The ODYSSEUS Survey. Characterizing magnetospheric geometries and hotspot structures in T Tauri stars

Magnetospheric accretion is a key process that shapes the inner disks of T Tauri stars, controlling mass and angular momentum evolution. It produces strong ultraviolet and optical emission that irradiates the planet-forming environment. In this work, we characterize the magnetospheric geometries, accretion rates, extinction properties, and hotspot structures of 67 T Tauri stars in the largest and most consistent study of ultraviolet and optical accretion signatures to date. To do so, we apply an accretion flow model to velocity-resolved H$α$ profiles for T Tauri stars from the HST/ULLYSES program with consistently-derived stellar parameters. We find typical magnetospheric truncation radii to be almost half of the usually-assumed value of 5 stellar radii. We then model the same stars' HST/STIS spectra with an accretion shock model, finding a diverse range of hotspot structures. Phase-folding multi-epoch shock models reveals rotational modulation of observed hotspot energy flux densities, indicative of hotspots that persist for at least 3 stellar rotation periods. For the first time, we perform a large-scale, self-consistent comparison of accretion rates measured using accretion flow and shock models, finding them to be consistent within $\sim$0.16 dex for contemporaneous observations. Finally, we find that up to 50% of the total accretion luminosity is at short wavelengths accessible only from space, highlighting the crucial role of ultraviolet spectra in constraining accretion spectral energy distributions, hotspot structure, and extinction.