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). In this 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 kappa(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 kappa(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 kappa(E)approximate to 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 kappa(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 kappa(E) increases and only when kappa(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

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.Comment: 4 pages, 4 figures, submitted to A&

Planetary transit observations at the University Observatory Jena: TrES-2

We report on observations of several transit events of the transiting planet TrES-2 obtained with the Cassegrain-Teleskop-Kamera at the University Observatory Jena. Between March 2007 and November 2008 ten different transits and almost a complete orbital period were observed. Overall, in 40 nights of observation 4291 exposures (in total 71.52 h of observation) of the TrES-2 parent star were taken. With the transit timings for TrES-2 from the 34 events published by the TrES-network, the Transit Light Curve project and the Exoplanet Transit Database plus our own ten transits, we find that the orbital period is P=(2.470614 +/- 0.000001) d, a slight change by ~0.6 s compared to the previously published period. We present new ephemeris for this transiting planet. Furthermore, we found a second dip after the transit which could either be due to a blended variable star or occultation of a second star or even an additional object in the system. Our observations will be useful for future investigations of timing variations caused by additional perturbing planets and/or stellar spots and/or moons.Comment: 10 pages, 13 figures, 5 tables, acceptes for publication in A

A study of the stability regions in the planetary system HD 74156 - Can it host earthlike planets in habitable zones?

Using numerical methods we thoroughly investigate the dynamical stability in the region between the two planets found in HD 74156. The two planets with minimum masses 1.56 M_JUP (HD 74156b) and 7.5 M_JUP (HD 74156c), semimajor axes 0.276 AU and 3.47 AU move on quite eccentric orbits (e=0.649 and 0.395). There is a region between 0.7 and 1.4 AU which may host additional planets which we checked via numerical integrations using different dynamical models. Besides the orbital evolution of several thousands of massless regarded planets in a three-dimensional restricted 4-body problem (host star, two planets + massless bodies) we also have undertaken test computation for the orbital evolution for fictive planets with masses of 0.1, 0.3 and 1 M_JUP in the region between HD74156b and HD74156c. For direct numerical integrations up to 10^7 years we used the Lie-integrator, a method with adaptive stepsize; additionally we used the Fast Lyapunov Indicators as tool for detecting chaotic motion in this region. We emphasize the important role of the inner resonances (with the outer planet) and the outer resonances (with the inner planet) with test bodies located inside the resonances. In these two "resonance" regions almost no orbits survive. The region between the 1:5 outer resonance (0.8 AU) and the 5:1 inner resonance (1.3 AU), just in the right position for habitability, is also very unstable probably due to three-body-resonances acting there. Our results do not strictly "forbid" planets to move there, but the existence of a planet on a stable orbit between 0.8 and 1.3 AU is unlikely.Comment: submitted to A&A, 4 pages, 5 figure

Fuzzy Characterization of Near-Earth-Asteroids

Due to close encounters with the inner planets, Near-Earth-Asteroids (NEAs) can have very chaotic orbits. Because of this chaoticity, a statistical treatment of the dynamical properties of NEAs becomes difficult or even impossible. We propose a new way to classify NEAs by using methods from Fuzzy Logic. We demonstrate how a fuzzy characterization of NEAs can be obtained and how a subsequent analysis can deliver valid and quantitative results concerning the long-term dynamics of NEAs.Comment: 11 pages, presented at the 7th Alexander von Humboldt Colloquium on Celestial Mechanics (2008), accepted for publication in "Celestial Mechanics and Dynamical Astronomy

Outer edges of debris discs: how sharp is sharp?

Ring-like features have been observed in several debris discs. Outside the main ring, while some systems exhibit smooth surface brightness profiles (SB) that fall off roughly as r**-3.5, others display large luminosity drops at the ring's outer edge and steeper radial SB profiles. We seek to understand this diversity of outer edge profiles under the ``natural'' collisional evolution of the system, without invoking external agents such as planets or gas. We use a statistical code to follow the evolution of a collisional population, ranging from dust grains (submitted to radiation pressure) to planetesimals and initially confined within a belt (the 'birth ring'). The system typically evolves toward a "standard" steady state, with no sharp edge and SB \propto r**-3.5 outside the birth ring. Deviations from this standard profile, in the form of a sharp outer edge and a steeper fall-off, occur only when two parameters take their extreme values: 1) When the birth ring is so massive that it becomes radially optically thick for the smallest grains. However, the required disc mass is here probably too high to be realistic. 2) When the dynamical excitation of the dust-producing planetesimals is so low ( <0.01) that the smallest grains, which otherwise dominate the total optical depth, are preferentially depleted. This low-excitation case, although possibly not generic, cannot be ruled out by observations. Our "standard" profile provides a satisfactory explanation for a large group of debris discs with outer edges and SB falling as r**-3.5. Systems with sharper outer edges, barring other confining agents, could still be explained by ``natural'' collisional evolution if their dynamical excitation is very low. We show that such a dynamically-cold case provides a satisfactory fit for HR4796AComment: Accepted for publication in A&A (abstract truncated here, full version in the pdf file); v2: typos corrected + rephrasing title of Section 5.1.2; v3 :final technical change

The orbit of 2010 TK7. Possible regions of stability for other Earth Trojan asteroids

Recently the first Earth Trojan has been observed (Mainzer et al., ApJ 731) and found to be on an interesting orbit close to the Lagrange point L4 (Connors et al., Nature 475). In the present study we therefore perform a detailed investigation on the stability of its orbit and moreover extend the study to give an idea of the probability to find additional Earth-Trojans. Our results are derived using different approaches: a) we derive an analytical mapping in the spatial elliptic restricted three-body problem to find the phase space structure of the dynamical problem. We explore the stability of the asteroid in the context of the phase space geometry, including the indirect influence of the additional planets of our Solar system. b) We use precise numerical methods to integrate the orbit forward and backward in time in different dynamical models. Based on a set of 400 clone orbits we derive the probability of capture and escape of the Earth Trojan asteroids 2010 TK7. c) To this end we perform an extensive numerical investigation of the stability region of the Earth's Lagrangian points. We present a detailed parameter study in the regime of possible stable tadpole and horseshoe orbits of additional Earth-Trojans, i.e. with respect to the semi-major axes and inclinations of thousands of fictitious Trojans. All three approaches underline that the Earth Trojan asteroid 2010 TK7 finds himself in an unstable region on the edge of a stable zone; additional Earth-Trojan asteroids may be found in this regime of stability.Comment: 11 pages, 16 figure