1. Secular, Kozai, mean-motion resonances D.N.C. Lin Department of Astronomy & Astrophysics University of California, Santa Cruz Lecture 4, AY 222 Apr 11th, 2012

2. A system of N planets

3. Normal modes 7au 9.7 au

4. Secular resonance Additional contributor to precession : disk self gravity It is possible for =0 => e can increase monotonically to large amplitudes. for a finite  lock-step precession & Sweeping secular resonance 18/24

5. Kozai resonance in inclined systems When => 19/24

6. High order terms and tidal contribution 20/24

7. Mean motion resonances Two secular frequencies: periastron precession and conjunction drift. A Hamiltonian approach: dq i /dt=  H/  p i dp i /dt=-  H/  q i Energy (=a) as well as Angular momentum exchange 21/24

8. Eccentricity excitation and transition to chaos 22/24 Consider N equal M  a planets with e excitation => nonlinear effect=> chaos

9. Overlapping resonances & dynamical instability Dynamical filling factor & gas damping 23/24

10. Many other competing forces Orbital changes may be due to: 1) Mass accretion and potential change 2)Planet-disk interaction, 3)Planet-magnetosphere interaction, 4)Planet-star tidal interaction, 5)Stellar radiation & wind, 6)Planetesimal scattering, 7) Planetary mass loss, 8)Perturbation by binary and field stars, 9)Higher order contributions 24/24