where all lengths are normalized to the major radius of Jupiter.
where , and all lengths are normalized to the major radius of Jupiter.
where is the column vector of the co-rotating coordinates, is the column vector of the inertial coordinates, and
Demonstrate that , where denotes a transpose, and
Hence, deduce that , or .
Show that
where is the column vector of the time derivatives of the co-rotating coordinates, is the column vector of the time derivatives of the inertial coordinates, and
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 is the value of the Jacobi constant, and and are the distances to the primary and secondary masses, respectively. The critical zero-velocity curve that passes through the point, when , has two branches. Defining polar coordinates such that and , show that when the branches intersect the unit circle at and . (Modified from Murray and Dermott 1999.)
Thus, parameterizes displacements from that are tangential to the unit circle on which the mass , and the , , and points, lie, whereas parameterizes radial displacements. Writing and , where , , are constants, demonstrate that
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
and |
and , are arbitrary constants. Demonstrate that, in the original inertial reference frame, the addition of the preceding normal mode to the unperturbed orbit of the tertiary mass (in the limit ) converts a circular orbit into a Keplerian ellipse of eccentricity . In addition, show that the perihelion point of the new orbit precesses (in the direction of the orbital motion) at the rate
Demonstrate that (in the co-rotating reference frame) the second normal mode takes the form
and |
and , are arbitrary constants. This type of motion, which entails relatively small amplitude radial oscillations, combined with much larger amplitude tangential oscillations, is known as libration.
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 .