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 6a): 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 6b. Are Stable Ternary Star Systems Possible?! Part b.
In this plot all 3 masses are equal (see
Figure 6a for 2 equal masses and
an “infinitesimal” perturbed mass). As in
Figure 5 and
Figure 6a, 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), 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 smaller in this plot with
all 3 masses equal (i.e. the lighter color near L4/L5 does not extend out
as far; the contour lines are almost irrelevant in this) than the
comparable region in
Figure 6a with an “infinitesimal” 3rd body. 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, since the
2 major masses are equal. The distance from L2 and L3 to its nearest major
mass is the same as the distance between the 2 major masses, symmetrical
since the 2 major masses are equal.
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 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.)
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