\chapter{Summary and Conclusions}
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%\textbf{Chapter 5}\\
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%\textbf{Summary and Conclusions}\\
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The level of homogeneity of elements distributed in a small set of PN was investigated. Also, each PN was modeled to constrain the stellar luminosity and temperature. To test the homogeneity, the co-spatial observations described in Paper~I were divided into different spatial regions and spectra were extracted. Individual line strengths were measured and these measurements were used to calculate the nebular temperatures, densities, and abundances. Comparisons of the nebular properties among regions of each PN were then made. 

Next, model of each PN were generated in order to determine the stellar properties using observational constraints. An rms value determined the effectiveness of each model to match each observation, and the rms value was used to choose the best parameters and estimate errors. Finally, three different density profiles (constant density, Gaussian density, and Gaussian with a power-law density profile) were used to see the effects each has on the best stellar parameters. 

The conclusions from this work are as follows. 
\begin{itemize}
\item The planetary nebulae in this sample are chemically homogeneous, which implies the shells of material that were ejected from their respective stars were well-mixed and the resulting nebula is as well. This is in line with current theoretical work on the formation of planetary nebulae. Specifically, the mixing timescale is much shorter than the nuclear timescale in the central star. The homogeneity also means that observations can be taken anywhere across the PN and the resulting abundance will accurately represent the nebula as a whole.
\item The constant density models can constrain the stellar parameters quite well depending on the planetary nebula. Also, the choice of density structure has only a small effect on the final stellar properties. This is advantageous since the model calculation time for non-constant density profiles is considerably higher than for constant density profiles.% For NGC 2440 and IC 2165, the non-constant density profiles were disfavored given the much larger rms values compared to the constant density rms. This implies that they are better represented by the constant density profile even with the large difference in morphology.
\item The progenitor (ZAMS) masses for the majority of this sample were around 1 M$_{\astrosun}$. This is expected since the observed distribution of white dwarf masses peaks at a progenitor mass around 1 M$_{\astrosun}$. The largest mass at 6.55 M$_{\astrosun}$ for NGC 5315 is somewhat questionable given the need to extrapolate from the post-AGB evolutionary models. This can only be improved with more high mass models being calculated.
\end{itemize}