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Register a new userOrbital Stability of Earth Trojans
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Lei Zhou
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The only discovery of Earth Trojan 2010 TK$_7$ and the subsequent launch of OSIRIS-REx motive us to investigate the stability around the triangular Lagrange points $L_4$ and $L_5$ of the Earth. In this paper we present detailed dynamical maps on the $(a_0,i_0)$ plane with the spectral number (SN) indicating the stability. Two main stability regions, separated by a chaotic region arising from the $ u_3$ and $ u_4$ secular resonances, are found at low ($i_0leq 15^circ$) and moderate ($24^circleq {i_0}leq 37^circ$) inclinations respectively. The most stable orbits reside below $i_0=10^circ$ and they can survive the age of the Solar System. The nodal secular resonance $ u_{13}$ could vary the inclinations from $0^circ$ to $sim 10^circ$ according to their initial values while $ u_{14}$ could pump up the inclinations to $sim 20^circ$ 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.
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Most of the major planets in the Solar System support populations of co-orbiting bodies, known as Trojans, at their L4 and L5 Lagrange points. In contrast, Earth has only one known co-orbiting companion. This paper presents the results from a search for Earth Trojans using the DECam instrument on the Blanco Telescope at CTIO. This search found no additional Trojans in spite of greater coverage compared to previous surveys of the L5 point. Therefore, the main result of this work is to place the most stringent constraints to date on the population of Earth Trojans. These constraints depend on assumptions regarding the underlying population properties, especially the slope of the magnitude distribution (which in turn depends on the size and albedo distributions of the objects). For standard assumptions, we calculate upper limits to a 90% confidence limit on the L5 population of $N_{ET}<1$ for magnitude $H<15.5$, $N_{ET}=60-85$ for $H<19.7$, and $N_{ET} $= 97 for $H=20.4$. This latter magnitude limit corresponds to Trojans $sim$300 m in size for albedo $0.15$. At H=19.7, these upper limits are consistent with previous L4 Earth Trojan constraints and significantly improve L5 constraints.
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We explore planetary migration scenarios for formation of high inclination Neptune Trojans (NTs) and how they are affected by the planetary migration of Neptune and Uranus. If Neptune and Uranuss eccentricity and inclination were damped during planetary migration, then their eccentricities and inclinations were higher prior and during migration than their current values. Using test particle integrations we study the stability of primordial NTs, objects that were initially Trojans with Neptune prior to migration. We also study Trans-Neptunian objects captured into resonance with Neptune and becoming NTs during planet migration. We find that most primordial NTs were unstable and lost if eccentricity and inclination damping took place during planetary migration. With damping, secular resonances with Neptune can increase a low eccentricity and inclination population of Trans-Neptunian objects increasing the probability that they are captured into 1:1 resonance with Neptune, becoming high inclination NTs. We suggest that the resonant trapping scenario is a promising and more effective mechanism explaining the origin of NTs that is particularly effective if Uranus and Neptune experienced eccentricity and inclination damping during planetary migration.
Previous work has demonstrated orbital stability for 100 Myr of initially near-circular and coplanar small bodies in a region termed the Earth-Mars belt from 1.08 au $< a <$ 1.28 au. Via numerical integration of 3000 particles, we studied orbits from 1.04-1.30 au for the age of the Solar system. We show that on this time scale, except for a few locations where mean-motion resonances with Earth affect stability, only a narrower `Earth-Mars belt covering $asim(1.09, 1.17)$ au, $e<0.04$, and $I<1^circ$ has over half of the initial orbits survive for 4.5 Gyr. In addition to mean-motion resonances, we are able to see how the $ u_3$, $ u_4$, and $ u_6$ secular resonances contribute to long-term instability in the outer (1.17-1.30 au) region on Gyr time scales. We show that all of the (rather small) near-Earth objects (NEOs) in or close to the Earth-Mars belt appear to be consistent with recently arrived transient objects by comparing to a NEO steady-state model. Given the $<200$ m scale of these NEOs, we estimated the Yarkovsky drift rates in semimajor axis, and use these to estimate that primordial asteroids with a diameter of 100 km or larger in the Earth-Mars belt would likely survive. We conclude that only a few 100-km sized asteroids could have been present in the belts region at the end of the terrestrial planet formation.
The most distant Kuiper belt objects exhibit the clustering in their orbits, and this anomalous architecture could be caused by Planet 9 with large eccentricity and high inclination. We then suppose that the orbital clustering of minor planets may be observed somewhere else in the solar system. In this paper, we consider the over 7000 Jupiter Trojans from the Minor Planet Center, and find that they are clustered in the longitude of perihelion $varpi$, around the locations $varpi_{mbox{{J}}}+60^{circ}$ and $varpi_{mbox{{J}}}-60^{circ}$ ($varpi_{mbox{{J}}}$ is the longitude of perihelion of Jupiter) for the L4 and L5 swarms, respectively. Then we build a Hamiltonian system to describe the associated dynamical aspects for the co-orbital motion. The phase space displays the existence of the apsidally aligned islands of libration centered on $Deltavarpi=varpi-varpi_{mbox{{J}}}approxpm60^{circ}$, for the Trojan-like orbits with eccentricities $e<0.1$. Through a detailed analysis, we have shown that the observed Jupiter Trojans with proper eccentricities $e_p<0.1$ spend most of their time in the range of $|Deltavarpi|=0-120^{circ}$, while the more eccentric ones with $e_p>0.1$ are too few to affect the orbital clustering within this $Deltavarpi$ range for the entire Trojan population. Our numerical results further prove that, even starting from a uniform $Deltavarpi$ distribution, the apsidal alignment of simulated Trojans similar to the observation can appear on the order of the age of the solar system. We conclude that the apsidal asymmetric-alignment of Jupiter Trojans is robust, and this new finding can be helpful to design the survey strategy in the future.
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