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Register a new userThe Stable Topology of the Planetary Systems of two 2:1 Resonant Companions:Application to HD 82943
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Ji Jianghui
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We have numerically explored the stable planetary geometry for the multiple systems involved in a 2:1 mean motion resonance, and herein we mainly study the HD 82943 system by employing two sets of the orbital parameters (Mayor et al. 2004; Ji et al. 2004). In the simulations, we find that all stable orbits are related to the 2:1 resonance that can help to remain the semi-major axes for two companions almost unaltered over the secular evolution for $10^{8}$ yr. In addition, we also show that there exist three possible stable configurations:(1) Type I, only $theta_{1} approx 0^{circ}$, (2) Type II, $theta_{1}approxtheta_{2}approxtheta_{3}approx 0^{circ}$ (aligned case), and (3) Type III, $theta_{1}approx 180^{circ}$, $theta_{2}approx0^{circ}$, $theta_{3}approx180^{circ}$ (antialigned case), where two resonant arguments are $theta_{1} = lambda_{1} - 2lambda _{2} + varpi_{1}$ and $theta_{2} = lambda_{1} - 2lambda_{2} + varpi_{2}$, the relative apsidal longitudes $theta_{3} = varpi_{1}-varpi_{2}=Deltavarpi$. And we find that other 2:1 resonant systems (e.g., GJ 876) may possess one of three stable orbits in their realistic motions. Moreover, we also study the existence of the assumed terrestrial bodies at $sim 1$ AU for HD 82943 and GJ 876 systems (see main texts).
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We present an analysis of the HD 82943 planetary system based on a radial velocity data set that combines new measurements obtained with the Keck telescope and the CORALIE measurements published in graphical form. We examine simultaneously the goodness of fit and the dynamical properties of the best-fit double-Keplerian model as a function of the poorly constrained eccentricity and argument of periapse of the outer planets orbit. The fit with the minimum chi_{nu}^2 is dynamically unstable if the orbits are assumed to be coplanar. However, the minimum is relatively shallow, and there is a wide range of fits outside the minimum with reasonable chi_{nu}^2. For an assumed coplanar inclination i = 30 deg. (sin i = 0.5), only good fits with both of the lowest order, eccentricity-type mean-motion resonance variables at the 2:1 commensurability, theta_1 and theta_2, librating about 0 deg. are stable. For sin i = 1, there are also some good fits with only theta_1 (involving the inner planets periapse longitude) librating that are stable for at least 10^8 years. The libration semiamplitudes are about 6 deg. for theta_1 and 10 deg. for theta_2 for the stable good fit with the smallest libration amplitudes of both theta_1 and theta_2. We do not find any good fits that are non-resonant and stable. Thus the two planets in the HD 82943 system are almost certainly in 2:1 mean-motion resonance, with at least theta_1 librating, and the observations may even be consistent with small-amplitude librations of both theta_1 and theta_2.
We carry out numerical simulations to explore the dynamical evolution of the HD 82943 and HD 37124 planetary systems,which both have two Jupiter-like planets. By simulating various planetary configurations in the neighborhood of the fitting orbits, we find three mechanisms to maintain the stability of these systems: For HD 82943,we find that the 2:1 mean motion resonance can act as the first mechanism for all the stable orbits. The second mechanism is the alignment of the periastron of the two planets of HD 82943 system. In the paper,we show one case is simultaneously maintained by the two mechanisms. Additionally,we also use the corresponding analytical models successfully to explain the different numerical results for the system. The third mechanism is the Kozai resonance which takes place in the mutual highly orbits of HD 37124. In the simulations,we discover that the argument of periastron $omega$ of the inner planet librates about $90^{circ}$ or $270^{circ}$ for the whole time span. The Kozai mechanism can explain the stable configuration of large eccentricity of the inner planet.
We present six years of new radial-velocity data from the Anglo-Australian and Magellan Telescopes on the HD 73526 2:1 resonant planetary system. We investigate both Keplerian and dynamical (interacting) fits to these data, yielding four possible configurations for the system. The new data now show that both resonance angles are librating, with amplitudes of 40 degrees and 60 degrees, respectively. We then perform long-term dynamical stability tests to differentiate these solutions, which only differ significantly in the masses of the planets. We show that while there is no clearly preferred system inclination, the dynamical fit with i=90 degrees provides the best combination of goodness-of-fit and long-term dynamical stability.
We perform numerical simulations to explore the dynamical evolution of the HD 82943 planetary system. By simulating diverse planetary configurations, we find two mechanisms of stabilizing the system: the 2:1 mean motion resonance between the two planets can act as the first mechanism for all stable orbits. The second mechanism is a dynamical antialignment of the apsidal lines of the orbiting planets, which implies that the difference of the periastron longitudes $theta_{3}$ librates about $180^{circ}$ in the simulations. We also use a semi-analytical model to explain the numerical results for the system under study.
The results of an extensive numerical study of the periodic orbits of planar, elliptic restricted three-body planetary systems consisting of a star, an inner massive planet and an outer mass-less body in the external 1:2 mean-motion resonance are presented. Using the method of differential continuation, the locations of the resonant periodic orbits of such systems are identified and through an extensive study of their phase-parameter space, it is found that the majority of the resonant periodic orbits are unstable. For certain values of the mass and the orbital eccentricity of the inner planet, however, stable periodic orbits can be found. The applicability of such studies to the 1:2 resonance of the extrasolar planetary system GJ876 is also discussed.
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