Planetary Dynamics Ge/Ay133

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

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 ↓

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

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…

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.

Might this explain some of the unusual exo-planets? Simulation of two-planet interactions, with slow outer planet migration: HD 12661 http://astron.berkeley.edu/~echiang/

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

More recent analyses suggest eUpsAndc is variable!

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