The Dynamics of Neptune Trojan: I. the Inclined Orbits

📝 Abstract

The stability of Trojan type orbits around Neptune is studied. As the first part of our investigation, we present in this paper a global view of the stability of Trojans on inclined orbits. Using the frequency analysis method based on the FFT technique, we construct high resolution dynamical maps on the plane of initial semimajor axis $a_0$ versus inclination $i_0 $. These maps show three most stable regions, with $i_0$ in the range of $(0^\circ,12^\circ), (22^\circ,36^\circ)$ and $(51^\circ,59^\circ)$ respectively, where the Trojans are most probably expected to be found. The similarity between the maps for the leading and trailing triangular Lagrange points $L_4$ and $L_5$ confirms the dynamical symmetry between these two points. By computing the power spectrum and the proper frequencies of the Trojan motion, we figure out the mechanisms that trigger chaos in the motion. The Kozai resonance found at high inclination varies the eccentricity and inclination of orbits, while the $\nu_8$ secular resonance around $i_0\sim44^\circ$ pumps up the eccentricity. Both mechanisms lead to eccentric orbits and encounters with Uranus that introduce strong perturbation and drive the objects away from the Trojan like orbits. This explains the clearance of Trojan at high inclination ( $>60^\circ $) and an unstable gap around $44^\circ$ on the dynamical map. An empirical theory is derived from the numerical results, with which the main secular resonances are located on the initial plane of $(a_0,i_0) $. The fine structures in the dynamical maps can be explained by these secular resonances.

💡 Analysis

The stability of Trojan type orbits around Neptune is studied. As the first part of our investigation, we present in this paper a global view of the stability of Trojans on inclined orbits. Using the frequency analysis method based on the FFT technique, we construct high resolution dynamical maps on the plane of initial semimajor axis $a_0$ versus inclination $i_0 $. These maps show three most stable regions, with $i_0$ in the range of $(0^\circ,12^\circ), (22^\circ,36^\circ)$ and $(51^\circ,59^\circ)$ respectively, where the Trojans are most probably expected to be found. The similarity between the maps for the leading and trailing triangular Lagrange points $L_4$ and $L_5$ confirms the dynamical symmetry between these two points. By computing the power spectrum and the proper frequencies of the Trojan motion, we figure out the mechanisms that trigger chaos in the motion. The Kozai resonance found at high inclination varies the eccentricity and inclination of orbits, while the $\nu_8$ secular resonance around $i_0\sim44^\circ$ pumps up the eccentricity. Both mechanisms lead to eccentric orbits and encounters with Uranus that introduce strong perturbation and drive the objects away from the Trojan like orbits. This explains the clearance of Trojan at high inclination ( $>60^\circ $) and an unstable gap around $44^\circ$ on the dynamical map. An empirical theory is derived from the numerical results, with which the main secular resonances are located on the initial plane of $(a_0,i_0) $. The fine structures in the dynamical maps can be explained by these secular resonances.

📄 Content

arXiv:0906.5075v1 [astro-ph.EP] 27 Jun 2009 Mon. Not. R. Astron. Soc. 000, 1–12 (2009) Printed 24 July 2021 (MN LATEX style file v2.2) The Dynamics of Neptune Trojan: I. the Inclined Orbits Li-Yong Zhou1⋆, Rudolf Dvorak2, Yi-Sui Sun1 1Department of Astronomy, Nanjing University, Nanjing 210093, China 2Institute for Astronomy, University of Vienna, T¨urkenschanzstr. 17, A-1180 Wien, Austria Accepted . Received ABSTRACT The stability of Trojan type orbits around Neptune is studied. As the first part of our investigation, we present in this paper a global view of the stability of Trojans on inclined orbits. Using the frequency analysis method based on the FFT technique, we construct high resolution dynamical maps on the plane of initial semimajor axis a0 versus inclination i0. These maps show three most stable regions, with i0 in the range of (0◦, 12◦), (22◦, 36◦) and (51◦, 59◦) respectively, where the Trojans are most probably expected to be found. The similarity between the maps for the leading and trailing triangular Lagrange points L4 and L5 confirms the dynamical symmetry between these two points. By computing the power spectrum and the proper frequencies of the Trojan motion, we figure out the mechanisms that trigger chaos in the motion. The Kozai resonance found at high inclination varies the eccentricity and inclination of orbits, while the ν8 secular resonance around i0 ∼44◦pumps up the eccentricity. Both mechanisms lead to eccentric orbits and encounters with Uranus that introduce strong perturbation and drive the objects away from the Trojan like orbits. This explains the clearance of Trojan at high inclination (> 60◦) and an unstable gap around 44◦on the dynamical map. An empirical theory is derived from the numerical results, with which the main secular resonances are located on the initial plane of (a0, i0). The fine structures in the dynamical maps can be explained by these secular resonances. Key words: Planets and satellites: individual: Neptune – Minor planets, asteroids – Celestial mechanics – Method: miscellaneous 1 INTRODUCTION In the restricted three-body model consisting of the Sun, a planet and an asteroid, the equilateral triangular Lagrange equilibrium points (L4 and L5) are stable for all planets in our Solar system. Asteroids in the vicinities of L4 and L5 of a parent planet are called Trojans after the group of asteroids found around Jupiter’s Lagrange points. Objects on Trojan like orbits around Mars and (temporarily) around Earth have been observed while the Trojan-type orbits of Saturn and Uranus have been proven unstable due to the perturbations from other planets. As for Neptune, the possibility of stable orbits around the Lagrange points have been verified in several papers, e.g. (Holman & Wisdom 1993; Weissman & Levison 1997; Nesvorn´y & Dones 2002), before the discovery of the first Neptune Trojan, 2001 QR322 (Pittichova et al. 2003). Up to now, 6 Neptune Trojans have been discovered1 around Neptune’s L4 point. We list their orbital properties in ⋆zhouly@nju.edu.cn 1 IAU: Minor Planet Center, http://www.cfa.harvard.edu/iau/ lists/NeptuneTrojans.html Table 1. It is suspected that there could be much more Trojan-type asteroids sharing the orbit with Neptune than with Jupiter, both in the sense of number and total mass (Sheppard & Trujillo 2006). After these discoveries, more papers were devoted to the issue of Neptune Trojans, fo- cusing either on the orbital stability and origin of spe- cific objects (Brasser et al. 2004b; Li, Zhou & Sun 2007) or on creating a global view of stable regions around the Lagrange points, e.g. (Marzari, Tricarico & Scholl 2003a; Dvorak et al. 2007; Dvorak, Lhotka & Schwarz 2008). Both the observing and the theoretical studies of Neptune Tro- jans could give important clues on how these objects were trapped into their current orbits, where and when the planet Neptune formed and how the orbits of the outer planets evolved in the early stage of the formation of the Solar sys- tem. Therefore, it is worth to investigate the orbital stability of fictitious Trojans in the whole parameter space. All the Trojans in Table 1 are on near-circular orbits (with quite small value of eccentricity) and two of them have high inclination values. The stability and origin of inclined orbit is an interesting topic (Li, Zhou & Sun 2007). As the first part of our investigation of the whole phase space, we study in this paper the orbital stability of Trojans on in- 2 Zhou, Dvorak & Sun Table 1. Orbits of 6 observed Neptune Trojans, given at epoch JD=2454800.5 with respect to the mean ecliptic and equinox at J2000.0. The perihelion argument ω, ascending node Ωand incli- nation i are in degrees. Designation M ω Ω i e a (AU) 2001 QR322 57.88 160.8 151.6 1.3 0.031 30.302 2004 UP10 341.28 358.5 34.8 1.4 0.028 30.212 2005 TN53 287.04 85.7 9.3 25.0 0.065 30.179 2005 TO74 268.10 302.6 169.4 5.3 0.052 30.190 2006 RJ103 238.64 27.1 120.8 8.2 0.028 30.077 2007 VL305 352.88 215.2 188.6 28.1

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