Florian Freistetter

Extrasolar Trojan planets close to habitable zones

We investigate the stability regions of hypothetical terrestrial planets around the Lagrangian equilibrium points L4 and L5 in some specific extrasolar planetary systems. The problem of their stability can be treated in the framework of the restricted three body problem where the host star and a massive Jupiter-like planet are the primary bodies and the terrestrial planet is regarded as being massless. From these theoretical investigations one cannot determine the extension of the stable zones around the equilibrium points. Using numerical experiments we determined their largeness for three test systems chosen from the table of the know extrasolar planets, where a giant planet is moving close to the so-called habitable zone around the host star in low eccentric orbits. The results show the dependence of the size and structure of this region, which shrinks significantly with the eccentricity of the known gas giant.

Planets in habitable zones: A study of the binary Gamma Cephei

The recently discovered planetary system in the binary y Cep was studied concerning its dynamical evolution. We confirm that the orbital parameters found by the observers are in a stable configuration. The primary aim of this study was to find stable planetary orbits in a habitable region in this system, which consists of a double star (a = 21.36 AU) and a relatively close (a = 2.15 AU) massive (1.7 m j u p sin i) planet. We did straightforward numerical integrations of the equations of motion in different dynamical models and determined the stability regions for a fictitious massless planet in the interval of the semimajor axis 0.5 AU < a < 1.85 AU around the more massive primary. To confirm the results we used the Fast Lyapunov Indicators (FLI) in separate computations, which are a common tool for determining the chaoticity of an orbit. Both results are in good agreement and unveiled a small island of stable motions close to I AU up to an inclination of about 15° (which corresponds to the 3:1 mean motion resonance between the two planets). Additionally we computed the orbits of earthlike planets (up to 90 earthmasses) in the small stable island and found out, that there exists a small window of stable orbits on the inner edge of the habitable zone in γ Cep even for massive planets.

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.

Fractal Dimensions as Chaos Indicators

The correlation dimension D 2 is used to develop a method of classification for phase space orbits. D 2 depends only on the mutual distances of the orbit’s points, therefore the time development of the orbit is reflected in the time development of the correlation dimension approximants. It is shown, that this technique allows to investigate the dynamical properties of a phase space orbit, a classification of chaotic and regular orbits and a detection of sticky orbits.

Dynamical evolution and collisions of asteroids with the earth

Abstract We study the dynamical evolution of asteroids which are a potential danger for collisions with the Earth. The work is undertaken on basis of a simplified model of the planetary system including the perturbations from Venus to Saturn. For a whole grid of initial conditions (0.7⩽a⩽1.45 and 0.1⩽e⩽0.8) we integrated the equations of motion with the Lie-series integration for 0.5 Myrs for 720 fictitious objects. The analysis of the data provided the following results: (a) the dynamical evolution for such objects which are initially in four different groups namely Subatens, Atens, Apollos and Amors (b) the flow between these groups and (c) the individual collision probabilities.

RECURRENCE OF ORDER IN CHAOS

The standard map x′ = x + y′, y′ = y + (K/2π)sin(2πx), where both x and y are given modulo 1, becomes mostly chaotic for K ≥ 8, but important islands of stability appear in a recurrent way for values of K near K = 2nπ (groups of islands I and II), and K = (2n + 1)π (group III), where n ≥ 1. The maximum areas of the islands and the intervals ΔK, where the islands appear, follow power laws. The changes of the areas of the islands around a maximum follow universal patterns. All islands surround stable periodic orbits. Most of the orbits are irregular, i.e. unrelated to the orbits of the unperturbed problem K = 0. The main periodic orbits of periods 1, 2 and 4 and their stability are derived analytically. As K increases these orbits become unstable and they are followed by infinite period-doubling bifurcations with a bifurcation ratio δ = 8.72. We find theoretically the connections between the various families and the extent of their stability. Numerical calculations verify the theoretical results.

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)

Our solar system as model for exosolar planetary systems

We investigate the dynamical behaviour of a simplified model of our planetary system (Mercury and the planets Uranus and Neptune were excluded) when we change the mass of the Earth via a mass factor κE ∈ [1, 300]. This is done to study the motions in this “model planetary system” as an example for extrasolar systems. It is evident that the new systems under consideration can only serve as a model for a limited number of exosystems because they have massive planets sometimes with large orbital eccentricities. We did these numerical experiments using an already well tested numerical integration method (LIE-integration) in the framework of the Newtonian equations of motions. We can show that these planetary systems are very stable up to several hundred earth masses, but for some specific values of κE they show a typical chaotic behaviour already in the semi-major axis. It is know from the inner Solar System that the planets move in a small region of weak chaos, but this behaviour (close to κE = 5) was quite unexpected. We then use a 1 order secular theory to explain the appearance of chaos. The results may serve for a better understanding of the dynamics of some extrasolar planetary systems.