2017MNRAS.468..549B

Motivated by the large number of compact extrasolar planetary systems discovered by the Kepler mission, this paper considers perturbations due to possible additional outer planets. The discovered compact systems sometimes contain multiple transiting planets, so that their orbital angular momentum vectors are tightly aligned. Since planetary orbits are susceptible to forced oscillations of their inclination angles, the highly aligned nature of these systems places constraints on possible additional (non-transiting) planets. If planets in the outer regions of these solar systems have sufficiently large mass or sufficiently small semi-major axis, they will induce the compact inner orbits to oscillate in and out of a transiting configuration. This paper considers the dynamics of the compact systems discovered to host five or more planets. In order not to perturb these systems out of a continually, mutually transiting state, additional planetary companions must generally have periastron p > 10 au. Specific constraints are found for each of the 18 planetary systems considered, which are obtained by marginalizing over other orbital parameters using three different choices of priors for the companion properties (a uniform prior, a transit-inspired prior and a non-transiting disc prior). A separate ensemble of numerical experiments shows that these compact systems generally cannot contain Jupiter analogues without disrupting the observed orbits. We also consider how these constraints depend on system properties and find that the surface density of the planetary system is one of the most important variables. Finally, we provide specific results for two systems, WASP-47 and Kepler-20, for which this analysis provides interesting constraints.