## Abstract

We use a secular model to describe the non-resonant dynamics of trans-Neptunian objects in the presence of an external ten-Earth-mass perturber. The secular dynamics is analogous to an “eccentric Kozai mechanism” but with both an inner component (the four giant planets) and an outer one (the eccentric distant perturber). By the means of Poincaré sections, the cases of a non-inclined or inclined outer planet are successively studied, making the connection with previous works. In the inclined case, the problem is reduced to two degrees of freedom by assuming a non-precessing argument of perihelion for the perturbing body. The size of the perturbation is typically ruled by the semi-major axis of the small body: we show that the classic integrable picture is still valid below about 70 AU, but it is progressively destroyed when we get closer to the external perturber. In particular, for \(a>150\) AU, large-amplitude orbital flips become possible, and for \(a>200\) AU, the Kozai libration islands at \(\omega =\pi /2\) and \(3\pi /2\) are totally submerged by the chaotic sea. Numerous resonance relations are highlighted. The most large and persistent ones are associated with apsidal alignments or anti-alignments with the orbit of the distant perturber.

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## Notes

- 1.
This is the same expression as the Eq. 2 by Batygin and Brown (2016a), except that they give the associated period \(2\pi /\nu _\varpi '\). Note that there is a typo error in their expression (the inverse of a sum is

*not*the sum of the inverses). - 2.
There is actually one very specific case where the generalised solution is not uniquely defined, namely when the crossing is exactly tangential. This can happen only if the mutual inclination of the asteroid and the planet is zero at the very moment of the orbital crossing. We will discard that case in this paper, since it has negligible probability to occur for an initially arbitrarily inclined small body.

- 3.
Gronchi and Milani (2001) stress also the symplectic property of Runge–Kutta–Gauss integrators. The handling of the discontinuity, though, requires necessarily an adjustable integration step, which breaks the symplecticity of the overall scheme.

- 4.
Correcting coefficients could actually be computed, but they would require to save a lot of information from the previous steps.

- 5.
We use here a broader definition of “classic” and “eccentric” Kozai mechanisms than Naoz et al. (2013) (right after their Eq. 26). Here, our definition holds for the non-truncated averaged Hamiltonian: it only indicates the orbit of the perturber, which is, respectively, circular or eccentric.

- 6.
Such a simple colour code can be a bit ambiguous, in particular for resonances between the oscillation frequency of one angle and the circulation frequency of the other: they are represented in blue even if one angle oscillates. This should not mislead the reader, though, since further indications are given in the captions and in the text.

- 7.
For a more straightforward comparison with Beust (2016), we could have taken directly the angle \(\varDelta \varpi =\omega +\delta h\) as canonical coordinate. However, the other resonances would have become harder to interpret (for instance \(\omega -\delta h\) turns to \(2\omega -\varDelta \varpi \)), and we would have lost the property of the equilibrium points of \(\omega \) and \(\delta h\), dividing prograde from retrograde resonances.

- 8.
These sections are made, respectively, for \(\delta h\) and \(\omega \) equal to \(\pi /2\), so a fixed point at \(3\pi /2\) means for both sections an equilibrium point of \(\varDelta \varpi =\omega +\delta h\) at 0.

- 9.
On 2017-06-06, the JPL Small-Body Database Search Engine reports 8 non-cometary objects with \(a>150\) AU, \(q>5\) AU and \(I>50^{\circ }\) (https://ssd.jpl.nasa.gov/sbdb_query.cgi).

- 10.
On the contrary, apsidal alignment or anti-alignment would have resulted in fixed points on the sections at \(\omega '\pm \pi /2\), where \(\omega '\) is given in Table 1.

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## Acknowledgements

We thank two anonymous referees who helped us to improve the paper. This work was partly funded by Paris Sciences et Lettres (PSL).

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Saillenfest, M., Fouchard, M., Tommei, G. *et al.* Non-resonant secular dynamics of trans-Neptunian objects perturbed by a distant super-Earth.
*Celest Mech Dyn Astr* **129, **329–358 (2017). https://doi.org/10.1007/s10569-017-9775-7

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### Keywords

- Secular model
- Trans-Neptunian object (TNO)
- Poincaré section