Bayesian Dynamic Mapping of an Exo-Earth from Photometric Variability

Photometric variability of a directly imaged exo-Earth conveys spatial information on its surface and can be used to retrieve a two-dimensional geography and axial tilt of the planet (spin-orbit tomography). In this study, we relax the assumption of the static geography and present a computationally tractable framework for dynamic spin-orbit tomography applicable to time-varying geography. First, a Bayesian framework of static spin-orbit tomography is revisited using analytic expressions of the Bayesian inverse problem with a Gaussian prior. We then extend this analytic framework to a time-varying one through a Gaussian process in the time domain, and present analytic expressions that enable efficient sampling from a full joint posterior distribution of geography, axial tilt, spin rotation period, and hyperparameters in the Gaussian process priors. Consequently, it only takes 0.3 s for a laptop computer to sample one posterior dynamic map conditioned on the other parameters with 3072 pixels and 1024 time grids, for a total of ∼3 × 10⁶ parameters. We applied our dynamic mapping method to a toy model and found that the time-varying geography was accurately retrieved along with the axial tilt and spin rotation period. In addition, we demonstrated the use of dynamic spin-orbit tomography with a real multicolor light curve of the Earth as observed by the Deep Space Climate Observatory. We found that the resultant snapshots from the dominant component of a principal component analysis roughly captured the large-scale, seasonal variations of the clear-sky and cloudy areas on the Earth.