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 .
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
Hence, deduce that
Finally, show that the Jacobi constant in the co-rotating frame,
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 , 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 , 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 .