\chapter{Improvements and Future Work}
\addcontentsline{lof}{chapter}{\thechapter \hspace{1.5mm} Improvements and Future Work}
Given that the sample size is only seven planetary nebulae, including more PNe would better explore the homogeneity of planetary nebulae. This would allow for a wider range of morphological types and progenitor masses to be tested. Also, probing the fainter outer structures would either corroborate the claims made here or show any discrepancies that would need explanations. A dedicated observing program would be required in most cases since few spatial studies have been made. Since HST time is quite competitive, a ground based program is being pursued. Lastly, it would be prudent to redo the observations for NGC 2440 to definitively decide on its homogeneity. 

The modeling method employed in this work is a basic gradient descent type approach for finding the minima of the parameters. By design, this method doesn't give a sense of errors in each parameter nor insurance that the global minimum is found. A better approach (to be implemented for future work) would be to use a method called Goodman and Weare's Affine Invariant Markov Chain Monte Carlo Ensemble sampling. This method utilizes a random sampling technique in a user defined number of chains. First, the starting points of each chain are randomly chosen from a distribution of possible parameters defined by the user. Next, a change in a parameter is proposed by sampling a normal distribution about the parameter with a user-defined standard deviation. Then, the likelihood of the current position and proposed change are calculated and compared. If the likelihood of the change is higher, it's accepted outright and accepted part of the time if it is lower. The acceptance for lower likelihoods is determined by first calculating the relative probabilities of the change to the original position (always $<$ 1), picking a random number between 0 and 1 to compare with, and keeping the change if the relative probability value is greater than the random number. In this way, the posterior probability distribution is sampled along each chain. From the various chains, the posterior probability distribution for a parameter can be sampled well enough to give highly probable values for that parameter. The best possible values can be determined quite efficiently using this method and using multiple starting locations help ensure that the global minimum is found.

Keeping with the theme of spatial variations, another project is currently underway looking at chemical differences between H II regions in six nearly face-on, nearby spiral galaxies (M 51, M 83, M 100, M 101, NGC 2403, and NGC 2997). H II regions are the birthplace of stars and tracers of the current chemical composition in spiral galaxies. Studying H II regions across a galaxy will map out the chemical distribution, which is an important constraint for determining mixing rates, infall rates, and star formation efficiencies. 

The main goal is to test the assumption that galaxies are chemically homogeneous around the galaxy for a fixed distance from the center. Secondary goals include calculating radial abundance gradients and star formation rates as well as improving the abundance measurements of carbon and neon for metal rich H II regions. These goals are attainable since the galaxies were observed at the McDonald Observatory using the VIRUS-P (Visible Integra-field Replicable Unit Spectrograph-Prototype, \citet{hil08}) instrument. This instrument has 246 fibers, each with a 4.16\ensuremath{''} diameter on the sky, spaced such that a 100\ensuremath{''}x102\ensuremath{''} area near the center of the galaxy can be covered in three exposures as seen for NGC 2403 in Figure~\ref{ngc2403}. These observations cover a full 360\degree~around the galaxy centers and include metal rich H II regions. The full wavelength range is in the optical (3480-6900 \AA) and emission line measurements are being carried out using the program ROBOSPECT (\citet{wat13}), which automates the measurements. Currently, calculations for the abundances of oxygen and nitrogen have been made for NGC 2403 using the ONS (Strong Line) empirical relation of \citet{pil16}. There is a clear radial and inconclusive azimuthal abundance gradient for both elements.  More analysis is underway for NGC 2403 as well as the other galaxies with completion expected in August.
\begin{figure}
\caption{The three exposures from VIRUS-P covering a 100$''$x102$''$ area of the central region of NGC 2403.} 
\includegraphics[scale=0.65]{ngc2403.png}
\label{ngc2403}
\end{figure}
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%\textbf{Chapter 6}\\
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%\textbf{Future Improvements}\\
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