DDA 2015 – p-ellipse Orbit Approximations, Lindblad Zones, and Resonant Waves in Galaxy Disks

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.

Curtis Struck (Iowa State)


p-ellipses are simple, yet very accurate formulae for orbits in power-law potentials, like those approximating galaxy disks. These precessing elliptical orbits reveal important systematics of orbits in such potentials, including simple expressions for the dependence of apsidal precession on eccentricity, and the fact that very few terms (or parameters) are needed for the approximation of even nearly radial orbits. The orbit approximations are also useful tools for addressing problems in galaxy dynamics. In particular, they indicate the existence of a range of eccentric resonances associated with the usual, near-circular Lindblad resonances. Collectively these change an isolated Lindblad resonance to a Lindblad Zone of eccentric resonances. A range of these resonances could be excited at a common paRern speed, aiding the formation of a variety of bars and spirals, out of eccentric orbits. Such waves would be persistent, and not wind up or disperse, since differences in their precession frequencies offset differences in the circular velocities at the radii of their parent orbits. The p-ellipse approximation further reveals how a non-axisymmetric component of the gravitational potential (e.g., due to bar self-gravity) significantly modifies precession frequencies, and similarly modifies the Lindblad Zones.


  • Precessing ellipses
    $\dfrac{1}{r} = \dfrac{1}{p}\left[1+e \cos (m \phi)\right]^{\frac{1}{2}+\delta}$
    yield good fits for orbits in different potentials
  • Apsidal precession:
    $\Delta \phi = \dfrac{\pi}{\sqrt{2(1-\delta)}}$
    $\rightarrow$ kinematic waves, bars
  • Nearly radial orbits: p-ellipse fits not so good
  • But can tweak to get good fits:
    • Fit to the extremal radii, not the position
    • Note that apsidal precession is not constant but depends on eccentricity
    • Add harmonic terms:
      $\dfrac{1}{r} = \dfrac{1}{p}\left[1+e_1 \cos (m \phi)+e_2 \cos (2 m \phi)\right]^{\frac{1}{2}+\delta}$
  • Example of (possibly) kinematic counter-rotating waves: NGC 4622
  • Summary
    • Accurate, simple approximations for orbits in a range of potentials
    • Can be extended to radial orbits
    • Turns Lindblad resonances into Lindblad zones
    • Allows formation of persistent kinematic waves of various types
      • but usually requires fine tuning

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