  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.
