chaos syndrome wrote: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?
Guest wrote: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...
chaos syndrome wrote: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.
chris wrote: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...
wcomer wrote: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.
Cormoran wrote: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.
granthutchison wrote:Evil Dr Ganymede wrote:
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.
marc wrote:
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.
