Discovering Planetary Nebula Geometries: Explorations with a Hierarchy of Models

Astronomical objects known as planetary nebulae (PNe) consist of a shell of gas expelled by an aging medium-sized star as it makes its transition from a red giant to a white dwarf. In many cases this gas shell can be approximately described as a prolate ellipsoid. Knowledge of the physics of ionization processes in this gaseous shell enables us to construct a model in three dimensions (3D) called the Ionization-Bounded Prolate Ellipsoidal Shell model (IBPES model). Using this model we can generate synthetic nebular images, which can be used in conjunction with Hubble Space Telescope (HST) images of actual PNe to perform Bayesian model estimation. Since the IBPES model is characterized by thirteen parameters, model estimation requires the search of a 13-dimensional parameter space. The 'curse of dimensionality,' compounded by a computationally intense forward problem, makes forward searches extremely time-consuming and frequently causes them to become trapped in local solutions. We find that both the speed and of the search can be improved by judiciously reducing the dimensionality of the search space. Our basic approach employs a hierarchy of models of increasing complexity that converges to the IBPES model. Earlier studies establish that a hierarchical sequence converges more quickly, and to a better solution, than a search relying only on the most complex model. Here we report results for a hierarchy of five models. The first three models treat the nebula as a 2D image, while the last two models explore its characteristics as a 3D object and enable us to characterize the physics of the nebula. This five-model hierarchy is applied to HST images of ellipsoidal PNe to estimate their geometric properties and gas density profiles.