DDA 2015 – The family of Quasi-satellite periodic orbits in the co-planar RTBP

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.

Session: New Approaches to Classical Dynamical Problems II

Alexandre Pousse (IMCCE – Observatoire de Paris)


In the framework of the Restricted Three-body Problem (RTBP), we consider a primary whose mass is equal to one, a secondary on circular or eccentric motion with a mass ε and a massless third body. The three bodies are in coplanar motion and in co-orbital resonance. We actually know three classes of regular co-orbital motions: in rotating frame with the secondary, the tadpole orbits (TP) librate around Lagrangian equilibria $L_4$ or $L_5$; the horseshoe orbits (HS) encompass the three equilibrium points $L_3$, $L_4$ and $L_5$; the quasi-satellite orbits (QS) are remote retrograde satellite around the secondary, but outside of its Hill sphere.

Contrarily to TP orbits which emerge from a fixed point in rotating frame, QS orbits emanate from a one-parameter family of periodic orbits, denoted family-f by Henon (1969). In the averaged problem, this family can be understood as a family of fixed points. However, the eccentricity of these orbits can reach high values. Consequently a development in eccentricity will not be efficient. Using the method developed by Nesvorný et al. (2002) which is valid for every values of eccentricity, we study the QS periodic orbits family with a numerical averaging.

In the circular case, I will present the validity domain of the average approximation and a particular orbit. Then, I will highlight an unexpected result for very high eccentricity on families of periodic orbits that originate from $L_3$, $L_4$ and $L_5$. Finally, I will sketch out an analytic method adapted to QS motion and exhibit associated results in the eccentric case.


  • Quasi-satellite motion (QS):
    • retrograde motion outside Hill sphere
    • co-orbital motion with the secondary libration around $\theta = \lambda\, – \lambda_{pl}$ (heliocentric coords)
    • Sidorenko et al 2013, Christou 2000, Kinoshita & Nakai 2007, etc.
  • Model:coplanarRTBP + averaging (to $2^{nd}$ order)
    • 2 DoF, in co-orbital resonance config
  • Circular case:
    • rotation symmetry $\rightarrow \Gamma = (1+u)(1-\sqrt{1-e^2}) = const.$
    • projection in $u-\theta$ plane then captures the dynamics
    • classic hyperbolic & elliptic fixed points, stable & unstable separatrices, chaotic regions
    • plot: frequencies vs $e_0$
    • “frozen ellipse”: $e_0 = 0.8352$
    • bifurcation, appearance of libration around $\theta = \pi$
      • Man Hoi Lee: 2 exoplanets exhibit this (Laughlin & Chambers)

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