DDA 2015 – Loners, Groupies, and Long-term Eccentricity Behavior – Insights from Secular Theory

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.

Christa Van Laerhoven (CITA)


Considering the secular dynamics of multi-planet systems provides substantial insight into the interactions between planets in those systems. Secular interactions are those that don’t involve knowing where a planet is along its orbit, and they dominate when planets are not involved in mean motion resonances. These interactions exchange angular momentum among the planets, evolving their eccentricities and inclinations. To second order in the planets’ eccentricities and inclinations, the eccentricity and inclination perturbations are decoupled. Given the right variable choice, the relevant differential equations are linear and thus the eccentricity and inclination behaviors can be described as a sum of eigenmodes. Since the underlying structure of the secular eigenmodes can be calculated using only the planets’ masses and semi-major axes, one can elucidate the eccentricity and inclination behavior of planets in exoplanet systems even without knowing the planets’ current eccentricities and inclinations. I have calculated both the eccentricity and inclination secular eigenmodes for the population of known multi-planet systems whose planets have well determined masses and periods. Using this catalog of secular character, I will discuss the prevalence of dynamically grouped planets (‘groupies’) versus dynamically uncoupled planets (‘loners’) and how this relates to the exoplanets ‘long-term eccentricity and inclination behavior. I will also touch on the distribution of the secular eigenfreqiencies.


  • Secular character of multi-planet system
  • planet-planet interactions
  • only need masses and semimajor axes (not eccentricity, not inclination) to set secular structure
  • two-planet system: two eccentricityeigenmodes
    • $h = e \sin \varpi$, $k = e \cos \varpi$ plot: $e$ is a vector
    • each $e$ vector is the sum of two eigenvectors
  • 3-planet system: “groupie”-ness and loners
    • groupies:
      • $e$ highly variable
      • $\varpi$ precession not uniform
    • loners:
      • $e$ does not vary by much
      • $\varpi$ precesses steadily
  • Kepler-11
    • outer planet is a loner — does not interact with others
    • 5 inner planets are groupies — interact strongly with each other
  • Summary: most planets are “groupies”, “loners” are rare.

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