\chapter{Discussion}
\addcontentsline{lof}{chapter}{\thechapter \hspace{1.5mm} Discussion}
\addcontentsline{lot}{chapter}{\thechapter \hspace{1.5mm} Discussion}
%\begingroup
%\fontsize{14}{16.8}\selectfont
%\textbf{Chapter 4}\\
%\endgroup
%\begingroup
%\fontsize{13}{15.6}\selectfont
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%\textbf{Discussion}\\
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%\endgroup
One of the major results of this study is that within the uncertainties, the elements He, C, N, O, Ne, Ar, and S are distributed homogeneously across each planetary nebula in this sample (Figures~\ref{fig:abun1}-\ref{fig:abun3} and Table~\ref{tab:abun1}). This is the first time these planetary nebulae have been shown to be chemically homogeneous at sub-arcsecond, effective spatial resolution. This abundance homogeneity for each PN supports current planetary nebula formation theory. During the asymptotic giant branch stage of a star's evolution, convection processes will mix recent nuclear fusion products like carbon, nitrogen, and helium into the surface layers which are eventually ejected during the formation of the PN. The relevant timescales are those of nuclear fusion and envelope convection. The nuclear timescale is of the order 10$^5$-10$^6$ years whereas the convection process is model dependent and the timescale is calculated by the taking the size of the envelope divided by the velocity of the convective atmospheric elements (\citet{ren81}). Models by \citet{bue97} (private communication) imply a convection timescale on the order of a year. Therefore, the mixing and subsequent ejection of matter, such as $^{12}$C, $^{13}$C and $^{14}$N, to form the planetary nebula is much shorter than the nuclear timescales, and each ejection should be homogeneous. 

Stellar parameters determined by various authors over the last three decades are compared to the best-fit constant density values in Table~\ref{tlstar} and Figure~\ref{tefflumfig}. The first column in Table~\ref{tlstar} gives the name of each sample PN followed in subsequent columns by the luminosities [in log(L/L$_{\astrosun}$)] and temperatures (in kK) for each source. As can be seen in Figure~\ref{tefflumfig}, the stellar parameters for IC 2165 agree within the 1$\sigma$ uncertainties with those from Paper~II and the temperature agrees with that of \citet{zha93}. The stellar parameters for IC 3568 are consistent with the other authors, except for the temperature of Paper~II (300 K smaller than the lower limit). The luminosity for NGC 2440 is significantly lower than the other published values, although the temperature matches (given the uncertainties) with that of \citet{zha93}. NGC 3242's temperature and luminosity are consistent with all authors except for \citet{zha93} and \citet{cor03}, respectively. For NGC 5315, the luminosity is significantly higher compared to the values from the other authors, except for the upper bound determined by \citet{sha89}. The temperatures for NGC 5315 and NGC 5882 are consistent with Paper~II and \citet{zha93}, respectively. The luminosities for NGC 5882 and NGC 7662 match with the values of \citet{fre08} and Paper~II, within uncertainties. Lastly, \citet{sha85} and \citet{fre08} have temperatures consistent with the value for NGC 7662. 
\begin{deluxetable}{lcccccc}
\tabletypesize{\footnotesize}
%%%\rotate
\tablecaption{Comparison: Stellar Parameters \label{tlstar}}
\tablewidth{0pt}
\tablehead{
\colhead{Object} & \colhead{Model} & \colhead{SK85/89\tablenotemark{a}} & \colhead{ZK93\tablenotemark{b}} & \colhead{CSSP03\tablenotemark{c}} & \colhead{F08\tablenotemark{d}} & \colhead{Paper~II\tablenotemark{e}} \\
\cline{1-7} \colhead{} & \multicolumn{5}{c}{Luminosity (log[L/L$_{\astrosun}$])} 
}
\startdata
IC 2165 & 3.17 & 3.76 & 3.95 & 3.51 & \nodata & 3.16\\
IC 3568 & 3.52 & 4.12 & 3.78 & 3.73 & \nodata & 3.88\\
NGC 2440 & 2.74 & 3.88 & 3.51 & \nodata & 3.32 & 3.10\\
NGC 3242 & 3.36 & 3.59 & 3.40 & 3.88 & 3.54 & 3.64 \\
NGC 5315 & 4.89 & $<$5.25 & 3.95 & \nodata & \nodata & 3.50\\
NGC 5882 & 3.45 & 3.72 & 3.80 & \nodata & 3.52 & 3.45\\
NGC 7662 & 3.24 & 3.89 & 3.76 & 3.99 & 3.42 & 3.42\\
\tableline
 & \multicolumn{5}{c}{Effective Temperature (kK)}\\
\tableline
IC 2165 & 110.0 & 118.0 & 112.1 & 154.9 & \nodata & 110.0\\
IC 3568 & 69.6 & 52.0 & 51.3 & 55.0 & \nodata & 51.0\\
NGC 2440 & 169.8 & 112.0 & 178.8 & \nodata & 208.0 & 198.0\\
NGC 3242 & 89.7 & 90.0 & 75.0 & 89.0 & 89.0 & 89.0 \\
NGC 5315 & 69.9 & 61.0 & 59.9 & \nodata & \nodata & 70.0\\
NGC 5882 & 78.7 & 70.0 & 73.0 & \nodata & 68.0 & 70.0\\
NGC 7662 & 109.9 & 113.0 & 96.8 & 100.0 & 111.0 & 95.0\\
\enddata
\tablenotetext{a}{\citet{sha85} and \citet{sha89}.}
\tablenotetext{b}{\citet{zha93}.}
\tablenotetext{c}{\citet{cor03}.}
\tablenotetext{d}{\citet{fre08}.}
\tablenotetext{e}{\citet{hen15}.}
\end{deluxetable}
\begin{landscape}
\begin{figure}
\caption{Comparison of the best fit constant density stellar temperature and luminosity for each planetary nebula to other values in the literature.\newline}
\includegraphics[scale=0.46]{Rtefflum.png}
\label{tefflumfig}
\end{figure}
\end{landscape}

Values of the final mass and radius as well as the main sequence lifespan, ZAMS mass, radius, luminosity, temperature, and spectral type can be estimated from the model-derived T and L values for each planetary nebula. Each of these derived properties is shown in Table~\ref{stellprop}. The final and ZAMS masses are estimated by interpolating between theoretical post-AGB evolutionary tracks on the H-R diagram shown in Figure~\ref{agbmods}. For NGC 5315, the mass estimates are from extrapolation since there are no post-AGB evolutionary tracks that have been successfully modeled with luminosities that high. Each track is from either \citet{vas94} (VW 1994, red solid lines), \citet{sch83} (S 1983, blue dash lines), or \citet{ber16} (MB 2016, green dash and purple dash-dot lines). Beneath each author on the plot is the ZAMS/final mass in solar masses following each evolutionary track from top to bottom. The ZAMS/final masses from different interpolations were averaged together when overlapping occurred since a single set of models was insufficient for all calculations. The final masses for five of the PNe match with the white dwarf mass peak, 0.565$M_{\astrosun}$, from \citet{lie05}. The current radius was calculated using the Stefan-Boltzmann law. Mass-luminosity and mass-radius relations from \citet{dem91} (appropriate for a given mass) were used to calculate the ZAMS luminosity and radius, respectively. The Stefan-Boltzmann law was also employed to calculate the ZAMS temperature. The main sequence lifespan was based on the approximate lifespan of the Sun, $\tau_{\astrosun}\approx$ M$_{\astrosun}$/L$_{\astrosun}$ = 9.5 Gyr, and the age of the Milky Way Galaxy, 13.2 Gyr, was used as a hard upper limit. Lastly, the ZAMS spectral type was determined from the ZAMS luminosity with the spectral type range within the brackets being based on the uncertainties in the luminosity.
%\begin{landscape}
\begin{figure}
\caption{Log \textit{L/L$_{sun}$} vs. log \textit{T$_{eff}$} for the entire planetary nebula sample. Post-AGB model tracks from \citet{vas94} (VW 1994, solid red lines, Z=0.016), \citet{sch83} (S 1983, blue dashed lines, Z=0.016) and \citet{ber16} (MB 2016, purple dash-dot lines, Z=0.02 and green dash lines, Z=0.01) are overlaid. Beneath each author on the plot is the ZAMS/final mass in solar masses following each evolutionary track from top to bottom.}
\includegraphics[scale=0.85]{RagbModels7PNe.eps}
\label{agbmods}
\end{figure}
%\end{landscape}
\begin{landscape}
\begin{deluxetable}{cccccccc}
\tabletypesize{\footnotesize}
\tablewidth{0in}
%%%\rotate
\tablecaption{Derived Stellar Parameters.\label{stellprop}}
\tablehead{
\colhead{Stellar Property} &
\colhead{IC 2165} &
\colhead{IC 3568} &
\colhead{NGC 2440} &
\colhead{NGC 3242} &
\colhead{NGC 5315} &
\colhead{NGC 5882} &
\colhead{NGC 7662}
}
\startdata
Final Mass (M$_{\astrosun}$)				& 0.56$^{+0.02}_{-0.01}$ & 0.55$^{+0.26}_{-0.01}$ & 0.66$^{+0.08}_{-0.06}$ & 0.56$^{+0.01}_{-0.01}$ & 1.17$^{+0.09}_{-0.09}$ & 0.56$^{+0.03}_{-0.01}$ & 0.56$^{+0.02}_{-0.01}$ \\
Current Radius (R$_{\astrosun}$) 				& 0.11$^{+0.08}_{-0.03}$ & 0.16$^{+0.36}_{-0.07}$ & 0.03$^{+0.01}_{-0.01}$ & 0.20$^{+0.10}_{-0.07}$ & 1.90$^{+0.53}_{-0.55}$ & 0.29$^{+0.15}_{-0.06}$ & 0.12$^{+0.04}_{-0.03}$ \\
Main Sequence Lifespan (Gyr) 				& 9.5$^{+3.7}_{-4.3}$ & 8.3$^{+4.9}_{-8.1}$ & 0.5$^{+0.8}_{-0.2}$ & 10.7$^{+2.5}_{-6.4}$ & 0.04$^{+0.01}_{-0.01}$ & 8.9$^{+3.5}_{-4.4}$ & 8.7$^{+3.4}_{-4.0}$ \\
ZAMS Mass (M$_{\astrosun}$)			& 0.99$^{+0.22}_{-0.15}$ & 1.04$^{+3.02}_{-0.23}$ & 2.81$^{+0.66}_{-0.81}$ & 0.95$^{+0.35}_{-0.09}$ & 6.55$^{+0.66}_{-0.65}$ & 1.02$^{+0.27}_{-0.11}$ & 1.02$^{+0.24}_{-0.11}$ \\
ZAMS Radius (R$_{\astrosun}$) 			& 1.05$^{+0.22}_{-0.15}$ & 1.10$^{+1.79}_{-0.23}$ & 2.36$^{+0.29}_{-0.40}$ & 1.01$^{+0.34}_{-0.09}$ & 3.78$^{+0.21}_{-0.21}$ & 1.08$^{+0.27}_{-0.11}$ & 1.08$^{+0.24}_{-0.11}$ \\
ZAMS $L_{star}$ ($L_{\astrosun}$) 		& 0.99$^{+1.21}_{-0.47}$ & 1.20$^{+247.43}_{-0.75}$ & 58.65$^{+75.32}_{-43.09}$ & 0.84$^{+1.98}_{-0.27}$ & 1618.68$^{+734.07}_{-546.60}$ & 1.09$^{+1.68}_{-0.39}$ & 1.12$^{+1.47}_{-0.40}$ \\
ZAMS Temperature (kK) 				& 5.6$^{+6.0}_{-4.0}$ & 5.8$^{+7.7}_{-0.7}$ & 10.4$^{+1.7}_{-2.2}$ & 5.5$^{+0.9}_{-0.3}$ & 18.9$^{+1.3}_{-1.3}$ & 5.7$^{+0.7}_{-0.3}$ & 5.7$^{+0.6}_{-0.3}$ \\
ZAMS Spectral Type 				& G2 [F6-G8] & G0 [B6-K3] & B8 [B7-A3] & G4 [G7-F5] & B2 [B2-B3] & G1 [F5-G5] & G1 [F5-G5] \\
\enddata
\end{deluxetable}
\end{landscape}
\citet{amn95} suggested that an increase in progenitor mass would increase the probability of low mass companions, which would determine the axisymmetric morphology of the planetary nebula. As can be seen in Table~\ref{stellprop}, the central stars of IC 2165, IC 3568, NGC 3242, NGC 5882, and NGC 7662 are each solar-like while those of NGC 2440 and NGC 5315 are super-solar. NGC 5882, NGC 2440, and NGC 5315 are the only PN in this sample to exhibit very asymmetric structures (Figures~\ref{fig:slits1}~\&~\ref{fig:slits2}), in support of the claim made by \citet{amn95}. 