  The Kepler mission and its successor K2 have brought forth a cascade of
transiting planets. Many of these planetary systems exhibit multiple members,
but a large fraction possess only a single transiting example. This
overabundance of singles has lead to the suggestion that up to half of Kepler
systems might possess significant mutual inclinations between orbits, reducing
the transiting number (the so-called "Kepler Dichotomy"). In a recent paper,
Spalding & Batygin (2016) demonstrated that the quadrupole moment arising from
a young, oblate star is capable of misaligning the constituent orbits of a
close-in planetary system enough to reduce their transit number, provided that
the stellar spin axis is sufficiently misaligned with respect to the planetary
orbital plane. Moreover, tightly packed planetary systems were shown to be
susceptible to becoming destabilized during this process. Here, we investigate
the ubiquity of the stellar obliquity-driven instability within systems with a
range of multiplicities. We find that most planetary systems analysed,
including those possessing only 2 planets, underwent instability for stellar
spin periods below ~3 days and stellar tilts of order 30 degrees. Moreover, we
are able to place upper limits on the stellar obliquity in systems such as
K2-38 (obliquity <20 degrees), where other methods of measuring spin-orbit
misalignment are not currently available. Given the known parameters of T-Tauri
stars, we predict that up to 1/2 of super-Earth mass systems may encounter the
instability, in general agreement with the fraction typically proposed to
explain the observed abundance of single-transiting systems.
