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The majority of extrasolar planetary systems, detected up to now, consists of a star and a giant planet revolving about it in an eccentric orbit. Therefore we use the elliptic restricted three-body problem as dynamical model. In our approach the giant planet moves in an elliptic orbit around the hosting star and we assume that the Earth-like planet (with negligible mass) moves under their gravitational influence.
We explored the stability properties of an Earth-like planet by investigating the stability structure of the a-e1 parameter planes, where a is the semi-major axis of the Earth-like planet and e1 is the eccentricity of the giant planet. We use dimensionless quantities, i.e. the semi-major axis of the giant planet is the distance unit: a1. Then asystem can be classified by its mass parameter:
μ = mp/(mstar+mp), where mp is the mass of the giant planet and mstar is the mass of the hosting star.
During our investigations we have accepted the working hypothesis that ordered behaviour of an Earth-like planets means stability, while chaotic behaviour means instability through chaotic diffusion of trajectories in the phase space. Thus in order to determine the stability properties of the a-e1 parameter planes, we can apply chaos detection methods. We mainly used the Relative Lyapunov Indicator (Sandor et al. 2000; Sandor et al. 2004) being an efficient and sensitive tool of chaos detection. In some cases we also applied the Fast Lyapunov Indicator (FLI) (Froeschle et al. 1997), and the Maximum Eccentricity Method (MEM) (see Dvorak et al. 2003).
We studied separately two cases: (i) the Earth-like planet revolves between the giant planet and the star (inner orbits), and (ii) the giant planet's orbit is between the star and Earth-like planet (outer orbits). In both cases the mass parameter μ has been changed between (i) 1×10-4 and 9×10-4 with Δ μ=10-4, (ii) between 1×10-3 and 9×10-3 with Δ μ=10-3, and (iii) between
1×10-2 and 5×10-2 with Δ μ=10-2. This results in 23 values for μ. We have also changed the mean anomaly of the giant planet between
M1=0° - 360° with Δ M1 = 45° for inner and outer orbits as well. Additionally, for inner orbits we changed M1 between 0° - 90° with &Delta M1 = 10° too. The argument of pericenter of giant planet as well as the mean anomaly of the Earth-like planet has been fixed to 0°. Thus our catalogue consists of 552 stability maps corresponding to each
(μ, M1) pair of the 23 values of μ and 12 values of M1 both for inner and outer orbits (23×12×2=552).
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In this figure we display the initial conditions for the giant and the Earth-like planets used during our numerical simulations. (Click enlarge!)
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By calculating a stability map, We changed the semi-major axis a of the test planet in two different intervals: (a) for orbits of the test planet ``inside" the orbit of the giant planet, we changed a between 0.1 and 0.9 with a stepsize of 10-3, and (b) for ``outside" orbits between 1.1 and 4.0 with a stepsize of 3.625×10-3. The eccentricity of the giant planet e1has been changed between 0 and 0.5 with a stepsize of 5×10-3. The initial eccentricity of the Earth-like planet has always been fixed to 0.
For a given (μ,M1) pair the stability map is prepared in such a way that for each set of the described initial conditions the RLI (or FLI) value of the corresponding orbit of the test planet is computed and visualized in the (a,e1) parameter plane. Due to the very small stepsize in a and e1, each stability map corresponds to more than 8×104 initial conditions thus providing a very fine resolution.
Light regions on the stability maps correspond to low values of the RLI (10-10), thus ordered, dynamically stable motion of the test planet. Dark shades correspond to large values of the RLI (10-5) and chaotic behaviour of the test planet. The dark unstable regions on the left are due to the proximity of the giant planet. Another common features of the stability maps are the "V"-shaped stipes of resonances. These resonances, depending on the relative orbital positions of the planets, can represent either ordered (stable for infinite time), or weakly chaotic (which may become unstable after very long time) behaviour. With the increase of e1 many resonances overlap giving raise to strongly chaotic and thus very unstable behaviour.
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