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, $\theta_{1}\approx 180^{\circ}$, $\theta_{2}\approx0^{\circ}$, $\theta_{3}\approx180^{\circ}$ (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).