2012ApJ...761..165W

We perform several suites of highly detailed dynamical simulations to investigate the architectures of the 24 Sextantis and HD 200964 planetary systems. The best-fit orbital solution for the two planets in the 24 Sex system places them on orbits with periods that lie very close to 2:1 commensurability, while that for the HD 200964 system places the two planets therein in orbits whose periods lie close to a 4:3 commensurability. In both cases, the proposed best-fit orbits are mutually crossing–a scenario that is only dynamically feasible if the planets are protected from close encounters by the effects of mutual mean-motion resonance (MMR). Our simulations reveal that the best-fit orbits for both systems lie within narrow islands of dynamical stability, and are surrounded by much larger regions of extreme instability. As such, we show that the planets are only feasible if they are currently trapped in mutual MMR–the 2:1 resonance in the case of 24 Sex b and c, and the 4:3 resonance in the case of HD 200964 b and c. In both cases, the region of stability is strongest and most pronounced when the planetary orbits are mutually coplanar. As the inclination of planet c with respect to planet b is increased, the stability of both systems rapidly collapses.