The Stable Topology of the Planetary Systems of two 2:1 Resonant Companions:Application to HD 82943

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, ${\theta }_1\approx {\theta }_2\approx {\theta }_3\approx 0^{\circ }$ (aligned case), and (3) Type III, ${\theta }_1\approx {180}^{\circ }$, ${\theta }_2\approx 0^{\circ }$, ${\theta }_3\approx {180}^{\circ }$ (antialigned case), where two resonant arguments are $\theta_{1} = \lambda_{1} - 2\lambda {2} + \varpi{1}$and$\theta_{2} = \lambda_{1} - 2\lambda_{2} + \varpi_{2}$, the relative apsidal longitudes $\theta_{3} = \varpi_{1}-\varpi_{2}=\Delta\varpi$. 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).