In principle, Fig. 5, central panel G, shows that (469219) 2016 HO3 may have been locked in a Kozai–Lidov resonance with ω librating about 270° for nearly 100 kyr and probably more. Because of the Kozai–Lidov resonance, both e (central panel E) and i (central panel F) oscillate with the same frequency but out of phase (for a more detailed view, see Fig. 4, panels E and F); when the value of e reaches its maximum the value of i is the lowest and vice versa ($\sqrt{1 - e^2} \cos i \sim$ constant, see Fig. 4, panel B). During the simulated time and for the nominal orbit, 469219 reaches perihelion and aphelion the farthest possible from the ecliptic. Fig. 5, G-panels, show that for other incarnations of the orbit of 469219, different from the nominal one, ω may librate about 90° as well during the simulated time interval. However, is this a true Kozai–Lidov resonance? Namouni (1999) has shown that the secular evolution of co-orbital objects is viewed more naturally in the erωr-plane, where er and ωr are the relative eccentricity and argument of perihelion computed as defined in Namouni's work (see equations 3 in Namouni 1999); these are based on the vector eccentricity and the vector inclination. Fig. 6 shows the multi-planet erωr-portrait for the nominal orbit of this object. It clearly resembles figs 13 and 19 in Namouni (1999). Asteroid 469219 librates around $\omega _{\rm r}=-90{^\circ }$ for Venus, the Earth, and Jupiter. This behaviour corresponds to domain III in Namouni (1999), horseshoe-retrograde satellite orbit transitions and librations (around $\omega _{\rm r}=-90{^\circ }$ or 90°). For a given cycle, the lower part corresponds to the horseshoe phase and the upper part to the quasi-satellite or retrograde satellite phase. This is not the Kozai–Lidov resonance; in this case, the Kozai–Lidov domain (domain II in Namouni 1999) is characterized by libration around $\omega _{\rm r}=0{^\circ }$ (or 180°) which is only briefly observed at the end of the backwards integrations (see Fig. 6). The Kozai–Lidov resonance is however in action at some stage in the orbits displayed in Figs 5 and 8. Our calculations show that the orbital evolution followed by 469219 is the result of the dominant secular perturbation of Jupiter as the periodic switching between co-orbital states ceases after about 8 kyr if Jupiter is removed from the calculations. Fig. 7 shows that, without Jupiter, 469219 switches between the Kozai–Lidov domain and that of horseshoe-quasi-satellite orbit transitions and librations (including both −90°and 90°). Jupiter plays a stabilizing role in the dynamics of objects following orbits similar to that of 469219. It is not surprising that Jupiter instead of the Earth or Venus is acting as main secular perturber of 469219. Ito & Tanikawa (1999) have shown that the inner planets share the effect of the secular perturbation from Jupiter; in fact, Venus and our planet exchange angular momentum (Ito & Tanikawa 2002). In their work, these authors argue that the inner planets maintain their stability by sharing and weakening the secular perturbation from Jupiter. Tanikawa & Ito (2007) have extended this analysis to conclude that, regarding the secular perturbation from Jupiter, the terrestrial planets form a collection of loosely connected mutually dynamically dependent massive objects. The existence of such planetary grouping has direct implications on the dynamical situation studied here; if Jupiter is removed from the calculations, the overlapping secular resonances and the recurrent dynamics disappear as well.
Citation: Namouni (1999)
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Citation: Namouni (1999)
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