Advertisement

Celestial Mechanics and Dynamical Astronomy

, Volume 129, Issue 3, pp 329–358 | Cite as

Non-resonant secular dynamics of trans-Neptunian objects perturbed by a distant super-Earth

  • Melaine Saillenfest
  • Marc Fouchard
  • Giacomo Tommei
  • Giovanni B. Valsecchi
Original Article

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.

Keywords

Secular model Trans-Neptunian object (TNO) Poincaré section 

Notes

Acknowledgements

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

References

  1. Bailey, E., Batygin, K., Brown, M.E.: Solar obliquity induced by planet nine. Astron. J. 152, 126 (2016)ADSCrossRefGoogle Scholar
  2. Bannister, M.T., Kavelaars, J.J., Gladman, B.J., Petit, J.-M., Burdullis, T., Gwyn, S.D.J., Chen, Y.-T., Alexandersen, M., Schwamb, M.: Minor planet electronic circular 2017-M22. Minor Planet Center (2017)Google Scholar
  3. Batygin, K., Brown, M.E.: Evidence for a distant giant planet in the Solar System. Astron. J. 151, 22 (2016a)ADSCrossRefGoogle Scholar
  4. Batygin, K., Brown, M.E.: Generation of highly inclined trans-Neptunian objects by planet nine. Astrophys. J. Lett. 833, L3 (2016b)Google Scholar
  5. Beust, H.: Orbital clustering of distant Kuiper belt objects by hypothetical Planet 9. Secular or resonant? Astron. Astrophys. 590, L2 (2016)ADSCrossRefGoogle Scholar
  6. Brown, M.E., Batygin, K.: Observational constraints on the orbit and location of planet nine in the outer Solar System. Astrophys. J. Lett. 824, 23 (2016)ADSCrossRefGoogle Scholar
  7. de la Fuente Marcos, C., de la Fuente Marcos, R.: Commensurabilities between ETNOs: a Monte Carlo survey. Mon. Not. R. Astron. Soc. 460, 64–68 (2016)CrossRefGoogle Scholar
  8. Fienga, A., Laskar, J., Manche, H., Gastineau, M.: Constraints on the location of a possible 9th planet derived from the Cassini data. Astron. Astrophys. 587, 8 (2016)ADSCrossRefGoogle Scholar
  9. Gallardo, T.: Atlas of the mean motion resonances in the Solar System. Icarus 184, 29–38 (2006a)ADSCrossRefGoogle Scholar
  10. Gallardo, T.: The occurrence of high-order resonances and Kozai mechanism in the scattered disk. Icarus 181, 205–217 (2006b)ADSCrossRefGoogle Scholar
  11. Gallardo, T., Hugo, G., Pais, P.: Survey of Kozai dynamics beyond Neptune. Icarus 220, 392–403 (2012)ADSCrossRefGoogle Scholar
  12. Gomes, R., Deienno, R., Morbidelli, A.: The inclination of the planetary system relative to the solar equator may be explained by the presence of Planet 9. Astron. J. 153, 27 (2016)ADSCrossRefGoogle Scholar
  13. Gomes, R.S., Soares, J.S., Brasser, R.: The observation of large semi-major axis Centaurs: testing for the signature of a planetary-mass solar companion. Icarus 258, 37–49 (2015)ADSCrossRefGoogle Scholar
  14. Gronchi, G.F.: Generalized averaging principle and the secular evolution of planet crossing orbits. Celest. Mech. Dyn. Astron. 83, 97–120 (2002)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. Gronchi, G.F., Milani, A.: Averaging on Earth-crossing orbits. Celest. Mech. Dyn. Astron. 71, 109–136 (1998)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. Gronchi, G.F., Milani, A.: Proper elements for Earth-crossing asteroids. Icarus 152, 58–69 (2001)ADSCrossRefGoogle Scholar
  17. Hamers, A.S., Portegies Zwart, S.F.: Secular dynamics of hierarchical multiple systems composed of nested binaries, with an arbitrary number of bodies and arbitrary hierarchical structure. First applications to multiplanet and multistar systems. Mon. Not. R. Astron. Soc. 459, 2827–2874 (2016)ADSCrossRefGoogle Scholar
  18. Hamers, A.S., Perets, H.B., Antonini, F., Portegies Zwart, S.F.: Secular dynamics of hierarchical quadruple systems: the case of a triple system orbited by a fourth body. Mon. Not. R. Astron. Soc. 449, 4221–4245 (2015)ADSCrossRefGoogle Scholar
  19. Harrington, R.S.: Dynamical evolution of triple stars. Astron. J. 73, 190–194 (1968)ADSCrossRefGoogle Scholar
  20. Hénon, M.: On the numerical computation of Poincaré maps. Phys. D 5, 412–414 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  21. Innanen, K.A., Zheng, J.Q., Mikkola, S., Valtonen, M.J.: The Kozai mechanism and the stability of planetary orbits in binary star systems. Astron. J. 113, 1915 (1997)ADSCrossRefGoogle Scholar
  22. Katz, B., Dong, S., Malhotra, R.: Long-term cycling of Kozai-Lidov cycles: extreme eccentricities and inclinations excited by a distant eccentric perturber. Phys. Rev. Lett. 107, 181101 (2011)ADSCrossRefGoogle Scholar
  23. Kozai, Y.: Secular perturbations of asteroids with high inclination and eccentricity. Astron. J. 67, 591 (1962)ADSMathSciNetCrossRefGoogle Scholar
  24. Laskar, J., Boué, G.: Explicit expansion of the three-body disturbing function for arbitrary eccentricities and inclinations. Astron. Astrophys. 522, 60 (2010)ADSCrossRefzbMATHGoogle Scholar
  25. Li, G., Naoz, S., Holman, M., Loeb, A.: Chaos in the test particle eccentric Kozai–Lidov mechanism. Astrophys. J. 791, 86 (2014a)ADSCrossRefGoogle Scholar
  26. Li, G., Naoz, S., Kocsis, B., Loeb, A.: Eccentricity growth and orbit flip in near-coplanar hierarchical three-body systems. Astrophys. J. 785, 116 (2014b)ADSCrossRefGoogle Scholar
  27. Lithwick, Y., Naoz, S.: The eccentric Kozai mechanism for a test particle. Astrophys. J. 742, 94 (2011)ADSCrossRefGoogle Scholar
  28. Milani, A., Nobili, A.M.: An example of stable chaos in the Solar System. Nature 357, 569–571 (1992)ADSCrossRefGoogle Scholar
  29. Naoz, S.: The eccentric Kozai–Lidov effect and its applications. Ann. Rev. Astron. Astrophys. 54, 441–489 (2016)ADSCrossRefGoogle Scholar
  30. Naoz, S., Farr, W.M., Lithwick, Y., Rasio, F.A., Teyssandier, J.: Secular dynamics in hierarchical three-body systems. Mon. Not. R. Astron. Soc. 431, 2155–2171 (2013)ADSCrossRefGoogle Scholar
  31. 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. 126, 369–403 (2016)MathSciNetCrossRefGoogle Scholar
  32. Saillenfest, M., Fouchard, M., Tommei, G., Valsecchi, G.B.: Study and application of the resonant secular dynamics beyond Neptune. Celest. Mech. Dyn. Astron. 127, 477–504 (2017)ADSMathSciNetCrossRefGoogle Scholar
  33. Takeda, G., Kita, R., Rasio, F.A.: Planetary systems in binaries. I. Dynamical classification. Astrophys. J. 683, 1063–1075 (2008)ADSCrossRefGoogle Scholar
  34. Teyssandier, J., Naoz, S., Lizarraga, I., Rasio, F.A.: Extreme orbital evolution from hierarchical secular coupling of two giant planets. Astrophys. J. 779, 166 (2013)ADSCrossRefGoogle Scholar
  35. 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)ADSCrossRefzbMATHGoogle Scholar
  36. Touma, J.R., Tremaine, S., Kazandjian, M.V.: Gauss’s method for secular dynamics, softened. Mon. Not. R. Astron. Soc. 394, 1085–1108 (2009)ADSCrossRefGoogle Scholar
  37. Walker, I.W., Emslie, A.G., Roy, A.E.: Stability criteria in many-body systems. I - an empirical stability criterion for co-rotational three-body systems. Celes. Mech. 22, 371–402 (1980)ADSCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  • Melaine Saillenfest
    • 1
    • 2
    • 3
  • Marc Fouchard
    • 1
    • 3
  • Giacomo Tommei
    • 2
  • Giovanni B. Valsecchi
    • 4
    • 5
  1. 1.IMCCEParisFrance
  2. 2.Dipartimento di MatematicaUniversità di PisaPisaItaly
  3. 3.PSL Research University, CNRS, Sorbonne Universités, UPMC Université Paris 06, LAL, Université de LilleParis/LilleFrance
  4. 4.IAPS-INAFRomeItaly
  5. 5.IFAC-CNRSesto FiorentinoItaly

Personalised recommendations