Á. Süli

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.

THE INFLUENCE OF GIANT PLANETS NEAR A MEAN MOTION RESONANCE ON EARTH-LIKE PLANETS IN THE HABITABLE ZONE OF SUN-LIKE STARS

We present a numerical study of several two-planet systems based on the motions of Jupiter and Saturn, in which the two giant planets move in low eccentric orbits close to a mean motion resonance. It is more likely to find two planets with similar characteristics in a system than a clone of the Jupiter-Saturn pair of our solar system. Therefore, we vary the distance between the two planets and their mass ratio by changing Saturns semimajor axis from 8 to 11 AU and increasing its mass by factors of 2-40. The different two-planets were analyzed for the interacting perturbations due to the mean motion resonances of the giant planets. We select several mass ratios for the gas giants, for which we study their influence on test bodies (with negligible mass) moving in the habitable zone (HZ) of a Sun-like star. The orbits are calculated for 2 × 107 yr. In all cases the HZ is dominated by a significant curved band, indicating higher eccentricity, which corresponds to a secular resonance with Jupiter. Interesting results of this study are finding (1) an increase of Venuss eccentricity for the real Jupiter and Saturn masses and the actual semimajor axis of Saturn; (2) an increase of the eccentricity of a test planet at Earths position when Saturns mass was increased by a factor of 3 or more; and (3) if the two giant planets are in 2:1 resonance, we observe a strong influence on the outer region of the HZ.

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 stability of the terrestrial planets with a more massive 'Earth'

Although the long-term numerical integrations of planetary orbits indicate that our planetary system is dynamically stable at least ±4 Gyr, the dynamics of our Solar system includes both chaotic and stable motions: the large planets exhibit remarkable stability on gigayear time-scales, while the subsystem of the terrestrial planets is weakly chaotic with a maximum Lyapunov exponent reaching the value of 1/5 Myr −1 .I nthis paper the dynamics of the Sun- Venus-Earth-Mars-Jupiter-Saturn model is studied, where the mass of Earth was magnified via a mass factor κ E. The resulting systems dominated by a massive Earth may serve also as models for exoplanetary systems that are similar to ours. This work is a continuation of our previous study, where the same model was used and the masses of the inner planets were uniformly magnified. That model was found to be substantially stable against the mass growth. Our simulations were undertaken for more than 100 different values of κ E for a time of 20 Myr, and in some cases for 100 Myr. A major result was the appearance of an instability window at κ E ≈ 5, where Mars escaped. This new result has important implications for theories of the planetary system formation process and mechanism. It is shown that with increasing κ E the system splits into two, well-separated subsystems: one consists of the inner planets, and the other consists of the outer planets. According to the results, the model becomes more stable as κ E increases and only when κ E 540 does Mars escape, on a Myr time-scale. We found an interesting protection mechanism for Venus. These results give insights also into the stability of the habitable zone of exoplanetary systems, which harbour planets with relatively small eccentricities and inclinations. Ke yw ords: celestial mechanics - Solar system: general.

Solving linearized equations of the N-body problem using the Lie-integration method

Several integration schemes exist to solve the equations of motion of the N-body problem. The Lie-integration method is based on the idea to solve ordinary differential equations with Lie-series. In the 1980s, this method was applied to solve the equations of motion of the N-body problem by giving the recurrence formulae for the calculation of the Lie-terms. The aim of this work is to present the recurrence formulae for the linearized equations of motion of N-body systems. We prove a lemma which greatly simplifies the derivation of the recurrence formulae for the linearized equations if the recurrence formulae for the equations of motions are known. The Lie-integrator is compared with other well-known methods. The optimal step-size and order of the Lie-integrator are calculated. It is shown that a fine-tuned Lie-integrator can be 30-40 per cent faster than other integration methods.

Detailed survey of the phase space around Nix and Hydra

We present a detailed survey of the dynamical structure of th e phase space around the new moons of the Pluto–Charon system. The spatial elliptic rest ricted three-body problem was used as model and stability maps were created by chaos indica tors. The orbital elements of the moons are in the stable domain both on the semimajor axis eccentricity and inclination spaces. The structures related to the 4:1 and 6:1 mean motion res ances are clearly visible on the maps. They do not contain the positions of the moons, co firming previous studies. We showed the possibility that Nix might be in the 4:1 resonance if its argument of pericenter or longitude of node falls in a certain range. The results stron gly suggest that Hydra is not in the 6:1 resonance for arbitrary values of the argument of perice nter or longitude of node.

On the influence of the Kozai mechanism in habitable zones of extrasolar planetary systems

Aims. We investigate the long-term evolution of inclined test particles representing a small Earth-like body with negligible gravitational effects (hereafter called massless test-planets) in the restricted three-body problem, and consisting of a star, a gas giant, and a massless test-planet. The test-planet is initially on a circular orbit and moves around the star at distances closer than the gas giant. The aim is to show the influences of the eccentricity and the mass of the gas giant on the dynamics, for various inclinations of the test-planet, and to investigate in more detail the Kozai mechanism in the elliptic problem. Methods. We performed a parametric study, integrating the orbital evolution of test particles whose initial conditions were distributed on the semi-major axis - inclination plane. The gas giants initial eccentricity was varied. For the calculations, we used the Lie integration method and in some cases the Bulirsch-Stoer algorithm. To analyze the results, the maximum eccentricity and the Lyapunov characteristic indicator were used. All integrations were performed for 10 5 periods of the gas giant. Results. Our calculations show that inclined massless test-planets can be in stable configurations with gas giants on either circular or elliptic orbits. The higher the eccentricity of the gas giant, the smaller the possible range in semi-major axis for the test-planet. For gas giants on circular orbits, our results illustrate the well-known results associated with the Kozai mechanism, which do not allow stable orbits above a critical inclination of approximately 40°. For gas giants on eccentric orbits, the dynamics is quite similar, and the massless companion exhibits limited variations in eccentricity. In addition, we identify a region around 35° consisting of long-time stable, low eccentric orbits. We show that these results are also valid for Earth-mass companions, therefore they can be applied to extrasolar systems: for instance, the extrasolar planetary system HD 154345 can possess a 35° degree inclined, nearly circular, Earth-mass companion in the habitable zone.

On the stability of the terrestrial planets as models for exosolar planetary systems

All results, achieved up to now, show the long term stability of our planetary system, although, especially the inner solar system is chaotic, due to some specific secular resonances. We study, by means of numerical integrations, the dynamical evolution of the planetary system where we concentrate on the stability of motion of the terrestrial planets Venus, Earth and Mars. Our model consists of a simplified planetary system with the inner planets Venus, Earth and Mars as well as Jupiter and Saturn. A mass factor κ was introduced to uniformly change the masses of the terrestrial planets; Jupiter and Saturn were involved in the system with their actual masses. We integrated the equations of motion with a Lie-integration method for a time interval of 107 years. It turned out that when 220 < κ < 245 and κ > 250 the system became unstable due to the strong interactions between the planets. We discuss the model planetary systems for small mass-factors 0.5 ≤ κ ≤ 10 and large ones 160 ≤ κ ≤ 270 with the aid of several different numerical tools. These results can be applied to recently discovered exoplanetary systems, which configuration is comparable to our own.

Dynamics of the TrES-2 system

The TrES-2 system harbors one planet which was discovered with the transit technique. In this work we investigate the dynamical behavior of possible additional, lower-mass planets. We identify the regions where such planets can move on stable orbits and show how they depend on the initial eccentricity and inclination. We find, that there are stable regions inside and outside the orbit of TrES-2b where additional, smaller planets can move. We also show that those planets can have a large orbital inclination which makes a detection with the transit technique very difficult (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)