Fitting Resonant Exoplanets Orbits. I. Properties and Applications of Fixed-Point Orbits

Intact chains of mean-motion resonances may preserve information about the formation and early dynamical evolution of multiplanet systems. Although most mature multiplanet systems are non-resonant, the resonant fraction may have been substantially higher at young ages. Establishing whether exoplanet data are consistent with a resonant configuration is often challenging because the planetary orbits must be sufficiently well constrained. Moreover, while resonance configurations promote stability, standard orbital fitting methods do not impose such a dynamical constraint, with stability tested only a posteriori. As the first step in a series of papers aimed at favoring stable orbital solutions during the fitting process, we develop a method that restricts solutions to stable fixed-point orbits (fPOs)---coplanar, exactly periodic configurations associated with resonant dynamics. We show that dissipative evolution drive resonant systems toward an fPO state, motivating for fPO-constrained fitting for those planet systems that have avoided large-scale dynamical rearrangements since formation. The fPO model greatly reduces the number of free parameters and yields stable solutions by construction. We determine the best-fitting fPos through $χ^2$ minimization and apply the method to astrometric data for the directly imaged PDS 70 and HR 8799 systems and to radial-velocity data for GJ 876. For HR 8799 we independently recover a previously reported resonant solution, illustrating the fidelity of our method. For PDS 70 we find solutions fall in two groups, with the group where the outer planet c mass exceeds planet b more aligned with previous measurements. Finally, For GJ 876 our fPO fit is on par with previous unconstrained fitting, which strengthens the case that its planets reside deep in resonance.