[SPACE] Rules considering Lagrange Points in Advanced Worldbuilding

[Agemegos]

Originally Posted by teviet

Hmm, I saw it as a problem in a classical mechanics textbook somewhere, but I can't remember which one. How about I just give you the proof?

Even better!

Theorem: The Trojan orbital configuration is an equilibrium for arbitrary masses.

Proof looks good to me. Do you know why it is so often written that the result only holds if the third mass is light? Is it that the configuration was discovered (by Lagrange) by considering a test mass?

Note: This only proves that the Trojan configuration is an equilibrium configuration for any set of masses, but says nothing about the stability of the configuration (i.e. whether small perturbations to the configuration will remain small or will grow over time). As has been pointed out, the stability of the Trojan configuration does depend on the masses of the objects.

I see (above) that you don't happen to know the stability limits. Do you happen to know whether the standard result for the stability limit does depend on an assumption that the third mass is arbitrarily light?

In reading about the Giant Impact Hypothesis I often come across statements that Theia's orbit in one of the Earth's Trojan points became unstable once it accumulated (through accretion) more than a limiting mass (eg http://www.search.com/reference/Giant_impact_hypothesis). Earth is supposed to have been nearly fully-formed, and Theia about the mass of Mars.

[teviet]

Originally Posted by Brett

Do you know why the statement that the result only holds if the third mass is light is so common? Is it that the configuration was discovered (by Lagrange) by considering a test mass?

Possibly. Myself, I don't recall encountering that statement before. But certainly the general 3-body problem in which the masses are not in a fixed co-rotating configuration is a lot harder (in fact impossible) to solve analytically when the third mass is nonzero.

Originally Posted by Brett

I see (above) that you don't happen to know the stability limits. Do you happen to know whether the standard result for the stability limit does depend on an assumption that the third mass is arbitrarily light?

It does. I'm not sure how much. I was intrigued enough to take a look at it, but all I concluded is that I am gradually losing the ability to perform basic algebra without Mathematica (it's only a 4d eigenvalue problem, how hard can it be?!?).

TeV

[jeff_wilson]

Originally Posted by Brett

Proof looks good to me. Do you know why it is so often written that the result only holds if the third mass is light? Is it that the configuration was discovered (by Lagrange) by considering a test mass?

L was supposedly looking for places in the rotating reference frame where the attraction of the two bodies and the centripetal acceleration summed to zero. The amount of mass of a third body or its absence would not alter the solution, but enough mass would invalidate the assumption that the second body is in a stable, near circular orbit around the first.