Long-term dynamics beyond Neptune: secular models to study the regular motions

Abstract

Two semi-analytical one-degree-of-freedom secular models are presented for the motion of small bodies beyond Neptune. A special attention is given to trajectories entirely exterior to the planetary orbits. The first one is the well-known non-resonant model of Kozai (1962) adapted to the transneptunian region. Contrary to previous papers, the dynamics is fully characterized with respect to the fixed parameters. A maximum perihelion excursion possible of 16.4 AU is determined. The second model handles the occurrence of a mean-motion resonance with one of the planets. In that case, the one-degree-of-freedom integrable approximation is obtained by postulating the adiabatic invariance, and is much more general and accurate than previous secular models found in the literature. It brings out in a plain way the possibility of perihelion oscillations with a very high amplitude. Such a model could thus be used in future studies to deeper explore that kind of motion. For complex resonant orbits (especially of type 1 : k), a segmented secular description is necessary since the trajectories are only "integrable by parts". The two models are applied to the Solar System but the notations are kept general so that it could be used for any quasi-circular and coplanar planetary system.

Three-dimensional structure of mean motion resonances beyond Neptune

“…That could be the case for 471325 (2011 KT19). Gallardo et al (2012) and Saillenfest et al (2016Saillenfest et al ( , 2017) studied the long term dynamical evolution of resonant motions due to the Lidov-Kozai mechanism where large changes in (e, i) take place along with variations of ω. In these cases, the resonances also have large variations in their behavior and strength, including notable changes in their topology.…”

Long-term orbital dynamics of trans-Neptunian objects

“…6, one can compute numerically various properties of the resonances, including the exact value of their width for any eccentricity and inclination. As shown by Gallardo (2006a) and Saillenfest et al (2016), an inner double island (left panels of Fig. 6) is always present for resonances with k p = 1, provided that the eccentricity is high enough.…”

“…This kind of semianalytical procedure is now widely used for resonant problems in celestial mechanics involving high eccentricities and/or high inclinations (see e.g. Milani and Baccili 1998;Gallardo 2006aGallardo ,b, 2019bSidorenko 2006Sidorenko , 2018Saillenfest et al 2016;Saillenfest and Lari 2017;Pichierri et al 2017;Batygin and Morbidelli 2017). We note that the evolution of the pair (U, u) is secular by nature (frequency ∝ ε P ), whereas the evolution of the pair (Σ, σ) is semi-secular (frequency ∝ √ ε P ).…”

The long-term evolution of known resonant trans-Neptunian objects

“…In this section, we outline the semi-analytical model introduced by Saillenfest et al (2016). We consider a massless particle affected by the Sun and N planets, which are supposed on circular and coplanar orbits (typically Jupiter, Saturn, Uranus and Neptune in their invariable plane).…”

“…Recently, Saillenfest et al (2016) presented a semi-analytical model designed to accurately describe, and in a plain way, the long-term dynamics of such objects. This model can be seen as a generalisation of the secular theory by Kozai (1985) and its extension by Gallardo et al (2012).…”