In this study, we performed a homogeneous analysis of the planets around FGK
dwarf stars observed by the Kepler and K2 missions, providing spectroscopic
parameters for 310 K2 targets -- including 239 Scaling K2 hosts -- observed
with Keck/HIRES. For orbital periods less than 40 days, we found that the
distribution of planets as a function of orbital period, stellar effective
temperature, and metallicity was consistent between K2 and Kepler, reflecting
consistent planet formation efficiency across numerous ~1 kpc sight-lines in
the local Milky Way. Additionally, we detected a 3X excess of sub-Saturns
relative to warm Jupiters beyond 10 days, suggesting a closer association
between sub-Saturn and sub-Neptune formation than between sub-Saturn and Jovian
formation. Performing a joint analysis of Kepler and K2 demographics, we
observed diminishing super-Earth, sub-Neptune, and sub-Saturn populations at
higher stellar effective temperatures, implying an inverse relationship between
formation and disk mass. In contrast, no apparent host-star spectral-type
dependence was identified for our population of Jupiters, which indicates
gas-giant formation saturates within the FGK mass regimes. We present support
for stellar metallicity trends reported by previous Kepler analyses. Using GAIA
DR3 proper motion and RV measurements, we discovered a galactic location trend:
stars that make large vertical excursions from the plane of the Milky Way host
fewer super-Earths and sub-Neptunes. While oscillation amplitude is associated
with metallicity, metallicity alone cannot explain the observed trend,
demonstrating that galactic influences are imprinted on the planet population.
Overall, our results provide new insights into the distribution of planets
around FGK dwarf stars and the factors that influence their formation and
evolution.Comment: 28 Pages, 12 Figures, 3 Tables; Accepted for Publication A