Warm Jupiters in TESS Full-Frame Images: A Catalog and Observed Eccentricity Distribution for Year 1

Warm Jupiters -- defined here as planets larger than 6 Earth radii with orbital periods of 8--200 days -- are a key missing piece in our understanding of how planetary systems form and evolve. It is currently debated whether Warm Jupiters form in situ, undergo disk or high eccentricity tidal migration, or have a mixture of origin channels. These different classes of origin channels lead to different expectations for Warm Jupiters' properties, which are currently difficult to evaluate due to the small sample size. We take advantage of the \TESS survey and systematically search for Warm Jupiter candidates around main-sequence host stars brighter than the \TESS-band magnitude of 12 in the Full-Frame Images in Year 1 of the \TESS Prime Mission data. We introduce a catalog of 55 Warm Jupiter candidates, including 19 candidates that were not originally released as \TESS Objects of Interest (TOIs) by the \TESS team. We fit their \TESS light curves, characterize their eccentricities and transit-timing variations (TTVs), and prioritize a list for ground-based follow-up and \TESS Extended Mission observations. Using hierarchical Bayesian modeling, we find the preliminary eccentricity distributions of our Warm-Jupiter-candidate catalog using a Beta distribution, a Rayleigh distribution, and a two-component Gaussian distribution as the functional forms of the eccentricity distribution. Additional follow-up observations will be required to clean the sample of false positives for a full statistical study, derive the orbital solutions to break the eccentricity degeneracy, and provide mass measurements.