On the Instability of Saturn's Hypothetical Retrograde Co-orbitals

We find an interesting fact that fictitious retrograde co-orbitals of Saturn, or small bodies inside the retrograde 1:1 resonance with Saturn, are highly unstable in our numerical simulations. It is shown that in the presence of Jupiter, the retrograde co-orbitals will get ejected from Saturn's co-orbital space within a timescale of 10 Myr. This scenario reminds us of the instability of Saturn Trojans caused by both the Great Inequality and the secular resonances. Therefore, we carry out in-depth inspections on both mechanisms and prove that the retrograde resonance overlap, raised by Great Inequality, cannot serve as an explanation for the instability of retrograde co-orbitals, due to the weakness of the retrograde 2:5 resonance with Jupiter at a low eccentricity. However, we discover that both $\nu_5$ and $\nu_6$ secular resonances contribute to the slow growth of the eccentricity, therefore, are possibly the primary causes of the instability inside Saturn's retrograde co-orbital space.