1. Planetary Dynamics
Ge/Ay133

3. Orbital elements (3-D),
& time evolution:

24. What ARE Lyapounov exponents and times?
Suppose that two orbits are separated in phase space by d, and that d follows
d = d0 e-g(t-t0)
G is the Lyapounov exponent, and 1/g is the Lyapounov time. Numerically,
G = lim ln(d/d0)/(t-t0)
As t → ∞. For regular orbits, g → 0.
Regular ↑
Chaotic ↓

25. Libration Rotation
Chaos in the outer S.S.:
Uranus’s orbit can go from librating to rotating behavior. Chaotic, but

26. The exact details are sensitive to the presence or
absence of remnant gas or planetesimal disks:
As disks dissipate, a stable system can become unstable…

27. Recall: Resonant systems common, & migrate more slowly in a disk…
From Armitage. Why? Couple of reasons, including wider gap opening and
exchange of momentum between the interacting planets & disk.

28. Might this explain some of the unusual exo-planets?
Simulation of two-planet interactions, with slow outer planet migration:
HD 12661

29. Are HD 12661 and/or Ups And “apsidally locked”?

30. More recent analyses suggest eUpsAndc is variable!

31. Need an impulsive scatterer. As disk disperses,
lose planet?