[SPACE] Rules considering Lagrange Points in Advanced Worldbuilding

[Sabersonic]

For my first thread on this board, I have a few inquiries related to the Advanced Worldbuilding in the creation of a customized star systems, or rather the lack of information for details that I require. Mainly Lagrangian/Liberation/Trojan points.

For Lagrangian Points subject, well I wanted to create a habitable moon of a Gas Giant that had Co-Orbital Trojan Moons along its orbit at the L4 and L5 locations after recalling and looking up information on natural occurring Trojan Moons of the Saturnian Moons Tethys and Dione as explained in this Wikipedia Article. Since my hardcover copy of GURPS Space 4th Edition did not have the information I sought, I attempted to search for the information through the forum. Although I have found some interesting notes about Lagrange Points such as

Originally Posted by dataweaver

(maximum mass for satellite with stable trojan points = 4% of primary's mass)

Originally Posted by Brett

The L4 and L5 points are only stable if the ratio of the masses of the two stars is greater than about 25.

However, I did not find the answers to my questions in determining if a Planet-Moon system could have Lagrangian Points or not and the maximum mass/moon type/moonlet that could occupy that zone. Is there a suggested dice formula to determine if such a system could even have Lagrangian points if they meet certain requirements or do I just dictate that those two bodies have those points and leave it as that?

[Diomedes]

Originally Posted by Sabersonic

However, I did not find the answers to my questions in determining if a Planet-Moon system could have Lagrangian Points or not and the maximum mass/moon type/moonlet that could occupy that zone. Is there a suggested dice formula to determine if such a system could even have Lagrangian points if they meet certain requirements or do I just dictate that those two bodies have those points and leave it as that?

They'll have the first three points, and the other two if the ratio of the masses are high enough (the aforementioned 25:1). As to the maximum mass that can be placed there, probably considerably smaller than either the primary or the satellite, to avoid disturbing their orbits.

[teviet]

Originally Posted by Diomedes

They'll have the first three points, and the other two if the ratio of the masses are high enough (the aforementioned 25:1). As to the maximum mass that can be placed there, probably considerably smaller than either the primary or the satellite, to avoid disturbing their orbits.

My guess is that the tertiary mass would not matter so much for stability, as long as both the secondary and tertiary were less than 1/25 of the primary. But this is a guess (based on a vague notion of how the rotating potential will change as the tertiary mass increases), not an analysis.

Nonetheless, among the many Trojan points in our Solar system, and the many bodies orbiting them, none have accumulated into a major-moon-sized mass, suggesting that such objects would be quite rare.

TeV

[malloyd]

Originally Posted by teviet

My guess is that the tertiary mass would not matter so much for stability, as long as both the secondary and tertiary were less than 1/25 of the primary. But this is a guess (based on a vague notion of how the rotating potential will change as the tertiary mass increases), not an analysis.

Since two such moons would be in *each other's* L4/L5 points (one is 60 degrees ahead of the other, hence the second is 60 degrees behind the first) there shouldn't be any mass ratio considerations, or you simply redefine which is in the point. Though possibly the combined mass of the two will need to be sufficiently low compared to the primary instead of just the individual masses.

Nonetheless, among the many Trojan points in our Solar system, and the many bodies orbiting them, none have accumulated into a major-moon-sized mass, suggesting that such objects would be quite rare.

You really would expect that. In order to move into the point and stay there something would have to be in an orbit that passed nearby without much energy difference, which means on anything like a geologic timescale it also passed near the main body lots of times too, not a recipe for staying in that orbit.

[Agemegos]

Strictly speaking the stability limits of the Trojan points are unknown, since the Lagrangian solution holds only for test masses (i.e. if the mass of the third body is ignored). No analytical solution exists for that case of the three-body problem. As for numerical simulations, I understand that they always lead to a collision or the expulsion of the smaller body for any mass ratio that is tried.

Obviously the orbits are close enough to stable for really large mass ratios, since there are Trojan asteroids and analogues in the orbit of Saturn. But I don't know of any Trojan-analogues in the orbit of Earth, which suggests that the ratio of masses of Earth to an asteroid visible at 1 AU is not enough. One of the leading theories for the formation of the Moon is that an object about the size of Mars formed in one of Earth's Trojan points, where its orbit was unstable and led in time (only 20–30 million years) to a collision.

I can't be quite definite, but I would have to guess that having a planet in the Trojan point of another planet, even a gas giant, is a space opera conceit rather than a hard SF plausibility.

[teviet]

Originally Posted by Brett

Strictly speaking the stability limits of the Trojan points are unknown, since the Lagrangian solution holds only for test masses (i.e. if the mass of the third body is ignored). No analytical solution exists for that case of the three-body problem.

No, the Lagrange solutions are equilibrium states for arbitrary masses. Their stability against small perturbations can be calculated analytically. (I haven't actually done it but the method is straightforward.) Klemperer rosettes are a similar N-body equilibrium state with nonzero masses.

But these are special cases that are either highly symmetric or stationary in the rotating reference frame. You are correct that the general (non-equilibrium) three-body problem has no analytic solution.

TeV