Planetary Nebula Modeling

Kevin H. Knuth
kevin.h.knuth@nasa.gov
http://www.huginn.com/knuth

This work is performed in collaboration with Arsen Hajian and his group at the United States Naval Observatory.

The Knuth Lab team members are:
Karen Huyser - Post-doctoral Fellow Education Associates NASA Ames, Stanford University Electrical Engineering Dept.
Bernd Fischer - NASA Ames (Automated Software Engineering Area)
Johann Schumann- NASA Ames (Automated Software Engineering Area)

Former Knuth Lab Members:
Domhnull Granquist-Fraser - Post-doctoral Fellow (currently at Lockheed Martin)

Last updated: 11 November 2004
INTRODUCTION
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.
 
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.
 
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, 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.
 

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.
 

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.

Back to top