where all lengths are normalized to the major radius of Jupiter.
where
where
Demonstrate that
Hence, deduce that
Show that
where
Demonstrate that
where
and
Hence, deduce that
Finally, show that the Jacobi constant in the co-rotating frame,
transforms to
in the inertial frame.
where
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|
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Thus,
and, hence, that
Show that the general solution to the preceding dispersion relation is a linear combination of two normal modes of oscillation, and that the higher frequency mode takes the form
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and | ![]() |
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and
Demonstrate that (in the co-rotating reference frame) the second normal mode takes the form
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and | ![]() |
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and
Finally, consider a Trojan asteroid trapped in the vicinity of the
point of the Sun-Jupiter system. Demonstrate
that the libration period of the asteroid (in the co-rotating frame) is approximately
years, whereas its perihelion precession period
(in the inertial frame) is approximately
years. Show that, in the co-rotating frame, the libration orbit is
an ellipse that is elongated in the direction of the tangent to the Jovian orbit in the ratio
.