Measuring the Orbital Parameters of Radial Velocity Systems in Mean-motion Resonance: A Case Study of HD 200964

The presence of mean-motion resonances (MMRs) complicates analysis and fitting of planetary systems that are observed through the radial velocity (RV) technique. MMR can allow planets to remain stable in regions of phase space where strong planet-planet interactions would otherwise destabilize the system. These stable orbits can occupy small phase space volumes, allowing MMRs to strongly constrain system parameters, but making searches for stable orbital parameters challenging. Furthermore, libration of the resonant angle and dynamical interaction between the planets introduces another long-period variation into the observed RV signal, complicating analysis of the periods of the planets in the system. We discuss this phenomenon using the example of HD 200964. By searching through parameter space and numerically integrating each proposed set of planetary parameters to test for long-term stability, we find stable solutions in the 7:5 and 3:2 MMRs in addition to the originally identified 4:3 MMR. The 7:5 configuration provides the best match to the data, while the 3:2 configuration provides the most easily understood formation scenario. In reanalysis of the originally published shorter-baseline data, we find fits in both the 4:3 and 3:2 resonances, but not in the 7:5. Because the time baseline of the data is shorter than the resonant libration period, the current best fit to the data may not reflect the actual resonant configuration. In the absence of a full sample of the longer libration period, we find that it is of paramount importance to incorporate long-term stability when the orbital configuration of the system is fit.