Bálint Érdi

The Phase Space Structure Around L4 in the Restricted Three-Body Problem

The phase space structure around L4 in the restricted three-body problem is investigated. The connection between the long period family emanating from L4 and the very complex structure of the stability region is shown by using the method of Poincaré’s surface of section. The zero initial velocity stability region around L4 is determined by using a method based on the calculation of finite-time Lyapunov characteristic numbers. It is shown that the boundary of the stability region in the configuration space is formed by orbits suffering slow chaotic diffusion.

Long Periodic Perturbations of Trojan Asteroids

The motion of the Trojan asteroids is studied in the elliptic restricted three-body problem of the Sun-Jupiter-asteroid system. Long periodic perturbations of the orbital elements are discussed. Relations between dynamical parameters are considered and comparisons are made with Biens and Schubarts results.

The Trojan Problem

There has been a renewed interest in the Trojan problem in recent years. Significant progress has been made in discovering and understanding dynamical features of motion of Jupiters Trojan asteroids. The dynamics of hypothetical Trojan-type asteroids of other major planets has also been the subject of several recent investigations. This paper offers an overview on the current status of researches on real and hypothetical Trojan asteroids of the major planets. Results of analytical and numerical works are surveyed. Questions of dynamical properties, long-term evolution of orbits, stability regions around the triangular Lagrangian points are discussed among other problems of the Trojans.

Stability of Co-Orbital Motion in Exoplanetary Systems

The stability of co-orbital motions is investigated in such exoplanetary systems, where the only known giant planet either moves fully in the habitable zone, or leaves it for some part of its orbit. If the regions around the triangular Lagrangian points are stable, they are possible places for smaller Trojan-like planets. We have determined the nonlinear stability regions around the Lagrangian point L4 of nine exoplanetary systems in the model of the elliptic restricted three-body problem by using the method of the relative Lyapunov indicators. According to our results, all systems could possess small Trojan-like planets. Several features of the stability regions are also discussed. Finally, the size of the stability region around L4 in the elliptic restricted three-body problem is determined as a function of the mass parameter and eccentricity.

Secondary resonances of co-orbital motions

The size distribution of the stability region around the Lagrangian point L 4 is investigated in the elliptic restricted three-body problem as the function of the mass parameter and the orbital eccentricity of the primaries. It is shown that there are minimum zones in the size distribution of the stability regions, and these zones are connected with the secondary resonances between the frequencies of librational motions around L 4 . The results can be applied to hypothetical Trojan planets for predicting values of the mass parameter and the eccentricity for which such objects can be expected or their existence is less probable.

Survey of the stability region of hypothetical habitable Trojan planets

Aims. In this work we study the dynamical possibility in extrasolar planetary systems that a terrestrial planet can exist in 1:1 mean motion resonance with a Jovian-like planet. We compiled a catalogue of hypothetical habitable Trojan planets, to be able to make a stability forecast for further extrasolar planetary systems discovered in the future. When speaking of habitability we also took the influence of the spectral type of the central star into account. Methods. We integrated some 10 6 orbits of fictitious Trojans around the Lagrangian points for up to 10 7 orbital periods of the primary bodies and checked the stability of the orbital elements and their chaoticity with the aid of the Lyapunov characteristic indicator and maximum eccentricity. The computations were carried out using the dynamical model of the elliptic, restricted three-body problem that consists of a central star, a gas giant moving in the habitable zone, and a hypothetical (massless) terrestrial planet. Results. Our investigations have shown that 7 exoplanetary systems can harbour habitable Trojan planets with stable orbits (HD 93083, HD 17051, HD 28185, HD 27442, HD 188015, HD 99109, and HD 221287, which is a recently discovered system). The comparison of the investigated systems with our catalogue showed matching results, so that we can use the catalogue in practice.

Symplectic mapping for Trojan-type motion in the elliptic restricted three-body problem

A symplectic mapping for Trojan-type motion has been developed in the secularly changing elliptic restricted three-body problem. The mapping describes well the characteristics of Trojan-type dynamics at small eccentricities. By using this mapping the boundary of the stability region has been studied for different values of the initial eccentricities of hypothetical Jupiters Trojans. It has been found that in the secularly changing elliptic case the chaotic diffusion at the border of the stability region is stronger than simply in the elliptic case. An explanation of this observation might be the destruction of the chain of islands of the 13:1 secondary resonance between the short and long period component of the Trojan-like motion, caused possibly by the indirect perturbations of Saturn.

A generalization of Szebehely's equation for three dimensions

Szebehelys equation for the potential generating a prescribed family of orbits in two dimensions is generalized for three-dimensional orbits. A simultaneous system of first-order linear partial differential equations is derived for the determination of the potential in the three-dimensional case. Solutions of this system are found in several cases including Keplers problem too.

Trojan planets in HD 108874? (Research Note)

Aims. It turned out recently that, in addition to a large planet with a semimajor axis a ∼ 1 AU and a low eccentricity (e ∼ 0.07), the extrasolar planetary system HD 108874 harbors another massive planet with 2.43 AU < a < 2.93 AU. The inner planet is orbiting the G5 host star in the habitable zone (=HZ); so that we could established stable regions for Earth-like Trojan planets. Methods. We integrated some 10 5 orbits of fictitious Trojans around the Lagrangian points for up to 107 years and checked the stability of the orbital elements and their chaoticity with the aid of the Fast Lyapunov Indicator. Results. It turns out that this multiplanetary system is the first one where - with the uncertainties in eccentricity and semimajor axes of the outer planet - the existence of Trojan terrestrial planets in stable orbits in the HZ is possible for some combinations of the orbital parameters.

The motion of the perihelion of Trojan asteroids

The problem is investigated by using the equations of Jupiters main perturbations in the eccentricity and in the perihelion longitude of Trojan asteroids. The limits and the period of the variation of the eccentricity and of the perihelion longitude are calculated for 30 Trojans. The perihelion is shown to circulate in 20 cases and to librate for 10 asteroids.