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Every planet apart from Mercury and Saturn has trojans in its L4 and/or L5 points.

Mercury is easy enough to explain: It is small, has an eccentric orbit that precesses, and any trojans it has would be heavily perturbed by the Sun's gravity.

However Saturn doesn't have any of those complications. The answer given to this question is that Jupiter's gravity perturbs any trojans at Saturn's L4 and L5 points.

But if Mars, which is much less massive than Saturn and comes closer to Jupiter, can have multiple trojans then there's no reason why Saturn shouldn't.

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They are pulled out of stable orbits by Jupiter.

The details are in https://ui.adsabs.harvard.edu/abs/2012jsrs.conf..225B/abstract

Full text https://syrte.obspm.fr/jsr/journees2011/pdf/baudisch.pdf

The planet Jupiter is solely responsible for the hole of instability for short time integrations ($T < 10^{7}$) compared to the age of the planetary system. On the long term scale this planet also destabilizes the whole region around the Saturnian libration points. If we find in the future Trojans of Saturn, these Trojans could only be captured asteroids, in orbits in the 1:1 MMR for a short time

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  • $\begingroup$ How come Mars' trojans aren't equally as destabilized by Jupiter's gravity then? Mars comes closer, and is on average closer, to Jupiter than Saturn. $\endgroup$
    – user177107
    Nov 9 '20 at 18:56
  • $\begingroup$ Because they are not or else they are short term captures. The paper shows that the only way this was discovered was by brute force numerical integration of the laws of motion. You can't deduce this from any general principle, only by simulation. $\endgroup$
    – James K
    Nov 9 '20 at 19:39
  • $\begingroup$ @JamesK there was a huge amount of mathematical work on the stability of three-body orbits and perturbation theory before numerical techniques were available. Just because people use numerical extensively now does not alone prove that there is no other way to do it. It just means that nobody wants to anymore. $\endgroup$
    – uhoh
    Nov 11 '20 at 0:10
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The answer is complex but likely relates to the properties of (near-)resonances in the Solar System, which can stabilise or destabilise orbits at quite a long range.

As noted in James K's answer, perturbations from Jupiter are the main reason for the absence of Saturnian Trojans. While the exact process is complex, one major factor appears to be Laplace's "Great Inequality": the near-5:2 resonance between Jupiter and Saturn. As remarked in de la Barre et al. (1996) (who use the term "Bruin" to describe Saturnian Trojans), this relationship is a major factor in the dynamics:

We numerically integrated various Bruin orbits using different Solar System models to develop a Hamiltonian perturbation theory for low-inclination Bruin orbits. Although only at the beginning stages of development, the theory already identifies three separatrices of Bruin motion due in part to the Great Inequality (GI) between Jupiter and Saturn. These GI separatrices are a major contributor to the unstable region near Saturn's L4 and L5 points.

In contrast, the Martian Trojan points can support orbits with lifetimes comparable to the age of the Solar System. Mars and Jupiter are not in a Great Inequality-like relationship, which may explain why the influence of Jupiter isn't as destructive.

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