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

Abstract

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 giant's 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 longtime 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.

Interesting dynamics at high mutual inclination in the framework of the Kozai problem with an eccentric perturber

“…This kind of behaviour is classified as mode 2 by Michtchenko et al (2006) and its existence cannot be deduced from the analysis of our representative plane. In the same way, no particular dynamics associated with ∼35° of mutual inclination, as the one reported by Funk et al (2011) and described in the next section, is visible on the representative planes of Fig. 2.…”

“…The main new feature of Fig. 6 is the sudden decrease of the maximal e 1 variation around a value close to 35° for all semimajor axis ratios and eccentricities of the gas giant, as observed in the numerical study of Funk et al (2011). Let us note that this region around 35° is not present for a circular orbit of the perturber and is much in evidence for high semimajor axis ratios, so that the change of dynamics is due to the third‐order terms of the octupole expansion.…”

“…As shown by Funk et al (2011), there exists a dynamical region around 35° of mutual inclination consisting of long‐time stable and particularly low‐eccentricity orbits of the small body. Using a 12th‐order Hamiltonian expansion in eccentricities and inclinations, in particular its action‐angle formulation obtained by Lie transforms in Libert & Henrard (2008), we have shown that this region corresponds to a commensurability of the two frequencies that are the precession rates of the arguments of the pericentre ω 1 and ω 2 .…”

“…Funk et al (2011) have realized a numerical study of the long‐term stability of inclined fictitious Earth‐mass planets moving under the attraction of an eccentric giant planet. The small body is initially on a quasi‐circular orbit.…”

On the stability of possible Trojan planets in the habitable zone: an application to the systems HD 147513 and HD 210277

“…Turnbull & Tarter (2003) have applied the Hill definition of stability and they have compiled a catalogue of habitable systems for the target selection for the search for extraterrestrial intelligence (SETI). Funk et al (2009, 2011) have conducted a stability study of inclined planets in the HZ. Rivera & Haghighipour (2007) have investigated whether the systems ν Andromedae, GJ 876, 47 UMa and 55 Cnc could harbour terrestrial planets in their HZs; they have found that 47 UMa and 55 Cnc are still the only good candidates.…”