
The Lagrange calculations apply only if the mass of the third body is effectively zero - that is, too small to shift the system's centre of gravity appreciably. I'd be worried that an Earth-sized object (0.003 Jupiter masses) would introduce significant deviations from the Lagrangian ideal, but whether they'd be show-stoppers is another matter. Maybe a numerical simulation would be interesting ...

Grant

[ajtribick]

Hmmm... anyone want to set up this situation in Gravity Simulator or something? I could never figure out that program.

[tony873004]

Hmmm... anyone want to set up this situation in Gravity Simulator or something? I could never figure out that program.

Sorry for dredging up a 6-week old thread, but I missed this one. In gravity simulator, here's an Earth-massed body in Jupiter's Lagrange 4 point. Jupiter is tracing the purple oval in the bottom right of the image, while the Earth-mass object is tracing the smaller oval in the top right corner of the image. This image is a rotating frame, attempting to keep Jupiter still. That's why it traces an oval instead of appearing to orbit the Sun. It is an oval rather than a point because Jupiter speeds up and slows down from its average speed due to a small amount of eccentricity in its orbit. I ran this simulation for a few hundred years, and the Earth-massed object still seems very stable in its Lagrange 4 point.

[edit] I also tried it after increasing the the trojan Earth's mass to .1 Jupiter masses, .5 Jm, and equal to Jupiter's mass. They all give the same results. Even a Jupiter-massed planet in Jupiter's Lagrange 4 or 5 would be stable. I ran the Jupiter-massed scenerio for 30,000 years and it show no sign of instablilty. The two masses have a small periodic affect on each other's eccentricities.[/edit]


To create this yourself,
A. start with the simulation solarsystem.grv (available on gravitysimulator.com). Press F8 to make sure your controls are visible. Move the scroll bar all the way to the top so you have an overhead view of the solar system. Use the Screen Scale to scale the screen a little larger than the size of Jupiter's orbit.

B. Reduce your time rate to 1, and pause the program (Time Rate's '||' button).

C. From the menu choose Objects > Edit Objects... choose Jupiter. Delete Jupiter.

D. Re-create Jupiter. From the menu choose Objects > Create Objects. Enter these settings:
__ number of objects: 1
__ Reference Object: Sun
__ Name: Jupiter
__ Eccentricity: .047
__ Mass: 317 +-0
__ Semi-Major Axis: 778000000 +-0
__ Inclination: 1.3011 +-0
__ Longitude of the ascending node: 180 +-0
__ Longitude of Perigee: 180 +-0
__ Mean Anomoly: 180 +-0
__ Size: not important.

E. Create Earth-mass object in Jupiter's L4 point. Follow the instructions in step D, except name it whatever you want, and give it a mean anomoly of 240 +-0 for the L4 point, or 120 +-0 for the L5 point, and a mass of 1.

(The reason for deleting Jupiter and then re-creating it is because it's easier to create Jupiter and Earth-mass trojan together so you can exactly set their mean anomolies 60 degrees apart.)

F. Un Pause the program (Time Rate's '>' button). Use Time rate to increase time to 32768.

G. From the menu, choose View>Rotating Frame Adjustment. In the Rotating Period box, choose Jupiter (if that's what you named it in step D. It might have 2 digits after its name. That's ok. It should be 2nd to last object). Click on the option Rotating Frame. This makes the program attempt to rotate the frame with Jupiter's period, but it never gets it quite right. Use the + and - buttons to tweak the rotating period so Jupiter's oval doesn't drift.