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 ) .
In the simulations , we find that all stable orbits are related to the 2:1 commensurability that can help to remain the semi-major axes for two companions almost unaltered over the secular evolution for 10 ^ { 7 } yr , and the apsidal phase-locking between two orbits can further enhance the stability for this system , because the eccentricities are simultaneously preserved to restrain the planets from frequent close encounters .
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 } \approx 0 ^ { \circ } , \theta _ { 3 } \approx 180 ^ { \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 ( where \lambda _ { 1 , 2 } are , respectively , the mean longitudes of the inner and outer planets ; \varpi _ { 1 , 2 } are the longitudes of periapse ) .
And we find that the other 2:1 resonant systems ( e.g. , GJ 876 or HD 160691 ) may possess one of three stable orbits in their realistic motions .
In addition , we also propose a semi-analytical model to study e _ { i } - \Delta \varpi Hamiltonian contours , which are fairly consistent with direct numerical integrations .
With the updated fit , we then examine the dependence of the stability of this system on the relative inclination , the planetary mass ratios , the eccentricities and other orbital parameters : in the non-coplanar cases , we find that stability requires the relative inclination being \sim 25 ^ { \circ } or less ; as to the planetary mass ratio , the stable orbits for HD 82943 requires \sin i \geq 0.50 for a fixed value or m _ { 1 } / m _ { 2 } \leq 2 where m _ { 2 } remains for the varying mass ratio ; concerning the eccentricities , the system can be always steady when 0 < e _ { 2 } \leq 0.24 and 0 < e _ { 1 } < 0.60 .
Moreover , we numerically show that the assumed terrestrial bodies can not survive near the habitable zones for HD 82943 due to the strong perturbations induced by two resonant companions , but these low-mass planets can be dynamically habitable in the GJ 876 system at \sim 1 AU in the numerical surveys .
Finally , we present a brief discussion on the origin of the 2:1 resonance for HD 82943 .