Originally Posted by

**StupendousMan**
Tony's excellent work prompted me to do what I should have done a few days ago: get up and walk to the library. I scanned through a book called "The Three-Body Problem" by Christian Marchal, published by Elsevier in 1990. Chapter 8 in that book discusses "Simple Solutions of the Three-Body Problem", one of which is the system Tony proposes: a very massive star (the Sun) and two much less massive bodies (Earth-1 and Earth-2), with the two small bodies orbiting the massive one in circular orbits of the same radius, separated by 60 degrees.

Marchal discusses the stability of these systems at some length, using first-order and higher-order analysis. He provides references to papers by C. Richa (thesis from University of Pierre and Marie Curie, 15 Oct 1980) and himself (seminar at the Bureau of Longitudes, 7 Mar 1968). In his discussion, he defines a couple of parameters which involves the relative masses of the three bodies:

N = sqrt( m1^2 + m2^2 + m3^3 - m1m2 - m1m3 - m2m3) / (m1 + m2 + m3)

which, in our case of m2 = m3, simplifies to (please check my algebra here!)

N = (m1 - m2) / (m1 + 2m2)

and the parameter

R = ( 3 - sqrt(12 N^2 - 3) ) / 6

Now, in the case Tony has suggested, in which m1 = Sun's mass and m2 = m3 = Earth's mass, the parameter N is a fraction very close to 1 (I find N=0.999994), and the parameter R is a number very close to zero (I find R=4 x 10^(-6)). Again, check my arithmetic!

The payoff for computing these values is that Marchal presents a nice graph (his Figure 13 on page 49) showing zones of stability for this "Trojan" arrangement. For circular orbits, there are two zones in which the orbit can be proved stable:

0 <= R < 0.02860...

and

0.02860... < R <= 0.03852....

Note that Tony's system falls well within this first zone of stability. In fact, this suggests a test: the arrangement should still be stable until the two "Earth-like" planets grow in mass so that R = 0.03852, which means N = 0.9428 or so. And that, in turn, if I can do algebra again, means that the system should be stable until each of the two "Earth-like" planets has a mass of m1*(0.01982). In other words, until the two Earth-like planets added together have a mass of about 4 percent of Sun.

Sorry for being so forceful in my incorrectness earlier :-(

Tony, thanks for sticking to your guns and helping me to learn something new today ....