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

Abstract

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 planets 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 the state-of-the-art published N-body integrator TTVFast, with a 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 Kepler’s and other observatories’ typical 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.