Orbital stability of Earth Trojans

¹ School of Astronomy and Space Science, Nanjing University, 163 Xianlin Avenue, Nanjing 210046, PR China
e-mail: zhouly@nju.edu.cn
² Key Laboratory of Modern Astronomy and Astrophysics in Ministry of Education, Nanjing University, Nanjing 210046, PR China
³ Universitätssternwarte Wien, Türkenschanzstr. 17, 1180 Wien, Austria

Received: 5 August 2018
Accepted: 4 December 2018

Abstract

The only discovery of Earth Trojan 2010 TK₇ and the subsequent launch of OSIRIS-REx have motived us to investigate the stability around the triangular Lagrange points of the Earth, L₄ and L₅. In this paper we present detailed dynamical maps on the (a₀, i₀) plane with the spectral number (SN) indicating the stability. Two main stability regions, separated by a chaotic region arising from the ν₃ and ν₄ secular resonances, are found at low (i₀ ≤ 15°) and moderate (24 ° ≤i₀ ≤ 37°) inclinations, respectively. The most stable orbits reside below i₀ = 10° and they can survive the age of the solar system. The nodal secular resonance ν₁₃ could vary the inclinations from 0° to ∼10° according to their initial values, while ν₁₄ could pump up the inclinations to ∼20° and upwards. The fine structures in the dynamical maps are related to higher degree secular resonances, of which different types dominate different areas. The dynamical behaviour of the tadpole and horseshoe orbits, reflected in their secular precession, show great differences in the frequency space. The secular resonances involving the tadpole orbits are more sensitive to the frequency drift of the inner planets, thus the instabilities could sweep across the phase space, leading to the clearance of tadpole orbits. We are more likely to find terrestrial companions on horseshoe orbits. The Yarkovsky effect could destabilize Earth Trojans in varying degrees. We numerically obtain the formula describing the stabilities affected by the Yarkovsky effect and find the asymmetry between the prograde and retrograde rotating Earth Trojans. The existence of small primordial Earth Trojans that avoid being detected but survive the Yarkovsky effect for 4.5 Gyr is substantially ruled out.