The Stable Planetary Geometry of the Exosystems in 2:1 Mean Motion Resonance

(Abridged) We have numerically explored the stable planetary geometry for the multiple systems involved in a 2:1 mean motion resonance, and herein we mainly concentrate on the study of the HD 82943 system by employing two sets of the orbital parameters (Mayor et al. 2004). We find that all stable orbits are related to the 2:1 commensurability for $10^{7}$ yr, and the apsidal phase-locking between two orbits can further enhance the stability for this system. For HD 82943, 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), here the lowest eccentricity-type mean motion 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 $\theta_{3} = \varpi_{1}-\varpi_{2}=\Delta\varpi$. In addition, we also propose a semi-analytical model to study $e_{i}-\Delta\varpi$ Hamiltonian contours. With the updated fit, we examine the dependence of the stability of this system on the orbital parameters. Moreover, we numerically show that the assumed terrestrial bodies cannot survive near the habitable zones for HD 82943 and low-mass planets can be dynamically habitable in the GJ 876 system at $\sim 1$ AU in the numerical surveys.