An Aside: ........ Resonances
Two types:

• Mean motion resonances: Between orbital periods.
• Secular resonances: Between precession periods.

---

Secular resonances:
Simplest System: ........ 2D with 2 planets.

• Model planets as elliptical rings.
  
  
• Two elliptical Rings will exert a torque on one another.
  • They precess.
  • Precession periods range from 50,000 to 2 million years in outer solar system.
  
  
• Include a massless particle and it will also precess.
  
  
• If particle precesses as same rate as a planet
  • Secular Resonance .
  • Angular momentum can transfer because the particle sees a non-axisymmetric potential.
  
  
• These can be very strong.

Also can happen with the nodes.

Notation: is periapse resonance, is nodal resonance, where X refers to a frequency (planet).