A Hierarchical Bayesian Framework for Inferring the Stellar Obliquity Distribution

Stellar obliquity, the angle between a planet's orbital axis and its host star's spin axis, traces the formation and evolution of a planetary system. In transiting-exoplanet observations, only the sky-projected stellar obliquity can be measured, but this can be deprojected using an estimate of the stellar obliquity. In this paper, we introduce a flexible, hierarchical Bayesian framework that can be used to infer the stellar obliquity distribution solely from sky-projected stellar obliquities, including stellar inclination measurements when available. We demonstrate that while a constraint on the stellar inclination is crucial for measuring the obliquity of an individual system, it is not required for robust determination of the population-level stellar obliquity distribution. In practice, the constraints on the stellar obliquity distribution are mainly driven by the sky-projected stellar obliquities. When applying the framework to all systems with measured sky-projected stellar obliquity, which are mostly hot Jupiter systems, we find that the inferred population-level obliquity distribution is unimodal and peaked at zero degrees. Misaligned systems have nearly isotropic stellar obliquities with no strong clustering near 90°. The diverse range of stellar obliquities prefers dynamic mechanisms, such as planet–planet scattering after a convergent disk migration, which could produce both prograde and retrograde orbits of close-in planets with no strong inclination concentrations other than that at 0°.