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&

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

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

The recently discovered planetary system in the binary GamCep 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 Mjup 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 1 AU up to an inclination of about 15 deg (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 GamCep even for massive planets.Comment: 4 pages, 5 figures, changed 2 references made minor changes due to referees advic

Where are the Uranus Trojans?

The area of stable motion for fictitious Trojan asteroids around Uranus' equilateral equilibrium points is investigated with respect to the inclination of the asteroid's orbit to determine the size of the regions and their shape. For this task we used the results of extensive numerical integrations of orbits for a grid of initial conditions around the points L4 and L5, and analyzed the stability of the individual orbits. Our basic dynamical model was the Outer Solar System (Jupiter, Saturn, Uranus and Neptune). We integrated the equations of motion of fictitious Trojans in the vicinity of the stable equilibrium points for selected orbits up to the age of the Solar system of 5 billion years. One experiment has been undertaken for cuts through the Lagrange points for fixed values of the inclinations, while the semimajor axes were varied. The extension of the stable region with respect to the initial semimajor axis lies between 19.05 < a < 19.3 AU but depends on the initial inclination. In another run the inclination of the asteroids' orbit was varied in the range 0 < i < 60 and the semimajor axes were fixed. It turned out that only four 'windows' of stable orbits survive: these are the orbits for the initial inclinations 0 < i < 7, 9 < i < 13, 31 < i < 36 and 38 < i < 50. We postulate the existence of at least some Trojans around the Uranus Lagrange points for the stability window at small and also high inclinations.Comment: 15 pages, 12 figures, submitted to CMD

Orbital characterization of the \beta Pictoris b giant planet

In June 2010, we confirmed the existence of a giant planet in the disk of the young star Beta Pictoris, located between 8 AU and 15 AU from the star. This young planet offers the rare opportunity to monitor a large fraction of the orbit using the imaging technique over a reasonably short timescale. Using the NAOS-CONICA adaptive-optics instrument (NACO) at the Very Large Telescope (VLT), we obtained repeated follow-up images of the Bpic system in the Ks and L' filters at four new epochs in 2010 and 2011. Complementing these data with previous measurements, we conduct a homogeneous analysis, which covers more than eight yrs, to accurately monitor the Bpic b position relative to the star. On the basis of the evolution of the planet's relative position with time, we derive the best-fit orbital solutions for our measurements. More reliable results are found with a Markov-chain Monte Carlo approach. The solutions favor a low-eccentricity orbit e < 0.17, with semi-major axis in the range 8--9 AU corresponding to orbital periods of 17--21 yrs. Our solutions favor a highly inclined solution with a peak around i=88.5+-1.7 deg, and a longitude of ascending node tightly constrained at Omega = -147.5+-1.5 deg. These results indicate that the orbital plane of the planet is likely to be above the midplane of the main disk, and compatible with the warp component of the disk being tilted between 3.5 deg and 4.0 deg. This suggests that the planet plays a key role in the origin of the inner warped-disk morphology of the Bpic disk. Finally, these orbital parameters are consistent with the hypothesis that the planet is responsible for the transit-like event observed in November 1981, and also linked to the cometary activity observed in the Bpic system.Comment: 10 pages, 12 figures, accepted to A&

Constrains on planets around beta Pic with Harps radial velocity data

Context. The {\beta} Pictoris system with its debris disk and a massive giant planet orbiting at \simeq 9 AU represents an ideal laboratory to study giant planet formation and evolution as well as planet-disk interactions. {\beta} Pic b can also help testing brightness-mass relations at young ages. Other planets, yet undetected, may of course be present in the system. Aims. We aim at putting direct constrains on the mass of {\beta} Pic b and at searching for additional jovian planets on orbits closer than typically 2 AU. Methods. We use high precision Harps data collected over 8 years since 2003 to measure and analyse {\beta} Pic radial velocities. Results. We show that the true mass of {\beta} Pic b is less than 10, 12, 15.5, 20 and 25 MJup if orbiting respectively at 8, 9, 10, 11 and 12 AU. This is the first direct constraint on the mass of an imaged planet. The upper mass found is well in the range predicted by brightness-mass relations provided by current "hot start" models. We also exclude the presence of giant planets more massive than 2.5 MJup with periods less than 100 days (hot Jupiters), more massive than 9 MJup for periods in the range 100-500 days. In the 500-1000 day range, the detection limit is in the brown dwarf domain. Beyond the intrinsic interest for {\beta} Pic, these results show the possibilities of precise RV measurements of early type, rapidly rotating stars.Comment: 6 pages, 9 figures, to appear in Astronomy and Astrophysic

Stability of Terrestrial Planets in the Habitable Zone of Gl 777 A, HD 72659, Gl 614, 47 Uma and HD 4208

We have undertaken a thorough dynamical investigation of five extrasolar planetary systems using extensive numerical experiments. The systems Gl 777 A, HD 72659, Gl 614, 47 Uma and HD 4208 were examined concerning the question of whether they could host terrestrial like planets in their habitable zones (=HZ). First we investigated the mean motion resonances between fictitious terrestrial planets and the existing gas giants in these five extrasolar systems. Then a fine grid of initial conditions for a potential terrestrial planet within the HZ was chosen for each system, from which the stability of orbits was then assessed by direct integrations over a time interval of 1 million years. The computations were carried out using a Lie-series integration method with an adaptive step size control. This integration method achieves machine precision accuracy in a highly efficient and robust way, requiring no special adjustments when the orbits have large eccentricities. The stability of orbits was examined with a determination of the Renyi entropy, estimated from recurrence plots, and with a more straight forward method based on the maximum eccentricity achieved by the planet over the 1 million year integration. Additionally, the eccentricity is an indication of the habitability of a terrestrial planet in the HZ; any value of e>0.2 produces a significant temperature difference on a planet's surface between apoapse and periapse. The results for possible stable orbits for terrestrial planets in habitable zones for the five systems are summarized as follows: for Gl 777 A nearly the entire HZ is stable, for 47 Uma, HD 72659 and HD 4208 terrestrial planets can survive for a sufficiently long time, while for Gl 614 our results exclude terrestrial planets moving in stable orbits within the HZ.Comment: 14 pages, 18 figures submitted to A&

The size of the stability regions of Jupiter Trojans

Context.We have known about the Trojan group of asteroids since 1906. With a suspected number comparable to that of the main-belt, they move around the Lagrangian points L4 and L5 of the Sun-Jupiter system. Aims. During the past century the Trojan asteroids were the topic of many studies of their orbital dynamics and stability. In recent years it has been possible to perform extensive numerical simulations of the motion of these asteroids and thus also to investigate the size and structure of the stability region around the Lagrangian points. The aim of this work is to find new numerical estimates of the size and shape of the stability region. Additionally, the problem of the asymmetry between the two Trojan groups (there seem to be more asteroids in L4 than in L5) is investigated. Methods.We investigate the extension of this regions in the ($\sigma - e$) and ($\sigma - i$) plane (where $\sigma=(\omega+\Omega+M)-(\omega_{\mathrm{JUP}}+\Omega_{\mathrm{JUP}}+M_{\mathrm{JUP}})$) and show how the initial inclination and initial eccentricity change the size and shape of this region. The problem of Trojan asymmetry is investigated by directly analyzing the evolution of orbital parameters. Results.We find both that there is a “critical” eccentricity of $e_{\mathrm{INI}} \approx 0.15$ that plays an important role in the dynamics of Trojan asteroids and influences the structure of the stability regions and that the region of the small amplitudes of $\Delta \sigma=\sigma_{\mathrm{max}}-\sigma_{\mathrm{min}}$ is larger for L4 due to the influence of Saturn and thus the L4 group of Trojans seems to be more stable than the L5 group