Trojan Motions Research Articles

Secondary resonances and the boundary of effective stability of Trojan motions

One of the most interesting features in the libration domain of co-orbital motions is the existence of secondary resonances. For some combinations of physical parameters, these resonances occupy a large fraction of the domain of stability and rule the dynamics within the stable tadpole region. In this work, we present an application of a recently introduced `basic Hamiltonian model' Hb for Trojan dynamics, in Paez and Efthymiopoulos (2015), Paez, Locatelli and Efthymiopoulos (2016): we show that the inner border of the secondary resonance of lowermost order, as defined by Hb, provides a good estimation of the region in phase-space for which the orbits remain regular regardless the orbital parameters of the system. The computation of this boundary is straightforward by combining a resonant normal form calculation in conjunction with an `asymmetric expansion' of the Hamiltonian around the libration points, which speeds up convergence. Applications to the determination of the effective stability domain for exoplanetary Trojans (planet-sized objects or asteroids) which may accompany giant exoplanets are discussed.

Dynamical spreading of small bodies in 1:1 resonance with planets by the diurnal Yarkovsky effect

A simple model is introduced to describe the inherent dynamics of Trojans in the presence of the diurnal Yarkovsky effect. For different spin statuses, the orbital elements of the Trojans (mainly semimajor axis, eccentricity and inclination) undergo different variations. The variation rate is generally very small, but the total variation of the semimajor axis or the orbit eccentricity over the age of the Solar system may be large enough to send small Trojans out of the regular region (or, vice versa, to capture small bodies in the regular region). In order to demonstrate the analytical analysis, we first carry out numerical simulations in a simple model, and then generalize these to two ‘real’ systems, namely the Sun–Jupiter system and the Sun–Earth system. In the Sun–Jupiter system, where the motion of Trojans is regular, the Yarkovsky effect gradually alters the libration width or the orbit eccentricity, forcing the Trojan to move from regular regionsto chaotic regions, where chaos may eventually cause it to escape. In the Sun–Earth system, where the motion of Trojans is generally chaotic, our limited numerical simulations indicate that the Yarkovsky effect is negligible for Trojans of 100 m in size, and even for larger ones. The Yarkovsky effect on small bodies captured in other 1:1 resonance orbits is also briefly discussed.

New Hamiltonian expansions adapted to the Trojan problem

A number of studies, referring to the observed Trojan asteroids of various planets in our Solar System, or to hypothetical Trojan bodies in extrasolar planetary systems, have emphasized the importance of so-called secondary resonances in the problem of the long term stability of Trojan motions. Such resonances describe commensurabilities between the fast, synodic, and secular frequency of the Trojan body, and, possibly, additional slow frequencies produced by more than one perturbing bodies. The presence of secondary resonances sculpts the dynamical structure of the phase space. Hence, identifying their location is a relevant task for theoretical studies. In the present paper we combine the methods introduced in two recent papers (Paez & Efthymiopoulos, 2015, Paez & Locatelli, 2015) in order to analytically predict the location of secondary resonances in the Trojan problem (SEE FILE FOR COMPLETE ABSTRACT)

Modeling Trojan dynamics: diffusion mechanisms through resonances

AbstractIn the framework of the ERTBP, we study an example of the influence of secondary resonances over the long term stability of Trojan motions. By the integration of ensembles of orbits, we find various types of chaotic diffusion, slow and fast. We show that the distribution of escape times is bi-modular, corresponding to two populations of short and long escape times. The objects with long escape times produce a power-law tail in the distribution.

Modeling resonant trojan motions in planetary systems

AbstractWe consider the dynamics of a small trojan companion of a hypothetical giant exoplanet under the secular perturbations of additional planets. By a suitable choice of action-angle variables, the problem is amenable to the study of the slow modulation, induced by secular perturbations, to the dynamics of an otherwise called ‘basic’ Hamiltonian model of two degrees of freedom (planar case). We present this Hamiltonian decomposition, which implies that the slow chaotic diffusion at resonances is best described by the paradigm of modulational diffusion.

Features of librational motions around L4

Trojan bodies are present in the Solar system in great number, as Trojan asteroids and also as Trojan moons. Thus it is possible that their presence is similar in extrasolar planetary systems too. We investigated features of librational motions around L4 with numerical methods on the mass parameter – eccentricity plane in the elliptic restricted three-body problem. We determined the lifetime of Trojan bodies, until they remained in the librational domain around L4, also illustrated the distribution of the three possible endgames of trojan motion. Finally we determined the frequences and resonances of the librational motion in the stable region.

The dynamics of Neptune Trojan - I. The inclined orbits

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.

A survey of orbits of co-orbitals of Mars

Many asteroids with a semimajor axis close to that of Mars have been discovered in the last several years. Potentially some of these could be in 1:1 resonance with Mars, much as are the classic Trojan asteroids with Jupiter, and its lesser-known horseshoe companions with Earth. In the 1990s, two Trojan companions of Mars, 5261 Eureka and 1998 VF 31, were discovered, librating about the L 5 Lagrange point, 60° behind Mars in its orbit. Although several other potential Mars Trojans have been identified, our orbital calculations show only one other known asteroid, 1999 UJ 7, to be a Trojan, associated with the L 4 Lagrange point, 60° ahead of Mars in its orbit. We further find that asteroid 36017 (1999 ND 43) is a horseshoe librator, alternating with periods of Trojan motion. This asteroid makes repeated close approaches to Earth and has a chaotic orbit whose behavior can be confidently predicted for less than 3000 years. We identify two objects, 2001 HW 15 and 2000 TG 2, within the resonant region capable of undergoing what we designate “circulation transition”, in which objects can pass between circulation outside the orbit of Mars and circulation inside it, or vice versa. The eccentricity of the orbit of Mars appears to play an important role in circulation transition and in horseshoe motion. Based on the orbits and on spectroscopic data, the Trojan asteroids of Mars may be primordial bodies, while some co-orbital bodies may be in a temporary state of motion.

On the accuracy of restricted three-body models for the Trojan motion

On the accuracy of restricted three-body models for the Trojan motion

On the accuracy of restricted three-body models for the Trojan motion

In this note we compare the frequencies of the motion of the Trojan asteroids in the Restricted Three-Body Problem (RTBP), the Elliptic Restricted Three-Body Problem (ERTBP) and the Outer Solar System (OSS) model. The RTBP and ERTBP are well-known academic models for the motion of these asteroids, and the OSS is the standard model used for realistic simulations. Our results are based on a systematic frequency analysis of the motion of these asteroids. The main conclusion is that both the RTBP and ERTBP are not very accurate models for the long-term dynamics, although the level of accuracy strongly depends on the selected asteroid.

The MATROS project: Stability of Uranus and Neptune Trojans. The case of 2001 QR322

We present in this paper an analysis of the long term stability of Trojan type orbits of both Uranus and Neptune. Employing the Frequency Map Analysis (hereinafter FMA) we measure the diffusion speed in the phase space for a large sample of Trojan orbits with short numerical integrations. High resolution diffusion maps are derived for different values of initial inclination. These maps outline where the most stable orbits can be found in the Trojan clouds of the two planets. The orbit of the newly discovered Neptune Trojan 2001 QR322 has been analysed in detail with the FMA method. In the phase space the body is located close to the border of a stable region for low inclination Neptune Trojans. Numerical integrations over 4.5 Gyr of clone orbits generated from the covariance matrix show that only 10% of the clones escape from the Trojan cloud. The proper frequencies of the Trojan motion computed with the FMA algorithm allow us to to derive a numerical secular theory. From this theory it is possible to locate in the phase space the main secular resonances that can perturb Trojan orbits of the two planets and lead to instability.

The role of secular resonances on trojans of the terrestrial planets

We investigate the role of secular resonances on the motion of terrestrial-planet trojans in tadpole motion for 5 Myr in the inclination range i ≤ 90°. We use a Fourier spectrum of the eccentricity vector, e, and angular momentum vector, L. When i > 40°, the trojans are in a Kozai resonance with Jupiter, which increases the eccentricity to planet-crossing values on a time-scale of 0.1 Myr; hence we found no stable trojan motion when i > 40°. Trojans of Mercury are unstable for a large range in inclination due to the resonances v 1 , v 1 2 and v 5 , and only those with low and high inclinations survived in our simulations. For Venus and Earth trojans, the v 3 and v 4 secular resonances with the Earth and Mars respectively increase the eccentricity to planet-crossing values for moderate inclinations. Venus trojans are also affected by the v 1 1 resonance with Mercury at high inclinations hence Venus and Earth trojans are stable only at low inclinations. Mars trojans are affected by the v 3 , v 4 , v 1 3 and v 1 4 resonances with the Earth and Mars for low inclinations which increase the eccentricity and suppress oscillations of the inclination. At i = 32°, the v 5 resonance destabilizes Mars trojans, although its width is less than a degree in inclination. The typical lifetime of inner-planet trojans trapped in a secular resonance is 1 Myr for Venus, Earth and Mars trojans, while it is 10 5 yr for Mercury trojans. Trojans in secular resonances are chaotic on time-scales of 10 3 -10 4 yr for all inner planets, while outside the resonances this becomes typically 10 6 yr.

Determination of the masses of Jupiter and Saturn from the motion of Trojans

The mass of a planet can be determined by the motion of its satellites or by perturbational effects on nearby bodies, such as comets, planets, minor planets. It is well known, for instance, that a body on a nearby orbit which is in close resonance with a planet is suitable for the determination of the planetary mass. In this case the orbital periods of the two bodies are equal or nearly equal to the ratio of small integers (i.e. 1: 1, 2: 1, 3:2).

Orbits of Trojan Asteroids

In this paper the orbital evolution of Trojan asteroids are studied by integrating numerically the equations of motion over the interval 1660–2060, perturbations from Venus to Pluto being taken into account. The comparison of the actual motion of Trojans in the solar system with the theory based on the restricted three-body problem are given.