Stability and 2:1 resonance in the planetary system HD 829431 †*

We have explored the secular dynamical evolution of the HD 82943 system with two resonant giant planets, by simulating various planetary configurations via direct numerical integration. We also studied their orbital motions in phase space. In the numerical integrations over 107 yr, we found that all the stable orbits are connected with the 2:1 resonance. Typically, there exists the libration of the two resonant arguments 1 and (or) 2 on the same timescale. Hence, both of the semi-major axes are strongly constrained to behave in a regular way, due to the confinement of the libration of the related angles. Using the analytical model we considered the motion of the inner planet in phase space for different values of the outer planet's eccentricity e2 and of the relative apsidal longitude . We found that the 2:1 orbital resonance is easily preserved when= 0† and when e2 is not too large. A moderate e2 can lock the two planets into deep resonance. The results by the analytical method agree well with those by the numerical simulation, both revealing the 2:1 resonance architecture.