  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 to not 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 marginalising over other
orbital parameters using three different choices of priors for the companion
properties (a uniform prior, a transit-inspired prior, and an non-transiting
disk prior). A separate ensemble of numerical experiments shows that these
compact systems generally cannot contain Jupiter-analogs 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.
