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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 AJ

We report the discovery of TOI-561, a multi-planet system in the galactic thick disk that contains a rocky, ultra-short period planet (USP). This bright ($V=10.2$) star hosts three small transiting planets identified in photometry from the NASA TESS mission: TOI-561 b (TOI-561.02, P=0.44 days, $R_b = 1.45\pm0.11\,R_\oplus$), c (TOI-561.01, P=10.8 days, $R_c=2.90\pm0.13\,R_\oplus$), and d (TOI-561.03, P=16.3 days, $R_d=2.32\pm0.16\,R_\oplus$). The star is chemically ([Fe/H]$=-0.41\pm0.05$, [$\alpha$/H]$=+0.23\pm0.05$) and kinematically consistent with the galactic thick disk population, making TOI-561 one of the oldest ($10\pm3\,$Gyr) and most metal-poor planetary systems discovered yet. We dynamically confirm planets b and c with radial velocities from the W. M. Keck Observatory High Resolution Echelle Spectrometer. Planet b has a mass and density of $3.2\pm0.8\,M_\oplus$ and $5.5^{+2.0}_{-1.6}\,$g$\,$cm$^{-3}$, consistent with a rocky composition. Its lower-than-average density is consistent with an iron-poor composition, although an Earth-like iron-to-silicates ratio is not ruled out. Planet c is $7.0\pm2.3\,M_\oplus$ and $1.6\pm0.6\,$g$\,$cm$^{-3}$, consistent with an interior rocky core overlaid with a low-mass volatile envelope. Several attributes of the photometry for planet d (which we did not detect dynamically) complicate the analysis, but we vet the planet with high-contrast imaging, ground-based photometric follow-up and radial velocities. TOI-561 b is the first rocky world around a galactic thick-disk star confirmed with radial velocities and one of the best rocky planets for thermal emission studies., Comment: Accepted at The Astronomical Journal; 25 pages, 10 figures