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8 FIGURES
(cont.)
Figure 4. The Lagrangian Points L1-L5.
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).
Lagrange’s theory assumes that L1-L5 are occupied only by “infinitesimal”
bodies (purely a calculational convenience, since no masses are truly
“infinitesimal”). Here the diagram assumes that
m <<
M.
NOTE: technically, the Lagrangian points L4 and L5 do not even exist unless
Lagrange’s restriction is met, i.e. that
m < ~0.04 M.
The relative positions of L1 and L2 (and even L3, but far less so) with
respect to the two non-infinitesimal masses depend on the mass ratio,
m/M,
but L4 and L5 always form equilateral triangles with
m and
M.
NOTE: it is implicit in the above diagram that the mass
m is
much smaller than mass M, i.e. the center of mass of both
masses together is very close to the center of mass of
M by itself.
This is the case with the Sun and Jupiter. However, the more general case
needs to be studied, and can be with the aid of computers.
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