Atmospheric characterization of hot Jupiters using hierarchical models of Spitzer observations

The field of exoplanet atmospheric characterization is trending towards comparative studies involving many planetary systems, and using Bayesian hierarchical modelling is a natural next step. Here we demonstrate two use cases. We first use hierarchical modelling to quantify variability in repeated observations by reanalysing a suite of 10 Spitzer secondary eclipse observations of the hot Jupiter XO-3 b. We compare three models: one where we fit 10 separate eclipse depths, one where we use a single eclipse depth for all 10 observations, and a hierarchical model. By comparing the widely applicable information criterion of each model, we show that the hierarchical model is preferred over the others. The hierarchical model yields less scatter across the suite of eclipse depths - and higher precision on the individual eclipse depths - than does fitting the observations separately. We find that the hierarchical eclipse depth uncertainty is larger than the uncertainties on the individual eclipse depths, which suggests either slight astrophysical variability or that single eclipse observations underestimate the true eclipse depth uncertainty. Finally, we fit a suite of published dayside brightness measurements for 37 planets using a hierarchical model of brightness temperature versus irradiation temperature. The hierarchical model gives tighter constraints on the individual brightness temperatures than the non-hierarchical model. Although we tested hierarchical modelling on Spitzer eclipse data of hot Jupiters, it is applicable to observations of smaller planets like hot Neptunes and super-Earths, as well as for photometric and spectroscopic transit or phase-curve observations.