Discovering Planetary Nebula Geometries: Explorations with a Hierarchy of Models

Abstract

Astronomical objects known as planetary nebulae (PNe) consist of a shell of gas expelled by an aging star. In cases where the gas shell can be assumed to be ellipsoidal, the PN can be easily modeled in three spatial dimensions. We utilize a model that joins the physics of PNe to this geometry and generates simulated nebular images. Hubble Space Telescope images of actual PNe provide data with which the model images may be compared. We employ Bayesian model estimation and search the parameter space for values that generate a match between observed and model images. The forward model is characterized by thirteen parameters; consequently 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 a local solution. We find that both the speed and quality of the search can be improved by reducing the dimensionality of the search space. Our basic approach utilizes a hierarchy of models of increasing complexity. 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, estimating its position, angular size, orientation and rim thickness. 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 real ellipsoidal PNe to estimate their geometric properties and gas density profiles.

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