An Accurate 3D Analytic Model for Exoplanetary Photometry, Radial Velocity, and Astrometry

Abstract

We developed and provide AnalyticLC, a novel analytic method and code implementation for dynamical modeling of planetary systems, including non-coplanar interactions, based on a disturbing function expansion to fourth order in eccentricities and inclinations. AnalyticLC calculates the system dynamics in 3D and the resulting model light curve, radial velocity, and astrometry signatures, enabling simultaneous fitting of these data. We show that for a near-resonant chain of three planets, where the two super-periods are close to each other, the TTVs of the pair-wise interactions cannot be directly summed to give the full system TTVs because the super-periods themselves resonate. We derive the simultaneous three-planet correction and include it in AnalyticLC. We compare the model computed by AnalyticLC to synthetic data generated by an N-body integrator, and evaluate its accuracy. Depending on the maximal order of expansion terms kept, AnalyticLC computation time can be up to an order of magnitude faster than state-of-the-art published N-body integrator TTVFast, with smaller enhancement seen at higher order. The advantage increases for long-term observations as our approach's computation time does not depend on the time-span of the data. Depending on the system parameters, the photometric accuracy is typically a few ppm, significantly smaller than typical Kepler's and other observatories' data uncertainty. Our highly efficient and accurate implementation allows full inversion of a large number of observed systems for planetary physical and orbital parameters, presented in a companion paper.