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

Abstract

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 $\hat{I}{¸}_1\approx 0^{\circ }$, (2) Type II, $\hat{I}{¸}_1\approx \hat{I}{¸}_2\approx \hat{I}{¸}_3\approx 0^{\circ }$ (aligned case), and (3) Type III, $\hat{I}{¸}_1\approx 180^{\circ }$, $\hat{I}{¸}_2\approx 0^{\circ }$, $\hat{I}{¸}_3\approx 180^{\circ }$ (antialigned case), where two resonant arguments are $\hat{I}{¸}_1=\hat{I}{»}_1-2\hat{I}{»}_2+{\varpi }_1$ and $\hat{I}{¸}_2=\hat{I}{»}_1-2\hat{I}{»}_2+{\varpi }_2$, the relative apsidal longitudes $\hat{I}{¸}_3={\varpi }_1-{\varpi }_2=\hat{I}”\varpi$. 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).

9 pages, 4 figures, submitted to Advances in Space Research, proceeding of COSPAR 2004, Paris