Abstract
Motivated by the dynamics of resonance capture, we study numerically the coorbital resonance for inclination \(0\le I\le 180^\circ \) in the circular restricted three-body problem. We examine the similarities and differences between planar and three dimensional coorbital resonance capture and seek their origin in the stability of coorbital motion at arbitrary inclination. After we present stability maps of the planar prograde and retrograde coorbital resonances, we characterize the new coorbital modes in three dimensions. We see that retrograde mode I (R1) and mode II (R2) persist as we change the relative inclination, while retrograde mode III (R3) seems to exist only in the planar problem. A new coorbital mode (R4) appears in 3D which is a retrograde analogue to an horseshoe-orbit. The Kozai–Lidov resonance is active for retrograde orbits as well as prograde orbits and plays a key role in coorbital resonance capture. Stable coorbital modes exist at all inclinations, including retrograde and polar obits. This result confirms the robustness the coorbital resonance at large inclination and encourages the search for retrograde coorbital companions of the solar system’s planets.
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Notes
We use standard Keplerian elements for the test particle’s orbit: a (semi-major axis), e (eccentricity), I (inclination), f (true anomaly), \(\omega \) (argument of pericentre), \(\Omega \) (longitude of pericentre), M (mean anomaly), \(\lambda \) (mean longitude). The planet’s circular orbit has semi-major axis \(a_\mathrm{J}=1\) and mean longitude \(\lambda _\mathrm{J}\).
This is equivalent to obtaining a retrograde configuration of mutual inclination \(I>90^\circ \) from a prograde configuration of mutual inclination \(I<90^\circ \) by inverting the motion of the planet with respect to the star.
The chaotic region due to the cumulative effect of disruptive close encounters with the planet.
Also known as quasi-satellite orbits (Mikkola et al. 2004). These have clockwise motion around the planet in the synodic frame and are not true satellites since the orbits are bound to the star.
If all bodies are treated as point-like objects and evolution is allowed to continue, no collision occurs. Instead orbital reversal takes place (Yu and Tremaine 2001).
The 1:1 resonance Hamiltonian is invariant with respect to the transformation \((\phi ,-\omega )\rightarrow (-\phi ,\omega )\). The CR3BP is invariant with respect to the transformation \(\omega \rightarrow \omega +180^{\circ }\) and with respect to changes in \(\Omega \).
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We thank Nelson Callegari Jr. for assistance with computational resources.
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Morais, M.H.M., Namouni, F. A numerical investigation of coorbital stability and libration in three dimensions. Celest Mech Dyn Astr 125, 91–106 (2016). https://doi.org/10.1007/s10569-016-9674-3
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DOI: https://doi.org/10.1007/s10569-016-9674-3