The secular approximation for the evolution of hierarchical triple configurations has proven to be very useful in many astrophysical contexts, from planetary to triple-star systems. Many gravitational triple systems are in a hierarchical configuration, two objects orbit each other in a relatively tight inner binary while the third object is on a much wider orbit. If the third object is sufficiently distant, an analytic, perturbative approach can be used to calculate the evolution of the system. In the secular approximation , the two orbits torque each other and exchange angular momentum, but not energy. Therefore the orbits can change shape and orientation (on timescales much longer than their orbital periods), but not semimajor axes. For example, for highly inclined triple systems, the Kozai-Lidov mechanism can produce large-amplitude oscillations of the eccentricities and inclinations.
In Naoz et al (2011a,b) we showed that the secular hierarchical triple body problem is qualitatively different from what was considered previously. Specifically, we showed that the inner orbit's can flip from prograde to retrograde and back, and can also reach extremely high eccentricities.
The movie shows the eccentric Kozai dynamical evolution of a hierarchical triple system. The inner orbit has a solar mass star and a test particle. The perturber (Jupiter mass planet in the example in the movie) lays in the horizontal plane. The two top panels show the evolution of the inner's orbit eccentricity (as 1-e) and the mutual inclination as a function of the inner orbit period (over 10000). The projection on the y-z plan show the projected normal to the orbit. After the orbit flip the normal to the orbit change orientation (and the ring is colored red). In the prograde state the ring color is green. The SMA of the inner orbit is 2AU, but I have removed the scale from the 3D plot because this behavior is typical.
Note that although it seems as if the ellipse is ''floating'' in space its focal point is actually constant and this effect is due to the angle of the camera.
Kozai (1962) studied the effects of Jupiter's gravitational perturbation on an inclined asteroid in our own solar system. In the assumed hierarchical configuration, treating the asteroid as a test particle, Kozai (1962) found that its inclination and eccentricity fluctuate on timescales much larger than its orbital period. Jupiter, assumed to be in a circular orbit, carries most of the angular momentum of the system. Due to Jupite's circular orbit and the negligible mass of the asteroid, the system's potential is axisymmetric and thus the component of the inner orbit's angular momentum along the total angular momentum is conserved during the evolution.More soon....