1. Ge/Ay133 Planetary Dynamics

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 = d 0 e -  (t-t0) G is the Lyapounov exponent, and 1/  is the Lyapounov time. Numerically, G = lim ln(d/d 0 )/(t-t 0 ) As t → ∞. For regular orbits,  → 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. Simulation of two-planet interactions, with slow outer planet migration: HD 12661 Might this explain some of the unusual exo-planets? http://astron.berkeley.edu/~echiang/

28. Are HD 12661 and/or Ups And “apsidally locked”? http://astron.berkeley.edu/~echiang/

29. More recent analyses suggest e UpsAndc is variable!

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