1. Three-Dimensional Modeling of Planetary Nebulae
Kevin H. Knuth (IC), Karen A. Huyser (IC), Arsen R. Hajian (USNO)
FAST HIERARCHICAL MODELS
We have developed a hierarchy of models that allow us to rapidly capture a critical subset of PN parameters: GAUSS captures the center position and general extent, SIGHAT captures the eccentricity and orientation, and Dual SIGHAT captures the shell thickness. These two-dimensional models significantly decrease the analysis time and increase the accuracy of the final results, in part by assuring that the final solutions are reasonable. With the help of Bernd Fischer and Johann Schumann, these models have been implemented using AutoBayes which, given a model specification, automatically generates the data analysis software.
BACKGROUND
Stars like our sun (initial masses between 0.8 to 8 solar masses) end their lives as swollen red giants surrounded by cool extended atmospheres. The nuclear reactions in their cores create carbon, nitrogen and oxygen, which are transported by convection to the outer envelope of the stellar atmosphere. As the star finally collapses to become a white dwarf, this envelope is expelled from the star to form a planetary nebula (PN) rich in organic molecules. The physics, dynamics, and chemistry of these nebulae are poorly understood and have implications not only for our understanding of the stellar life cycle but also for organic astrochemistry and the creation of prebiotic molecules in interstellar space.
PARAMETERIZING THE PN GAS DENSITY
Each PN is modeled as a prolate ellipsoidal shell of gas. It is assumed that the PN is ionization-bounded (top), which means that the ionizing radiation from the central star is all absorbed before it reaches the outer boundary of the shell.
Since the shell is optically thin, the visible intensity is proportional to the density squared. The greater the column density, the brighter the nebula.
A typical nebula is not uniformly dense.  It has been compressed radially by hot winds from the central star, and may exhibit latitudinal density variations from any of a variety of causes.
Radial density variations are modeled as a power law with exponent g. Latitudinal density variations, which dramatically affect the ionization boundary shape, are modeled by a pole-to-equator ratio b and a latitudinal density gradient a.
In the near future we will be exploring more detailed models of PN densities.
Ionization Bounded Nebula
Spherical Uniform Density
a = 0 b = 1 g = 0
Hourglass shape caused by a 1 to 10 polar to equatorial density ratio
a = 4 b = g = 0
RESOLVING 3D PN CHARACTERISTICS
The 3D characteristics of the PN, along with the viewing angle, have a significant impact on the nature of the resulting projection of the nebula on the night sky. Our algorithms use this sensitive dependence to learn the three-dimensional nebular structure from the data.
INFERRING 3D STRUCTURE
We are inferring three-dimensional planetary nebula (PN) models — including the size, shape, expansion rate, orientation, nebular mass distribution, and distance from Earth — using data consisting of images obtained over time from the Hubble Space Telescope (HST) and long-slit spectra obtained from Kitt Peak National Observatory and Cerro Tololo Inter-American Observatory. These images are taken from a single viewpoint in space, which creates a very challenging tomographic reconstruction.
We employ Bayesian model estimation using a parameterized physical model of the nebula, which incorporates much prior information about the known physics of how the PN is illuminated by the ionizing radiation from the central star. The model (lower right) is used to make a prediction (upper right), which is then compared with the real data (HST image, middle left) to determine how to improve the model. This methodology is extremely powerful and allows us to incorporate multiple disparate data types.
FROM MODEL TO IMAGE
Given a parameterized model of the distribution of the nebular gas, line integrals summing the squared gas density are computed along the line of sight for every image pixel. The result is a synthetic HST image that represents the prediction our model makes.
PUBLICATIONS
Hajian, A.R., Movit, S.M., Balick, B., Terzian, Y., Palen, S., Knuth, K.H., Bond, H., & Panagia, N An atlas of [N II] and [O III] spectra of planetary nebulae. Submitted to: ApJ Supp.
Fischer B., Hajian A.R., Knuth K.H., Schumann J Automatic derivation of statistical data analysis algorithms: Planetary nebulae and beyond. In press: Y. Zhai, G.J. Erickson (eds.), Bayesian Inference and Maximum Entropy Methods in Science and Engineering, Jackson Hole WY 2003, AIP Conference Proceedings, American Institute of Physics, Melville NY.
Knuth K.H., Hajian A.R Hierarchies of models: toward understanding planetary nebulae. In: C. Williams (ed.), Bayesian Inference and Maximum Entropy Methods in Science and Engineering, Moscow ID 2002, AIP Conference Proceedings 659, American Institute of Physics, Melville NY, pp
PROPOSALS
Time-resolved proper motions in planetary nebulae, A.R. Hajian (PI), D.G. Currie (Co-I), K.H. Knuth (Co-I), Y. Terzian (Co-I), B. Balick (Co-I), B.N. Dorland (Co-I). STScI (Hubble Space Telescope) Cycle 13 GO Proposal. In review.
Learning the three-dimensional structures of planetary nebulae, K.H. Knuth (PI), A.R. Hajian (Co-I). OSS Applied Information Systems Research (AISR) Proposal. To be submitted: 3/10/04.
ACKNOWLEDGEMENTS
This work builds on significant past efforts by Steve Movit (Penn State) and Domhnull Granquist-Fraser (Lockheed Martin Corp)
THE MOST PROBABLE MODEL
Bayes’ Theorem can be used to calculate the most probable set of model parameters or, more interestingly, to explore the probability of various models. The posterior probability of the model parameters depends in part on the likelihood function, which quantifies our belief that the model could have resulted in the HST images. The likelihood depends upon the difference between a real image obtained from the HST and the synthetic image obtained from the model. The gradient of the probability computed using this difference tells us how to change the model parameters to improve the results.