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Newton’s Great...
Oversight
Galileo’s Falling Bodies and
Lagrange’s Trojan Asteroids
With Their Tadpole and Horseshoe Orbits
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4 TROJAN POINTS AND
THEIR TADPOLE AND HORSESHOE ORBITS
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SECTIONS
4.1 Trojan Asteroids in
Tadpole Orbits
4.2 Trojan Asteroids in
Horseshoe Orbits
4.3 Trojan Space Debris?! Trojan Atmospheres?!
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4.1 Trojan Asteroids in
Tadpole Orbits
(See also
APPENDIX.) No asteroids are known to be equilibrated precisely in a
Trojan point proper. They tend to orbit the Trojan point in an orbit that
is elongated, non-elliptical, and not even symmetrical. It bends so that
its center of curvature is roughly the center of mass of the whole
ensemble (near the largest mass, e.g. the Sun), its “head” curves so that
it points toward the 2nd largest mass (e.g. Jupiter, which is
roughly 1/1000 the mass of the Sun), and its “tail” curves around toward
L3. (See
Figure 4.) At some point, someone decided it was shaped like a
“tadpole”, and that’s what they are now called.
Figure 5, a contour plot derived from the falling rate difference,
gives a rough idea of the shape. It can take hundreds of years for one of
Jupiter’s Trojan asteroids to complete a tadpole orbit of its Trojan
point.
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SECTIONS
4.1 Trojan Asteroids in
Tadpole Orbits
4.2 Trojan Asteroids in
Horseshoe Orbits
4.3 Trojan Space Debris?! Trojan Atmospheres?!
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4.2 Trojan Asteroids in
Horseshoe Orbits
Figure 5 also shows how the tadpole orbits for L4 and L5, which are
mirror symmetrical, start to “grow” or extend around the other side of the
more massive body until they blend into a “horseshoe” shaped orbit.
Asteroids are actually known to follow these horseshoe orbits in which
they orbit BOTH Trojan points! They have enough energy to escape the
tadpoles, but not enough to escape the horseshoe. Although it is possible
that we might someday find one in a horseshoe orbit, all the known Trojan
asteroids of Jupiter are found in the tadpoles, but with more found in the
leading tadpole around L4. This is probably a clue to something, but
what?!
Figure 5 suggests that there is a lot more to learn about Trojan
points and their tadpole and horseshoe orbits.
The Earth “companion” asteroid,
3753 Cruithne, is in a very unusual horseshoe orbit
(Earth relative),
one that is inclined at an extreme angle to the orbit of Earth around the
Sun. Cruithne’s orbit, which takes about 770 years to complete and
occasionally takes it almost directly above Earth, indicates that
the 2 dimensional study seen in
Figure 5 does not give the whole picture, by any means. In fact, this
orbit seems to be a combination of a modified horseshoe orbit with
“retrograde satellite motion” — one of several new classes of co-orbital
motion found recently by Fathi Namouni, Apostolos Christou, and Carl
Murray of Queen Mary and Westfield
College in London (and probably other colleagues) — in which an asteroid
slowly orbits a planet at a great distance, perhaps even half the distance
between the planet and the Sun. (For more, see
http://focus.aps.org/v4/st16.html.) Fascinating...
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4.3
Stability?!
The question of orbital stability comes up in relation to Trojan
asteroids. Our whole concept of such stability has changed drastically
since Lagrange.
He predicted that the Trojan asteroids would “stay close” to the Trojan
points, but he did not mention that they could be relatively stable in the
exaggerated tadpole orbits and even more exaggerated horseshoe orbits that
are now being studied. Astronomers now consider them to be orbitally
stable if they are in tadpole or horseshoe orbits.
And astronomers are also studying asteroids that enter such an orbit,
bounce around the horseshoe for a few thousand years, then exit. They, too, are now
considered orbitally “stable”, at least when they are “in” such an orbit.
Our whole concept of orbital stability is — or should be — currently changing
rapidly. This may open
the door to a renewed study of rejected possibilities, even the
previously ridiculed (as impossible because “unstable”) “alternate/mirror image
Earth theory” (the “alternate Earth” that remains permanently invisible — from the Earth
— since it’s opposite the Earth on the other
side of the Sun. This is more humorous than likely since e.g. the
gravitational effects would probably have been seen in the orbit of
Mercury, but studying it with a new concept of stability might yield
insights. After all, the same people who ridicule the idea have... oversighted
the falling rate difference of lighter and heavier bodies for over 300
years.
A possibility that is more likely, of course,
is that e.g. “stable” ternary star
systems will be accepted theoretically, and eventually discovered, even
ones not within the
limits/restrictions calculated by
Lagrange. (See
Figure 6a and
Figure 6b,
and Trojan Point
Astronomy in the 21st Century.)
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SECTIONS
4.1 Trojan Asteroids in
Tadpole Orbits
4.2 Trojan Asteroids in
Horseshoe Orbits
4.3 Trojan Space Debris?! Trojan Atmospheres?!
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4.4 Trojan Space Debris?!
Trojan Atmospheres?!
We know that there are hundreds — and probably
thousands — of Trojan asteroids large enough to see by telescope
associated with Jupiter’s L4 and L5. It should be obvious that low
relative-velocity bodies of much smaller size will also tend to become
trapped in the “concentric” tadpole orbits, which form a valley
surrounding a Trojan point, so that they orbit the Trojan point. Viscosity
or drag from the tenuous Solar System atmosphere and from other “space
debris” will cause these orbits to decay extremely slowly, but reasonably
surely.
Perhaps this needs further explanation. Despite the
fact that we think in terms of “empty space, there is actually an
atmosphere (non-zero partial pressure) and even much space debris in our
Solar System. Matter comes from the solar wind, and there is a whole range
of plasmic, atomic, molecular and particulate matter ranging up to
official asteroid and even planet size that has a small but non-zero
viscosity.
This viscosity can act over extremely long periods of
time to slow even larger asteroids, so their orbits around a Trojan point
inevitably decay. This atmosphere and this space debris is also in some
orbit or other, so it may in fact be traveling with a Trojan body and not
affecting it viscously, but there will tend to be much more in some other
orbit that does. The collisions will by and large not be perfectly
elastic, far from it, and even if they are, the orbit will eventually
decay entropically. Bodies will from time to time be ejected when given
enough energy by the other bodies there to attain escape velocity, leaving
the bodies with which it interacted with lower energies and
decaying orbits. Perturbations from planets could also cause bodies to
escape the tadpole.
The tadpole orbits around Trojan points will not only
tend to trap low energy space debris, they are even capable of maintaining
an atmosphere. A low temperature, low pressure atmosphere will tend to
collect and be held there, even without the actual presence of a larger
gravitational body at the Trojan point proper, or orbiting in the tadpole.
So the study of Trojan bodies overall is far from simple.
It is almost certainly worth sending space probes to
the Trojan tadpoles of the Earth-Moon system. They are roughly the same
distance away as the moon, extremely close by space exploration standards,
much closer than the asteroid EROS to which we sent the NEAR space
probe at a cost of $224 million. (The Near Earth Asteroid Rendezvous
mission was renamed NEAR-Shoemaker in honor of the late astronomer, Gene
Shoemaker.) It could probably be done with noticeably less expense. The
atmosphere could be studied, and debris would be concentrated and much
easier and cheaper to find there and bring back to Earth than in space in
general.
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