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 time-scale 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 of both mechanisms and prove that the retrograde resonance overlap, raised by great inequality, cannot serve as an explanation for the instability of the retrograde co-orbitals, due to the weakness of the retrograde 2:5 resonance with Jupiter at low eccentricity. However, we discover that both ν₅ and ν₆ secular resonances contribute to the slow growth of the eccentricity and are therefore possibly the primary causes of the instability inside Saturn’s retrograde co-orbital space.