\chapter{Introduction}
\addcontentsline{lof}{chapter}{\thechapter \hspace{1.5mm} Introduction}
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%\textbf{Chapter 1}\\
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%\textbf{Introduction}\\
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All stars begin their lives fusing hydrogen-1 into helium-4. Fusion of any element requires very high temperatures such that the Coulomb repulsion force felt between two nuclei is overcome. This commonly occurs in the cores of stars, and while the star is fusing hydrogen in its core, it is said to be on the main sequence. The main sequence spans a specific region of the Hertzsprung-Russell (H-R) diagram, a plot of stellar luminosity vs. temperature (see Figure~\ref{hrdiagram}). The end products of hydrogen fusion are a helium-4 nucleus and energy in the form of light. This released energy heats the star's interior, increasing its outward pressure to the point where it counters the inward gravitational pressure. This stabilizes the size and temperature of the star. 

\begin{figure}
\caption{The Hertzsprung-Russell diagram showing the regions of stellar evolution that stars reside in for long periods of time. Each region is a different evolutionary stage, and each point is a star.}
\includegraphics[scale=0.85]{hrdiagram.png}
\label{hrdiagram}
\end{figure}

Hydrogen fusion occurs in two main channels: the proton-proton (PP) chain reaction and the carbon-nitrogen-oxygen (CNO) cycle. The PP chain is dominant for stars of mass $<$ 1.3 M$_{\astrosun}$, while the CNO cycle is dominant for stars with masses above 1.3 M$_{\astrosun}$ (\citet{sal05}). Only hydrogen-1 and its fusion products (deuterium and helium-3) are needed for the production of helium-4. The CNO cycle, however, requires carbon to act as a catalyst for the conversion of hydrogen-1 into helium-4. The CNO cycle starts with carbon-12, fuses it with hydrogen-1, and produces nitrogen-13. Nitrogen-13 is an unstable isotope and will decay to carbon-13 in a matter of minutes. Then, hydrogen-1 will fuse with carbon-13 to create nitrogen-14. Nitrogen-14 has the smallest fusion cross-section in the CNO cycle, making its fusion rate with hydrogen-1 very small. This creates a bottle-neck in the CNO cycle, resulting in an overall increase of this isotope in the core. Also due to this bottleneck, carbon-13 increases in the core. 

The star will be on the main sequence for the majority of its life (millions to billions of years, depending on its mass), leaving once all hydrogen in the core has been fused into helium. At this point, the core will collapse until either the force of gravity is balanced by the force associated with electron degeneracy pressure\footnotemark~(the core is not hot enough to fuse helium, stellar mass $<$ 2 M$_{\astrosun}$), or the core becomes hot enough to fuse helium (stellar mass $>$ 2 M$_{\astrosun}$, \citet{kar14}). 
\footnotetext[1]{Electron degeneracy pressure occurs because the Pauli exclusion principle doesn't allow electrons to simultaneously occupy the same quantum state. This restricts how compressed the matter within the core can be.}

While the core is collapsing, it is also getting hotter as the gravitational energy is being converted to thermal energy. This causes a shell of hydrogen surrounding the core to become hot enough to begin fusing. With the now hotter core and shell fusion occurring just outside the core (closer to the surface of the star), the star will expand to return to equilibrium. The net result is an increase in luminosity and decrease in temperature, as the star enters the giant region of the H-R diagram (see Figure~\ref{hrdiagram}). 

During the ascent to the giant region, convection within the star's envelope will extend downward and mix with the hydrogen fusing shell. The envelope will therefore become enriched with hydrogen fusion products in what is called the first dredge-up. For PP chain dominated shells, the primary product is helium-4, but for CNO cycle dominated shells, the products are helium-4, nitrogen-14 and carbon-13. Also, the envelope will be so extended after the equilibrium between gravity and the hotter internal conditions is re-established that mass loss occurs via stellar winds. These stellar winds are generated by turbulence within the envelope (possibly associated with Alfv\'en waves; \citet{sch05}) that has enough mechanical energy to escape the gravitational potential well of the star. Depending on the mass of the star, this may result in as much as 30\% of the star's original mass to be lost (\citet{kar14}). This mass loss enriches the interstellar medium in the vicinity of the star with these nuclear products.

Eventually, the cores of stars with masses $<$ 2 M$_{\astrosun}$ will become hot enough from helium deposition from the hydrogen fusing shell that helium will begin to fuse. Since the core is initially in a degenerate state, the core will not increase in size in response to an increase in temperature, resulting in a runaway of helium fusion within the core. This results in what is called a helium flash, where a large amount of helium is fused in a matter of seconds. The flash is short because the core becomes non-degenerate again and expands until equilibrium is restored and quiescent helium-4 fusion ensues. This restructuring will affect the surface where the temperature will increase and the luminosity will slightly decrease, but the star remains in the giant region. While it is fusing helium-4 in its core, it is said to be on the horizontal branch and will remain there for millions to hundreds of millions of years, depending on its mass. 

Helium fusion follows the triple-alpha process, where three helium-4 nuclei are combined to make carbon-12. Occasionally, carbon-12 will fuse with helium-4 to produce oxygen-16 as well. This builds up a core of carbon and oxygen. 

After all of the helium-4 in the core is fused, the core will collapse in all cases to a degenerate state. Now there will be a shell of hydrogen fusion making helium and a shell of helium fusion making carbon and oxygen for stars with masses $<$ 2 M$_{\astrosun}$. For all others, the hydrogen fusion shell is extinguished first and reignited once the helium fusion shell is exhausted. For stars above 4 M$_{\astrosun}$, a second dredge-up will occur, enriching the surface with more helium and nitrogen-14 (\citet{kar14}). Again, the star will become more luminous and cooler, entering the supergiant region of the H-R diagram (see Figure~\ref{hrdiagram}).

As the star remains in the supergiant region, the helium shell is replenished with helium-4 by the hydrogen fusion shell. While the helium shell is building up, the small region closest to the core will be degenerate. Eventually, the helium shell will become hot enough to begin fusing. Like before, the degenerate portion of the helium shell will fuse in a matter of seconds in what is called a helium shell flash. This causes the star to expand and the hydrogen shell to cease fusion, and is known as a thermal pulse. Also, convection in the form of the third dredge-up mixes helium and hydrogen fusion products (i.e. carbon-12, oxygen-16, and helium-4) to the surface. After this thermal pulse, the star will contract again until the hydrogen fusion shell is re-ignited. The hydrogen shell begins building up the helium shell until it reaches a high enough temperature again and the process repeats. It will continue these pulsations for upwards of hundreds of thousands of years, depending on the mass.

If the star has a mass above 5 M$_{\astrosun}$, a phase known as hot bottom burning may occur (\citet{kar14}). For stars of this mass, the hydrogen-rich convective envelope may reach all the way to the hydrogen burning shell in between dredge-up events. In this case, the bottom of the hydrogen-rich convection envelope will reach temperatures hot enough for the CNO cycle to occur and, due to the convection, carbon-12 will be converted to nitrogen-14 and mixed throughout the envelope as the nitrogen is produced. This reduces the total abundance of carbon as the total nitrogen abundance is increased. However, after hot bottom burning ceases, there may be more dredge-up episodes to mix in more carbon from the helium fusion shell.

With each thermal pulse, the surface of the star will expand, cool, and lose mass to stellar winds. If the pulsation occurs such that shock waves propagate through the stellar atmosphere and drive the upper layers out far enough, the gas will cool enough for heavy elements to condense into grains (\citet{vil02}). Then radiation pressure and the coupling of the grains and gas will drive off the outer layers of the star with a speed of approximately 20-30 km/s. Eventually, the entire envelope is thrown off, exposing internal layers that are on the order of 10$^5$ K. These hot layers emit copious amounts of radiation with photon energies high enough to ionize hydrogen. 

The previously expelled gas, which is composed of mostly hydrogen but now enriched with fusion products and s-process elements, absorbs the ionizing photons, causing the gas temperature to increase from scattering between the ions (protons, electrons, etc.) in the gas. The free electrons become thermalized quickly from scattering with hydrogen and helium (H and He e$^-$-scattering cross section $\gg$ photoionization cross section). Cooling of the gas happens in the form of electron free-free, free-bound, and bound-bound emission of photons. Electron free-free photon emission, or Bremsstrahlung emission, occurs when a free election is decelerated by an atomic ion, emitting a photon. Electron free-bound emission, or recombination emission, is the result of a free electron recombining with an atomic ion and emitting a photon. The last emission, electron bound-bound emission, happens after a collision between ions excites a bound electron to a higher energy level. The excited electron eventually transitions to lower levels, emitting photons. This last emission process is the dominant cooling mechanism in the gas. Metals within the gas, such as oxygen and carbon, cool the gas by this collisional process, emitting photons whose energies are below the ionizing threshold of hydrogen and escape the gas. At steady-state, the absorption of ionizing photons and the emission of photons that escape the gas become equal, resulting in the fairly constant emission nebula known as the planetary nebula. 

Thus, a planetary nebula, the subject of this dissertation, is a low density, light emitting gas that has recently been ejected from a 0.8 - 8 M$_{\astrosun}$ star as it nears the end of its life. After thousands to a few tens of thousands of years, the nebula will become diffuse enough that it no longer radiates and the gas will become part of the interstellar medium. Typically, planetary nebulae have masses in the range of 0.1-1.0 M$_{\astrosun}$, densities of order 10$^2$-10$^4$ cm$^{-3}$, and temperatures between 8000 and 20,000 K. They are concentrated on the sky along the Galactic plane and toward the Galactic center. The emission lines commonly used to identify these objects while they are still radiating are the strong forbidden lines of oxygen at wavelengths 4959 and 5007 \AA. When these emission lines were first observed, they were erroneously thought to originate from a mysterious element called ``nebulium'' that only existed in extra-terrestrial conditions. In 1928, the physicist and astronomer Ira Bowen determined they were in fact electronic transitions of O$^{+2}$.

Throughout the lives of these progenitor stars, there are two main times when a large amount of material is returned to the interstellar medium. The first is during the ascent to the giant region of the H-R diagram. The second, the focus of this dissertation, is during the formation of planetary nebulae. Both of these events enrich the interstellar medium, affecting the chemical composition of galaxies and the formation of new stars. For planetary nebulae, this enrichment or stellar yield is primarily in the form of carbon, nitrogen, and helium. The carbon will especially influence the effectiveness of the CNO cycle and subsequent lifespans of the next generation of stars that form from the enriched material. Also, since all known life uses carbon and nitrogen, understanding the production and distribution of these elements is important to astrobiology research as well. However, more massive stars also enrich the interstellar medium with these elements. Whether planetary nebula forming stars or more massive stars are the dominant contributors of carbon and nitrogen is uncertain and an active area of research. Therefore, a complete and accurate model of the evolution of stars occupying the mass range between 0.8 and 8 M$_{\astrosun}$ is particularly important for understanding the contribution that these stars make to the buildup of carbon and nitrogen in the interstellar medium. 

One requirement for a good model is the proper understanding of the mass ejection process, where some of the details of this process should be revealed by studying the spatial variations in the matter that comprises a planetary nebula (PN). If PNe are chemically inhomogeneous, it would raise questions as to the validity of the current formation theory for planetary nebulae where the nebulae are assumed to be homogeneous. This would impact the stellar yield by adding a directional dependence that would need to be accounted for in chemical evolution modeling of the Galaxy. Seven PNe (IC 2165, IC 3568, NGC 2440, NGC 3242, NGC 5315, NGC 5882, and NGC 7662) are analyzed here with the purpose of addressing chemical homogeneity in planetary nebulae. 

The planetary nebulae in this study were first presented in \citet{duf15} (hereafter Paper~I), where the main goal was to measure critical emission lines of carbon and nitrogen for accurate abundance calculations of these elements. Among the selection criteria for the planetary nebulae were a high surface brightness and, where practical, large angular size. These criteria make them ideal candidates for studying the chemical distribution of their matter. 

Each of the seven planetary nebulae can be seen in Figures~\ref{fig:slits1}~\&~\ref{fig:slits2}. IC 2165 is a slightly elliptical planetary nebula with shells detected at 2$''$, 4$''$, and 20$''$ (not visible in Figure~\ref{fig:slits1}) from the central star ~(\citet{cor03}). IC 3568 has only two detected shells at radii of 4$''$ and 9$''$ (not visible in Figure~\ref{fig:slits1}) and is very circular in appearance (\citet{bal92} and \citet{cor03}). The shape of NGC 2440 is more complicated, with three bipolar structures emanating at various angles from the central star~(\citet{lop98}), while NGC 3242 exhibits a multi-shell structure. The brightest features for NGC 3242 include an inner 28\ensuremath{''}x20\ensuremath{''} shell surrounded by a 46\ensuremath{''}x40\ensuremath{''} halo (not visible in Figure~\ref{fig:slits1};~\citet{rui11}). NGC 5315 is an irregularly shaped, compact PN of radius 3$''$~(\citet{ack92}). An elliptical shell with radial dimensions of 5.5$''$ x 3$''$, a more circular shell of radius 7.5$''$, and a halo (not visible in Figure~\ref{fig:slits2}) that extends to 90$''$ are observed for NGC 5882 (\citet{cor00} and \citet{wel87}). NGC 7662 is another triple-shell PN with two elliptical shells of radii 9$''$ x 6.2$''$ (only visible part in Figure~\ref{fig:slits2}) and 15.4$''$ x 13.7$''$ encompassed by a 67$''$ circular shell~(\citet{gue04}).
\begin{figure}
\caption{The locations of the extracted spectra where all seven gratings spatially overlap one another from STIS. In addition to the individual regions, a Full region for each object was extracted over all the regions shown. From left to right, top to bottom: IC2165, IC 3568, NGC2440, NGC 3242. The double slits in NGC 2440 are the result of a pointing error discussed in the text.\newline}
\centering 
\includegraphics[scale=0.31]{4PneRegionsThicker.png}
\label{fig:slits1}
\end{figure} 
\begin{figure}
\caption{The same as Figure~\ref{fig:slits1}. From left to right, top to bottom: NGC 5315, NGC 5882, NGC 7662.\newline}
\centering 
\includegraphics[scale=0.31]{3PneRegionsThicker.png}
\label{fig:slits2}
\end{figure} 

Given these morphologies, the central star of each PN has experienced mass ejections which may have been inhomogeneous in their elemental composition, either from shell to shell or at different positions within one ejection. Each shell could have a unique composition if each ejection occurred with a different core temperature, resulting in a change in fusion rates around the core and convection rates to the surface. If the convection process was naturally asymmetrical or influenced by a companion star (stripping material), an ejection may result in an uneven distribution of elements. \textit{Therefore, the main goal of this project is to look for spatial differences in the abundances of carbon, nitrogen, helium, oxygen, neon, sulfur, and argon relative to hydrogen across each planetary nebula, using the spectroscopic data presented in Paper~I.} 

Turning to the problem of determining the central star properties, many authors have inferred the stellar temperature and luminosity using techniques such as photoionization modeling of the nebula (see e.g. \citet{hen15}, hereafter Paper~II) and the Zanstra method\footnotemark~ (see e.g. Shaw and Kaler \citep{sha85}, \citep{sha89};~\citet{zha93}). Consequently, there is often a large range in the derived stellar parameters for any one object. In particular, the Zanstra method gives very different temperatures for the same star depending on if hydrogen or helium lines are used in the method. Since the mass of a star is the primary factor that determines its luminosity and temperature, having uncertain measurements of these stellar parameters results in unreliable initial/final mass calculations. 

\textit{This leads to the second goal of this dissertation: to constrain the stellar temperature and luminosity of each planetary nebula by modeling each with the photoionization code Cloudy, using the observations as constraints.} However, unlike previous photoionization modeling work extant in the literature, a physically motivated spectral energy distribution is used here for the central star, and the modeled emission lines are calculated as seen through the slit instead of over the entire nebula.
\footnotetext[1]{The Zanstra method assumes all ionizing photons from the Lyman continuum are absorbed by the nebula, which makes it possible to determine the stellar temperature from a nebular line such as H$\beta$.}

The observations and analysis are presented in Chapter 2 followed by results in Chapters 3 and 4. Chapter 5 contains the summary and conclusions while Chapter 6 addresses improvements and future work.  
