8 FIGURES
(cont.)
Temporary explanation of what's what (compare to
Figure 4): on the central vertical line: L4 and L5 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 lies between the second and third vertical lines left of the
central vertical line; m1(largest mass) lies just inside
the first vertical line left of the central line; L1 lies between the
central vertical line and the first vertical line right of the central
line; m2 (second largest mass) lies just inside the
first vertical line right of the central line; L2 lies just outside
the first vertical line to the right of the central line.
Figure 5. “Tadpole” and “horseshoe” orbits.
This contour plot gives some idea of what “tadpole” and “horseshoe” orbits
are about. It plots a function of the falling rate difference that
indicates the rate at which a (static) triangle of masses, with the
“infinitesimal” body at L4 (or L5) perturbed (to the point on the plot),
further degrades from equilateral. The degradation is least quick in the
tadpoles, quicker in the horseshoes, to very much quicker outside them.
The largest mass is set to “1”; the second largest mass is 0.01 m1,
i.e. smaller than the
~0.04 m1 that Lagrange’s analysis indicated was the
upper level for m2 that allowed stability (reminder:
with the 3rd mass “infinitesimal”); the smallest mass is
0.0001 m1, and so approximates “infinitesimal”.
The “centers” of the mirror-symmetric tadpoles — L4 and L5 — lie on the
vertical x = 0 line, roughly 0.866 above and below the y = 0 line.
L1, L2 and L3 all lie on the horizontal y = 0 line. On this plot,
L1 is about x = 0.3, L2 about
x = 0.9, and L3
about x = -1.5.Regarding L3, the more equal the 2 smaller
masses are, the more likely that they — one of them at L3 — will be the
same distance from the largest mass.
NOTE how the tadpoles “grow” till they meet and form an oddly shaped
horseshoe, one which covers L3. L3 is actually in a place where the
(degenerate) triangle degrades roughly as fast as it does in the tadpoles.
This suggests that L3 should be thought of as stable if the horseshoe
orbits are considered stable!... unless e.g. the velocity through L3 is
critical to its returning, and could not generally be obtained by
“wandering away, perturbed” from L3.
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.
NOTE: we think of the asteroid as being in a tadpole or horseshoe orbit
with regard to e.g. the Earth or Jupiter, since we are looking at its
orbit from an Earth or Jupiter relative reference frame. If we look at the
orbit from a frame that is where Earth would be if there were no asteroid,
and look at both the Earth and the asteroid orbiting, the Earth will also
be seen to have its own “tadpole” or “horseshoe” type movement, but
smaller as its mass is larger. The asteroid’s orbit will also have a
somewhat different shape. If the masses are more equal, the orbits can
take on very different shapes. See examples of possible shapes in
Figure 6b, which has all 3 masses equal.
ALSO NOTE: technically, the Lagrangian points would not even exist
if Lagrange’s restriction on relative mass, i.e. that
m2 <
~0.04 m1, is not met.
FURTHER, NOTE that this 2 dimensional plot is inadequate to give a picture
of e.g. the dynamics of the Earth companion asteroid,
3753 Cruithne, with its highly inclined orbit
that takes it directly above Earth.
(Plotted with Mathcad
2000.)
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