\chapter{Are Planetary Nebulae Chemically Homogeneous?}
\addcontentsline{lof}{chapter}{\thechapter \hspace{1.5mm} Are Planetary Nebulae Chemically Homogeneous?}
\addcontentsline{lot}{chapter}{\thechapter \hspace{1.5mm} Are Planetary Nebulae Chemically Homogeneous?}
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%\textbf{Chapter 3}\\
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%\textbf{Results}\\
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%\textbf{3.1 Element Distribution Within Each Planetary Nebula}
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With the abundances calculated from ELSA, a direct comparison between regions can be made to look for possible differences. Figures~\ref{fig:abun1}-\ref{fig:abun3} compare the total abundances of each region with the values scaled to the abundance of the full region in each planetary nebula. The bottom two panels of Figure~\ref{fig:abun3} also compare the [O III] temperature and C III] density of each region with each value scaled in the same manner as the abundances. Out of all the regions, only region 2 of NGC 2440 and region 1 of NGC 5315 show abundances that are significantly lower than the respective full regions. For region 2 of NGC 2440, this can likely be attributed to both the lack of co-spatiality between the UV and optical observations as well as NGC 2440's high ionization stratification (see e.g. \citet{lop98} and \citet{ric91}). Spurious abundances would result from combining areas where the distributions of the ionized states for a particular element are different. For region 1 of NGC 5315, a higher electron temperature is measured compared with the temperature of its full region, causing the inferred abundances to be lower. However, the temperature is consistent (given the errors) with the full region, and substituting the full region's temperature value in for the abundance calculations of Region 1 yields abundances consistent with the full region's values.

\begin{landscape}
\begin{figure}
\caption{Helium, carbon, and nitrogen abundances for each region listed in Table~\ref{tab:abun1} scaled to the full region of each PN. The abundances at nearly all positions in each object are consistent within errors, with outliers discussed in the text.\newline}
\includegraphics[scale=0.47]{Abun1.png}
\label{fig:abun1}
\end{figure}
\end{landscape}
\begin{landscape}
\begin{figure}
\caption{Same as Figure~\ref{fig:abun1} but for oxygen, neon and argon.\newline}
\includegraphics[scale=0.47]{Abun2.png}
\label{fig:abun2}
\end{figure}
\end{landscape}
\begin{landscape}
\begin{figure}
\caption{Top panel: Same as Figure~\ref{fig:abun1} but for sulfur. Middle and bottom panels show the [O III] electron temperature and C III] electron density listed in Table~\ref{ionabun1} for each region in a PN.\newline}
\includegraphics[scale=0.47]{Abun3.png}
\label{fig:abun3}
\end{figure}
\end{landscape}

A literature search over the last twenty years shows that only NGC 2440, NGC 3242, and NGC 5882 from this sample have been investigated previously for abundance variations.~\citet{per98} presented results on the abundances of helium, oxygen, nitrogen, neon, argon, and sulfur for NGC 2440. They made long-slit observations from 3600-9600 \AA~along the major axis using the Boller \& Chivens spectrograph and the 1.52m ESO telescope at La Silla, Chile. The observations were divided into seven regions spanning the slit length and width of 76.1$''$ and 2$''$, respectively. There were no observed spatial variations between the seven regions within their uncertainties for every element. \citet{mon13} addressed the idea of chemical inhomogeneity in NGC 3242. They observed the nebula between 3900-7000~\AA~using 6400 fibers onboard the instrument VIMOS-IFU, covering an area of 54$''$x54$''$. They measured accurate oxygen and helium abundances from the most abundant ionic species in this range. For other elements in the study, i.e., nitrogen, sulfur and chlorine, only less abundant ionic species were observable, resulting in large (as high as 6x) discrepancies in the total abundances measured at different locations in the nebula. \citet{mon13} concluded that helium and oxygen were homogeneous throughout NGC 3242. NGC 5882 was observed by~\citet{gue99} over the wavelength range 3400-10,300 \AA~using the same instrument as the one employed by~\citet{per98}. They compared the outermost halo to the inner two shells as a whole. They found the abundances of helium, oxygen, and neon to be consistent within errors between the halo and shells. 

Table~\ref{tab:abuncomp} compares the full region abundances for each planetary nebula with other values in the literature. Each PN has two comparison values from either authors with quoted uncertainties or average values with standard deviations from multiple authors. The majority of the abundances match within 1$\sigma$ and nearly all are consistent within 2$\sigma$ with the other authors. The largest discrepancy is seen in the nitrogen abundance of NGC 2440 when compared to the value of \citet{kwi03}. This is most likely caused by the fact that the nitrogen abundance of \citet{kwi03} was calculated using the ICF method (ICF value of 9.71) whereas this value is the sum of the observed ionic species, which includes the most populated ionic states.
\begin{landscape}
\begin{deluxetable}{lcccccc}
\tabletypesize{\footnotesize}
\setlength{\tabcolsep}{0.07in}
\tablewidth{0in}
%\rotate
\tablecaption{Comparison: Total Elemental Abundances\label{tab:abuncomp}}
\tablehead{
\multicolumn{7}{c}{Comparison: Total Elemental Abundances}
}
\startdata
 & \multicolumn{3}{c}{IC 2165 Full} & \multicolumn{3}{c}{IC 3568 Full}\\
\tableline
Element & This Work & \citet{kwi03} & BH\tablenotemark{a} & This Work & \citet{hen04} & \citet{kwi98} \\
\tableline
He/H (10$^{-2}$) 	& 11.30$^{+1.65}_{-1.65}$ & 9.00$^{+3.00}_{-3.00}$ & 10.83$^{+0.66}_{-0.66}$ & 9.53$^{+1.36}_{-1.36}$ & 12.00$^{+4.00}_{-4.00}$ & 11.00$^{+1.65}_{-1.65}$ \\
C/H (10$^{-4}$) 	& 4.14$^{+0.87}_{-0.87}$ & \nodata & 6.54$^{+1.76}_{-1.76}$ & 4.66$^{+2.71}_{-2.71}$ & \nodata & 0.97$^{+0.49}_{-0.49}$ \\
N/H (10$^{-4}$) 	& 1.18$^{+0.29}_{-0.29}$ & 1.33$^{+0.19}_{-0.19}$ & 1.20$^{+0.32}_{-0.32}$ & \nodata & \nodata & \nodata\\
O/H (10$^{-4}$) 	& 3.39$^{+0.64}_{-0.64}$ & 3.11$^{+0.44}_{-0.44}$ & 2.28$^{+0.31}_{-0.31}$ & 4.06$^{+1.26}_{-1.26}$ & 3.77$^{+0.53}_{-0.53}$ & 3.14$^{+0.47}_{-0.47}$ \\
Ne/H (10$^{-5}$) 	& 5.36$^{+1.26}_{-1.26}$ & 6.90$^{+0.93}_{-0.93}$ & 5.66$^{+2.23}_{-2.23}$ & 8.15$^{+1.26}_{-1.26}$ & 7.16$^{+1.13}_{-1.13}$ & 4.08$^{+0.61}_{-0.61}$ \\
S/H (10$^{-6}$) 	& 1.80$^{+0.31}_{-0.31}$ & 2.70$^{+0.31}_{-0.31}$ & 3.62$^{+1.29}_{-1.29}$ & \nodata & \nodata & \nodata\\
Ar/H (10$^{-7}$) 	& 10.10$^{+3.42}_{-3.42}$ & 16.00$^{+2.18}_{-2.18}$ & 13.23$^{+4.57}_{-4.57}$ & 12.70$^{+3.91}_{-3.91}$ & 10.56$^{+3.39}_{-3.39}$ & \nodata\\
\tableline
\\
 & \multicolumn{3}{c}{NGC 2440 Full} & \multicolumn{3}{c}{NGC 3242 Full}\\
\tableline
Element & This Work & \citet{kwi03} & \citet{hyu98} & This Work & \citet{hen00} & \citet{mil02}\\
\tableline
He/H (10$^{-2}$) 	& 12.60$^{+0.84}_{-0.84}$ & 10.00$^{+3.00}_{-3.00}$ & 12.00$^{+1.20}_{-1.20}$ & 10.60$^{+0.45}_{-0.45}$ & 8.00$^{+1.00}_{-1.00}$ & 11.00$^{+3.30}_{-3.30}$\\
C/H (10$^{-4}$) 	& 2.18$^{+0.44}_{-0.44}$ & \nodata & 4.00$^{+2.00}_{-2.00}$ & 2.35$^{+0.36}_{-0.36}$ & 4.12$^{+1.66}_{-1.66}$ & \nodata\\
N/H (10$^{-4}$) 	& 4.62$^{+0.77}_{-0.77}$ & 10.67$^{+1.51}_{-1.51}$ & 9.50$^{+2.38}_{-2.38}$ & \nodata & \nodata & \nodata\\
O/H (10$^{-4}$) 	& 4.30$^{+0.65}_{-0.65}$ & 5.19$^{+0.73}_{-0.73}$ & 4.40$^{+0.70}_{-0.70}$ & 4.44$^{+0.36}_{-0.36}$ & 3.38$^{+0.51}_{-0.51}$ & 4.37$^{+1.31}_{-1.31}$\\
Ne/H (10$^{-5}$) 	& 7.03$^{+1.24}_{-1.24}$ & 10.70$^{+1.56}_{-1.56}$ & 9.00$^{+1.35}_{-1.35}$ & 9.62$^{+0.91}_{-0.91}$ & 8.10$^{+1.35}_{-1.35}$ & 9.00$^{+2.70}_{-2.70}$\\
S/H (10$^{-6}$) 	& 1.52$^{+0.16}_{-0.16}$ & 2.80$^{+0.52}_{-0.52}$ & 2.00$^{+0.60}_{-0.60}$ & \nodata & \nodata & \nodata\\
Ar/H (10$^{-7}$) 	& 48.00$^{+6.47}_{-6.47}$ & 40.60$^{+5.71}_{-5.71}$ & 21.00$^{+4.20}_{-4.20}$ & \nodata & \nodata & \nodata\\
\tableline
\\
& \multicolumn{3}{c}{NGC 5315 Full} & \multicolumn{3}{c}{NGC 5882 Full}\\
\tableline
Element & This Work & \citet{pei04} & TKSDT\tablenotemark{b} & This Work & \citet{kwi03} & \citet{gue99} \\
\tableline
He/H (10$^{-2}$) 	& 11.90$^{+0.17}_{-0.17}$ & 11.75$^{+0.11}_{-0.11}$ & 10.71$^{+1.20}_{-1.20}$ & 11.50$^{+0.73}_{-0.73}$ & 12.00$^{+4.00}_{-4.00}$ & 11.10$^{+0.70}_{-0.70}$\\
C/H (10$^{-4}$) 	& 5.20$^{+0.65}_{-0.65}$ & 7.08$^{+1.47}_{-1.47}$ & 5.96$^{+5.40}_{-5.40}$ & 2.72$^{+1.09}_{-1.09}$ & \nodata & \nodata\\
N/H (10$^{-4}$) 	& 7.86$^{+1.01}_{-1.01}$ & 6.92$^{+2.39}_{-2.39}$ & 3.61$^{+1.75}_{-1.75}$ & 2.34$^{+0.83}_{-0.83}$ & 1.80$^{+0.27}_{-0.27}$ & 1.40$^{+0.40}_{-0.40}$\\
O/H (10$^{-4}$) 	& 5.04$^{+0.33}_{-0.33}$ & 7.41$^{+0.51}_{-0.51}$ & 6.02$^{+1.55}_{-1.55}$ & 7.63$^{+1.49}_{-1.49}$ & 5.48$^{+0.77}_{-0.77}$ & 4.93$^{+0.45}_{-0.45}$\\
Ne/H (10$^{-5}$) 	& 12.90$^{+1.01}_{-1.01}$ & 20.41$^{+4.70}_{-4.70}$ & 16.66$^{+4.95}_{-4.95}$ & 17.80$^{+4.21}_{-4.21}$ & 14.80$^{+2.19}_{-2.19}$ & 8.40$^{+1.20}_{-1.20}$\\
S/H (10$^{-6}$) 	& 20.70$^{+3.23}_{-3.23}$ & 32.35$^{+8.94}_{-8.94}$ & 17.87$^{+3.31}_{-3.31}$ & 11.90$^{+4.29}_{-4.29}$ & 7.20$^{+1.10}_{-1.10}$ & 7.00$^{+0.70}_{-0.70}$\\
Ar/H (10$^{-7}$) 	& 37.80$^{+1.70}_{-1.70}$ & 56.21$^{+10.35}_{-10.35}$ & 49.58$^{+14.20}_{-14.21}$& 28.50$^{+5.34}_{-5.34}$ & 27.90$^{+3.84}_{-3.84}$ & 31.50$^{+1.08}_{-1.08}$\\
\tableline
\tablebreak
& \multicolumn{3}{c}{NGC 7662 Full} & & &\\
\tableline
Element & This Work & \citet{kwi03} & LHBA\tablenotemark{c} \\
\tableline
He/H (10$^{-2}$) 	& 10.00$^{+0.76}_{-0.76}$ & 10.00$^{+3.00}_{-3.00}$ & 10.05$^{+1.15}_{-1.15}$\\
C/H (10$^{-4}$) 	& 4.08$^{+0.43}_{-0.43}$ & \nodata & 6.27$^{+2.48}_{-2.48}$\\
N/H (10$^{-4}$) 	& 0.77$^{+0.11}_{-0.11}$ & 0.74$^{+0.13}_{-0.13}$ & 1.00$^{+0.42}_{-0.42}$\\
O/H (10$^{-4}$) 	& 4.47$^{+0.42}_{-0.42}$ & 4.19$^{+0.59}_{-0.59}$ & 4.59$^{+1.70}_{-1.70}$\\
Ne/H (10$^{-5}$) 	& 8.99$^{+0.92}_{-0.92}$ & 7.10$^{+0.84}_{-0.84}$ & 7.58$^{+1.80}_{-1.80}$\\
S/H (10$^{-6}$) 	& 2.59$^{+0.68}_{-0.68}$ & 4.30$^{+0.42}_{-0.42}$ & 5.71$^{+1.52}_{-1.52}$\\
Ar/H (10$^{-7}$) 	& 17.10$^{+2.16}_{-2.16}$ & 24.60$^{+3.35}_{-3.35}$ & 14.80$^{+4.46}_{-4.46}$\\
\enddata
\tablenotetext{a}{Values and uncertainties are averages and standard deviations, respectively, from \citet{boh13} and \citet{hyu94}.}
\tablenotetext{b}{Values and uncertainties are averages and standard deviations, respectively, from \citet{tsa03}, \citet{kin94}, \citet{sam92}, \citet{def91}, and \citet{tor81}.}
\tablenotetext{c}{Values and uncertainties are averages and standard deviations, respectively, from \citet{liu04}, \citet{hyu97}, \citet{bar86}, and \citet{all83}.}
\end{deluxetable}
\end{landscape} 

Looking at the temperatures of each region, only region 6 of IC 2165 is significantly lower than its full region temperature. This region probes the middle shell of IC 2165, as opposed to the other regions that probe the innermost shell. Therefore, this may indeed be cooler gas since it is farther from the central star and may be optically thick such that the effect of heating by radiation hardening\footnotemark~ doesn't occur. Lastly, most of the density variations appear to coincide with brighter areas of each PN for the higher densities, and dimmer areas for the lower densities. This is consistent with the assumption that the brightness is proportional to the density squared and may be the result of clumps of ejected material that form prior/during the formation of the planetary nebula. The only regions with densities that are unusual are Regions 2, 1 and 7 of IC 2165, IC 3568 and NGC 3242, respectively. A plausible explanation is that there is a small knot of enhanced density in each region that is emitting the observed light, though a literature search turns up no corroboration.
\footnotetext[1]{Radiation hardening refers to optically thin gas exterior to optically thick gas being illuminated by preferentially higher-energy photons since the optically thick gas absorbed the photons near the hydrogen series limit.}

To summarize, first, dereddened emission line intensities for various regions across each planetary nebula were presented. Next, these emission lines were used to calculate nebular properties of temperature and density as well as ionic abundances. Last, total abundances for He, C, N, O, Ne, S and Ar were calculated from the ionic abundances for each region. \textit{The major result is that each of the seven planetary nebulae appears to exhibit a chemically homogeneous distribution of these elements.} 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. 
