Study and application of the resonant secular dynamics beyond Neptune


We use a secular representation to describe the long-term dynamics of transneptunian objects in mean-motion resonance with Neptune. The model applied is thoroughly described in Saillenfest et al. (Celest Mech Dyn Astron, doi:10.1007/s10569-016-9700-5, 2016). The parameter space is systematically explored, showing that the secular trajectories depend little on the resonance order. High-amplitude oscillations of the perihelion distance are reported and localised in the space of the orbital parameters. In particular, we show that a large perihelion distance is not a sufficient criterion to declare that an object is detached from the planets. Such a mechanism, though, is found unable to explain the orbits of Sedna or \(2012\text {VP}_{113}\), which are insufficiently inclined (considering their high perihelion distance) to be possibly driven by such a resonant dynamics. The secular representation highlights the existence of a high-perihelion accumulation zone due to resonances of type 1:k with Neptune. That region is found to be located roughly at \(a\in [100;300]\) AU, \(q\in [50;70]\) AU and \(I\in [30;50]^{\circ }\). In addition to the flux of objects directly coming from the Scattered Disc, numerical simulations show that the Oort Cloud is also a substantial source for such objects. Naturally, as that mechanism relies on fragile captures in high-order resonances, our conclusions break down in the case of a significant external perturber. The detection of such a reservoir could thus be an observational constraint to probe the external Solar System.

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  1. 1.

    A parameter \(\eta _0=0\) is also attainable for \(e=1\) (whatever the inclination), but since we are interested in perihelion distances always beyond Neptune, we will not deal with parabolic orbits in this paper.

  2. 2.

    A fixed libration centre is very unsuitable in this region because it varies actually between 0 and \(2\pi \). That comment holds also for their Fig. 11f–h beyond about 50 AU. In the lower part, on the contrary, a resonance island centred around \(60^{\circ }\) does exist (even if it actually shifts and deforms a bit).

  3. 3.

    Strictly speaking, the non-resonant Hamiltonian is defined with the semi-major axis as a constant parameter. However, the resonant interaction does not cause a to vary enough for the Kozai islands to be notably distorted.

  4. 4.

    Some resonances other than 1:k can actually present two resonance islands, but the corresponding ranges of parameters for \(q>a_\text {N}\) are so narrow that we will dismiss that case in the present paper.

  5. 5.

    As the semi-secular Hamiltonian \(\mathcal {K}\) is symmetric in \(\omega \) with respect to \(\pi /2\), the secular level curves for oscillations inside the left island are obtained by the transformation \(\omega \rightarrow \pi -\omega \).

  6. 6.

    The discontinuity line spans in \(\omega \in [0;\pi /2]\) or \([\pi /2;\pi ]\) if the particle occupies the right or the left resonance islands, respectively.

  7. 7.

    The orbital elements are taken from AstDyS database (, except from \(2013\text {RF}_{98}\) which comes from the JPL database.

  8. 8.

    Our secular model can be applied as well to high-perihelion objects with smaller semi-major axes. In particular, it could be used to explore in a plain way the close past and future of the resonant objects recently described by Sheppard et al. (2016). This is left for future papers.

  9. 9.

    In particular, we did not manage to lock \(2007\text {TG}_{422}\) in resonance in our numerical integrations (semi-major axis \(\approx \)493 AU) even by putting it exactly at the resonance centre. This is different for high inclinations, for which it is not rare to observe stable resonance captures with \(a_0 > 500\) AU even for perihelion distances near Neptune. However, the probability to find a real body with that kind of orbit is likely very low.

  10. 10.

    Naturally, as the final trajectory relies on a high-amplitude oscillation of the resonant angle, the proximity of the separatrices can lead to an accidental extra transition. However, this proves to be very rare and only temporary, as seen in the next section.


  1. Batygin, K., Brown, M.E.: Evidence for a distant giant planet in the solar system. Astron. J. 151, 22 (2016)

    ADS  Article  Google Scholar 

  2. Duncan, M.J., Levison, H.F., Budd, S.M.: The dynamical structure of the Kuiper belt. Astron. J. 110, 3073 (1995)

    ADS  Article  Google Scholar 

  3. Fouchard, M., Rickman, H., Froeschlé, C., Valsecchi, G.B.: On the present shape of the Oort cloud and the flux of “new” comets. Icarus (accepted for publication) (2016)

  4. Francis, P.J.: The demographics of long-period comets. Astrophys. J. 635, 1348–1361 (2005)

    ADS  Article  Google Scholar 

  5. Gallardo, T., Hugo, G., Pais, P.: Survey of Kozai dynamics beyond Neptune. Icarus 220, 392–403 (2012)

    ADS  Article  Google Scholar 

  6. Gladman, B., Holman, M., Grav, T., Kavelaars, J., Nicholson, P., Aksnes, K., et al.: Evidence for an extended scattered disk. Icarus 157, 269–279 (2002)

  7. Gomes, R.S.: The origin of TNO 2004 XR 190 as a primordial scattered object. Icarus 215, 661–668 (2011)

    ADS  Article  Google Scholar 

  8. Gomes, R.S., Gallardo, T., Fernández, J.A., Brunini, A.: On the origin of the high-perihelion scattered disk: the role of the Kozai mechanism and mean motion resonances. Celest. Mech. Dyn. Astron. 91, 109–129 (2005)

    ADS  Article  MATH  Google Scholar 

  9. Hénon, M.: On the numerical computation of Poincaré maps. Phys. D Nonlinear Phenom. 5, 412–414 (1982)

    ADS  Article  MATH  Google Scholar 

  10. Henrard, J.: Dynamics Reported—Expositions in Dynamical Systems, vol. 2. Springer, Berlin (1993)

    Google Scholar 

  11. Holman, M.J., Wisdom, J.: Dynamical stability in the outer solar system and the delivery of short period comets. Astron. J. 105, 1987–1999 (1993)

    ADS  Article  Google Scholar 

  12. Jílková, L., Portegies Zwart, S., Pijloo, T., Hammer, M.: How Sedna and family were captured in a close encounter with a solar sibling. Mon. Not. R. Astron. Soc. 453, 3157–3162 (2015)

    ADS  Article  Google Scholar 

  13. Kozai, Y.: Secular perturbations of asteroids with high inclination and eccentricity. Astron. J. 67, 591 (1962)

    ADS  MathSciNet  Article  Google Scholar 

  14. Kozai, Y.: Secular perturbations of resonant asteroids. Celest. Mech. 36, 47–69 (1985)

    ADS  Article  MATH  Google Scholar 

  15. Laskar, J.: The chaotic motion of the solar system—a numerical estimate of the size of the chaotic zones. Icarus 88, 266–291 (1990)

    ADS  Article  Google Scholar 

  16. Milani, A., Baccili, S.: Dynamics of Earth-crossing asteroids: the protected Toro orbits. Celest. Mech. Dyn. Astron. 71, 35–53 (1998)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  17. Saillenfest, M., Fouchard, M., Tommei, G., Valsecchi, G.B.: Long-term dynamics beyond Neptune: secular models to study the regular motions. Celest. Mech. Dyn. Astron. (2016). doi:10.1007/s10569-016-9700-5

  18. Sheppard, S.S., Trujillo, C., Tholen, D.J.: Beyond the Kuiper belt edge: new high perihelion trans-Neptunian objects with moderate semimajor axes and eccentricities. Astrophys. J. Lett. 825, 13 (2016)

    ADS  Article  Google Scholar 

  19. Thomas, F., Morbidelli, A.: The Kozai resonance in the outer solar system and the dynamics of long-period comets. Celest. Mech. Dyn. Astron. 64, 209–229 (1996)

    ADS  Article  MATH  Google Scholar 

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The authors thank the two anonymous referees for their wise and very stimulating comments. They brought a valuable contribution to the article, allowing a deeper understanding in several parts of the work.

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Correspondence to Melaine Saillenfest.

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Saillenfest, M., Fouchard, M., Tommei, G. et al. Study and application of the resonant secular dynamics beyond Neptune. Celest Mech Dyn Astr 127, 477–504 (2017).

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  • Secular model
  • Mean-motion resonance
  • Transneptunian object (TNO)
  • High-perihelion TNOs
  • Resonant secular model