Newton’s Great... Oversight
Galileo’s Falling Bodies and Lagrange’s Trojan Asteroids
With Their Tadpole and Horseshoe Orbits

8          FIGURES (cont.)

Temporary explanation of what's what (compare to Figure 4, but remember that the mass ratios are different, so L3, L1 and L2 lie in different locations than in Figure 5, but in the same as in Figure 6b): on the central vertical line: L4 and L5 still lie on either side of the central horizontal line, just past the first horizontal line in either direction away from the central line; on the central horizontal line: L3 now lies... somewhere out past the first vertical line left of the central vertical line; m₁(largest mass) still lies just inside the first vertical line left of the central line; L1 now lies on the central vertical line, halfway in between the two largest masses;  m₂ (second largest mass) still lies just inside the first vertical line right of the central line;  L2 now lies... somewhere out past the first vertical line right of the central vertical line.

Figure 6a. Are Stable Ternary Star Systems Possible?! Part a.

In this plot, the 2 “unperturbed” masses are equal, m₁ = m₂, and the 3^rd, perturbed mass at L4 (or L5) is again 0.0001 m₁, and so approximates “infinitesimal”. (See Figure 6b for 3 equal masses.) As in Figure 5, this contour plot shows a function of the falling rate difference that indicates the rate at which a (static) triangle, here of equal masses, with the body at L4 (or L5) perturbed (to the point on the plot), further degrades from equilateral. The “tadpoles” no longer have that characteristic, asymmetric shape, but rather have “degenerated” to a seemingly simpler, symmetric shape, since the 2 “unperturbed” masses are in fact equal. Again, the degradation is least quick in the degenerated “tadpoles”, quicker in the degenerated “horseshoe” (which almost seems to have formed a circular orbit around both masses; actually, see Figure 6b since it shows this much better), to very much quicker outside them.

It may be difficult to tell from this plot, but the region of possible stable “equilibrium” around L4 (and L5) seems larger in this plot with an “infinitesimal” 3^rd body (i.e. the lighter color near L4/L5 extends further out; the contour lines are almost irrelevant in this) than the comparable region in Figure 6b with all 3 masses equal. Comparing Figures 6a and 6b suggests that a wide range of ternary star systems or binary star systems with a planet at L4 or L5 are possible.

NOTE that L1 has moved to a place of symmetry between L4 and L5, symmetrical since the 2 major masses are equal. L2 and L3 do not even appear on this plot as they are ~1.4 times the distance between the 2 major masses, but beyond on either side.

NOTE ALSO: a contour plot is like a topographical map: the shape of the contour lines depends on the “elevation” of the intersections of various “cutting planes” with the function’s 3-dimensional surface. The contour lines of the same surface can look quite different if contour-plotted slightly differently. Some of the difference between the contours of Figure 6a and Figure 6b is merely that, some that the region of possible stability around the Trojan points is narrower when all 3 masses are equal.

ALSO NOTE: technically, the Lagrangian points do not even exist in the case of 2 (or 3) equal masses, since Lagrange’s restriction, i.e. that m < ~0.04 M, is not met.

To examine stability carefully, it would be necessary to look at the total dynamics of the system, but computers should make that quite feasible, even without making the simplifying assumption — currently considered necessary — of “infinitesimality” for any of the masses. Even if it is unstable, a ternary star system could take an astronomically significant time to degrade. And it’s good to remember: the scientists who think ternary star systems are unstable and therefore impossible also think lighter and heavier bodies always fall at precisely the same rate.

(Plotted with Mathcad 2000.)