Note: I had L5 leading but now have skimed your link and see it is L4 that is leading so tried to switch 4 and 5. - hope I got them all changed.
Here for the benefit of others from DH's link is equation 1149 results in words:
"We thus conclude that the L4 and L5 Lagrange points are stable equilibrium points, in the co-rotating frame, provided that mass m2 {the Earth} is less than about 4% of mass m1 {the sun} ..." I.e. small m3 can orbit at either L4 or L5 forever if there were no Jupiter etc.
Billy: What makes you think there is a mass that abolishes the triangular libration points (L4 and L5)? Did you read the second link cited in post #27? In particular, equations 9 and 10 describe the stability conditions. A system with Earth and Theia at the triangular points will be stable so long as the combined mass of Earth and Theia is less than about 1/25 of the mass of the Sun, or about 13,000 Earth masses. ...
I only suspected that they could be abolished, called it an assumption, etc. The post 30 footnote is what I more believe might be true; but it too may be wrong as only based on intuition, and the following thoughts, not analysis:
What you are telling me is that 6 masses (A,B,C,D,E,F in cyclic order with B leading A etc.) each 2000 times Earth mass in circular orbit around the sun at the corners of regular hexagon (neglecting all other planets etc.) are stable. Each is in a stable L point of two others, so I can certainly accept they are at least “meta-stable.”) I believe that they are completely stable, probably even if Jupiter is added to the system.)
As the eccentricity increases from zero, I strongly think they would go unstable, even if the major axis sun line was the same for all as the in orbit speed is changing as they speed up towards the perigee point on that line. Consider the instant when E (heavy Earth) has least speed at apogee. Then L4 of D is slightly in front of E approximately (if not exactly) at L5 of F which has been speeding up for two months. It would seem to naive me the E should be slightly falling down into this “double well” which I think is slightly closer to the sun than E’s undisturbed ellipse. That I think should increase the eccentricity of E. – a positive feedback that drives the system unstable.
Now consider the less extreme case of Sun and only Earth and mass M (= to the moon mass) which on average is at E’s L4 in same, but slightly eccentric, orbit at the instant when M is at apogee. Then the following E (always faster for the last half year than M) is less than 60 degrees behind M so E’s L4 is in front of M and slightly closer to the sun (as before with six planets). This case seems to be unstable also (I think, but may be wrong) like the case with six 2000 Earth masses with large e, but with much small growth rate for the instability. That is why my intuition suggested to me that the alternative moon formation idea of my prior post needs to be seriously considered (some analysis). However, with Earth’s current small eccentricity and moon much smaller mass, a moon mass at E’s L4 might not escape from E’s L4 well. That is why I suggested, my alternative idea for moon formation might need to call on Jupiter to help push it out of the well.
You might try running a simulation with the real earth and moon masses but e = 0.9 and see my intuition is still wrong. If it is not, lower e enough to take too long to exhibit the instability; perhaps by noting that M’s separation for E’s L4 is starting to decrease again. – Hinting that M is in some orbit about E’s L4. If that “stable e” is not too much larger than Earth’s e, maybe it is interesting enough to put Jupiter into the simulation and see if M can get helped out of the well with e near Earth’s current e?
One of the advantage of my slightly less than total ignorance about all this is that it lets me think of things you would not normally.
I hope this post makes enough sense so that you can at least tell where my intuitive ideas come from.
