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SECTIONS
HOMOGRAPHIC SOLUTIONS
Asteroid 3753 CRUITHNE
SOHO
L3 STABILITY?! |
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(See
Figure 4
for diagram of Langrangian/Trojan points.)
WARNING
NOTE: if you have spent much time
searching the Internet for info on Lagrangian points, you have probably
been confused by the lack of consistency in the numbering of the points.
About the only consistency that you will find there is that there is
agreement that L1-L3 are the collinear points, and L4-L5 are the
equilateral triangle points. Sigh... If it helps, the numbering used here is
the same as that found in e.g. the
Encyclopedia of the Solar System (hyperlink
unfortunately no longer seems valid), pp. 815-7; Weissman, McFadden,
Johnson, Eds.; Academic Press, 1999. For an example of alternate numbering
(with a pedigree), see
http://www.astro.queensu.ca/~wiegert/etrojans/etrojans.html (which
unfortunately no longer seems valid).
Using his perturbation theory (a major extension of differential
equations), Lagrange found 5 points — referred to collectively as “LAGRANGIAN
POINTS” — where an “infinitesimal” body
would theoretically maintain its position relative to the 2
non-infinitesimal bodies as they all move through space (only if
completely unperturbed in the case of L1,
L2, and L3;
all 3 of these are considered positions of “unstable equilibrium”). These
are the “homographic solutions” to the equations of motion.
HOMOGRAPHIC
SOLUTIONS are (here) solutions of the equations of motion that
retain the same shape — but not necessarily the same size or orientation
— as the system evolves through time. I.e. the ratios of the distances
between each pair of points remain the same. In particular, for the (only)
solutions involving
non-collinear points, the equilateral triangle formed by the 3 points
remains geometrically similar to its initial configuration; i.e. it
remains an equilateral triangle as it rotates, expands and contracts.
Three of these solutions, L1,
L2, and L3
are collinear — i.e. all 3 bodies lie on a straight line as they move
through space, revolving around their common center of gravity. The other
two, L4 and L5,
are the TROJAN POINTS at the
equilateral triangle positions (which we have just found — incompletely
—
by questioning Galileo’s hypothesis that lighter and heavier bodies fall
at the same rate). Asteroids that orbit these points are known as “TROJAN
ASTEROIDS”. Jupiter’s
Trojan asteroids can take hundreds of years to complete such an orbit.
(They are mostly in “tadpole” orbits, but probably at least some in
“horseshoe” orbits. See below.)
Although L1-L3 are theoretically stable if “unperturbed” (this is the
meaning of the homographic solution result), they are known to be unstable
in fact (or thought to be; see
below). Infinitesimal bodies placed there will eventually “wander
away” under the influence of “perturbations” induced by e.g. the
gravitational influence of other planets.
The points
L4 and L5 are considered stable, i.e. if the perturbations are small
enough, infinitesimal bodies placed there will stay near the Trojan points
in relatively stable “TADPOLE ORBITS”
(see
Figure 5), elongated, non-elliptical, asymmetrically curved orbits. If
the perturbations on Trojan bodies are large enough and concerted enough,
their tadpole orbits might even “grow” in such a way that (mixing
metaphors) the L4 and L5 tadpoles meet and form still stable “HORSESHOE
ORBITS” (as seen in the reference frame
of e.g. the Earth). (See
Figure 5.) The more energy the 3rd body has, the larger the
tadpole or horseshoe orbit (and the easier it might be to perturb it
sufficiently for it to escape).
Figure 5 shows these “concentric” tadpole and horseshoe orbits.
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SECTIONS
HOMOGRAPHIC SOLUTIONS
Asteroid 3753 CRUITHNE
SOHO
L3 STABILITY?! |
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Asteroid
3753 CRUITHNE
is an example of an “Earth companion”, a body in a very peculiar horseshoe
orbit relative to Earth. As of this writing, two of the
web sites that give interesting info about Cruithne are:
http://aries.phys.yorku.ca/~wiegert/3753.html
(which unfortunately no longer seems valid) and
http://focus.aps.org/v4/st16.html
SOHO
— L1
has become famous because that is where SOHO, the Solar and Heliospheric
Observatory, is stationed. Due primarily to the Earth-Sun mass ratio, L1
is about 1% closer to the Sun than the Earth, so about 930,000 miles
sunward from Earth. (L2 is about 1% further away from the Sun.) The lack
of stability means that it must use fuel to keep itself sufficiently near
L1. Actually it orbits L1 in what is called a “Halo Orbit”. (NASA has web
pages that give interesting details.) This special orbit actually keeps it
somewhat away from L1, which is in direct line of sight with the
Sun, i.e. enough away from solar interference to send data back. Roughly
every 27 days SOHO must readjust its orbit.
L3
is almost precisely the same distance away from the Sun as Earth
(approximately millionths of a percent difference), and on the opposite
side. This is because the gravitational effect of Earth is very small at
twice its distance to the Sun. L3 is the place where science fiction has
placed some of its alternate Earths, but scientists are sure that
this is not feasible because, as with SOHO in L1, its instability means
that a body (at least an “infinitesimal” body) would drift away from L3 if
energy were not expended to keep it there. But let us remind ourselves:
scientists are sure about this in the same way they have been
sure for almost 400 years that lighter and heavier bodies fall at
precisely the same rate!
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SECTIONS
HOMOGRAPHIC SOLUTIONS
Asteroid 3753 CRUITHNE
SOHO
L3 STABILITY?! |
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L3 STABILITY?!:
By
the way, you may have noted that the L3 point is in the “band” of the
“concentric” horseshoe orbits, and that this might technically make it a
point of “stable” equilibrium since the tadpole orbits are considered part
of the stability of L4 and L5, and they expand and merge into the
horseshoe orbits. Good for you! Science — and mathematics — need to have
their inconsistencies pointed out. (Now, what about L1 and L2?!...)
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