DDA 2015 – Instabilies in Near-Keplerian Systems

This is one of a series of notes taken during the 2015 meeting of the AAS Division on Dynamical Astronomy, 3-7 May, at CalTech. An index to this series (all the papers presented at the meeting) is here.

Anne-Marie Madigan (UC Berkeley) (invited)


Closed orbits drive secular gravitational instabilities, and Kepler potentials are one of only two potentials in which bound orbits are closed. Though the Kepler potential is common in astrophysics — relevant for stars orbiting massive black holes in the centers of galaxies, for planets orbiting stars, and for moons orbiting planets — few instabilities have been explored beyond the linear regime in this potential. I will present two new instabilities which grow exponentially from small initial perturbations and act to reorient eccentric orbits in near-Keplerian disks. The first results from forces in the plane of the disk and acts to spread orbits in eccentricity. The second instability results from forces out of the disk plane and drives orbits to high inclination. I will explain the dynamical mechanism behind each and make observational predictions for both planetary systems and galactic nuclei.


  • Why Kepler potentials?
    • Only two potentials yield closed orbits: $\psi \sim -\frac{1}{r}$, $\psi \sim r^2$
    • More general, richer dynamics (than quadratic potentials)
  • Eccentric disk instability
    • Madigan 2009
    • Galactic center vs. Andromeda nucleus
      • single peak vs. double peak (in luminosity)
      • Presence/absence of nuclear star cluster changes direction of apsidal precession.
      • Andromeda: apsidally aligned orbits $\rightarrow$ double luminosity peak
    • Prograde precession case
      • torque from disk grav. reducesang. momentum, increasing eccentricity.
        • produces oscillations
        • but stable disks
      • Andromeda
    • Retrograde precession:
  • Inclination instability
    • Madigan 2015
    • Thick disk of stars in Galactic center
    • Dwarf planets (inner Oort Cloud)are clustered in $\omega$. How?
      • $\cos \omega = \dfrac{\sin i_a}{\sin i_b}$ (inclinations wrt major and minor axes)
      • Dwarf planets are clustered in $\omega$ because high eccentricity orbits in a disk are unstable.
    • In Galactic center, ~80% of young stars are not in disk plane (Yelda 2014). How did they get there?
    • Set by initial inclinations.
    • Two-body diffusion stage, then ~sudden instability.
    • Instability grows exponentially.