Section 4 is finished. Not great though

LaTeX/thesis.tex

@@ -162,6 +162,7 @@ Monotonic Basin Hopping}
 		right at the moment of closest approach to the (flyby) target body. Because of this,
 		optimization with impulsive trajectories and gravity assists are common.
 
+		% TODO: Might need to remove the HOCP stuff
 		However, there is no physical reason why low-thrust trajectories can't also incorporate gravity
 		assists. The optimization problem becomes much more complicated. The separate problems of
 		optimizing flyby parameters (planet, flyby date, etc.) and optimizing the low-thrust control
@@ -470,23 +471,108 @@ Monotonic Basin Hopping}
 			trajectory within the Two-Body Problem, with only linearly-increasing computation time.
 
 	\section{Interplanetary Trajectory Considerations} \label{interplanetary}
-		Highlight the problems with the 2BP in co-ordinating influences of extra bodies over an
-		interplanetary journey.
+
+		The question of interplanetary travel opens up a host of additional new complexities. While
+		optimizations for simple single-body trajectories are far from simple, it can at least be
+		said that the assumptions of the Two Body Problem remain fairly valid. In interplanetary
+		travel, the primary body most responsible for gravitational forces might be a number of
+		different bodies, dependent on the phase of the mission. In the ideal case, every relevant
+		body would be considered as an ``n-body'' perturbation during the entire trajectory. For
+		some approaches, this method is sufficient and preferred. However, for other uses, a more
+		efficient model is necessary. The method of patched conics can be applied in this case to
+		simplify the model.
+
+		Interplanetary travel does not simply negatively impact trajectory optimization. The
+		increased complexity of the search space also opens up new opportunities for orbit
+		strategies. The primary strategy investigated by this thesis will be the gravity assist, a
+		technique for utilizing the gravitational energy of a planet to modify the direction of
+		solar velocity.
 
 		\subsection{Patched Conics}
-			Describe the method of patched conics.
+
+			The first hurdle to deal with is the problem of reconciling the Two-Body problem with
+			the presence of multiple and varying planetary bodies. The most common method for
+			approaching this is the method of patched conics. In this model, we break the
+			interplanetary trajectory up into a series of smaller sub-trajectories. During each of
+			these sub-trajectories, a single primary is considered to be responsible for the
+			trajectory of the orbit, via the Two-Body problem.
+
+			The transition point can be calculated a variety of ways. The most typical method is to
+			calculate the gravitational force due to the two bodies separately, via the Two-Body
+			models. Whichever primary is a larger influence on the motion of the spacecraft is
+			considered to be the primary at that moment. This effectively breaks the trajectory into
+			a series of orbits defined by the Two-Body problem (conics), patched together by
+			distinct transition points.
 
 		\subsection{Gravity Assist Maneuvers}
-			Describe how a gravity assist maneuver would work in the framework of patched conics. Also
-			discuss the advantages of such a maneuver.
+
+			As previously mentioned, there are methods for utilizing the orbital energy of the other
+			planets in the Solar System. This is achieved via a technique known as a Gravity Assist,
+			or a Gravity Flyby. During a gravity assist, the spacecraft enters into the
+			gravitational sphere of influence of the planet and, because of its excess velocity,
+			proceeds to exit the sphere of influence. Relative to the planet, the speed of the
+			spacecraft increases as it approaches, then decreases as it departs. From the
+			perspective of the planet, the velocity of the spacecraft is unchanged. However, the
+			planet is also orbiting the Sun.
+
+			From the perspective of a Sun-centered frame, though, this is effectively an elastic
+			collision. The overall momentum remains the same, with the spacecraft either gaining or
+			losing some in the process (dependent on the directions of travel). The planet also
+			loses or gains momentum enough to maintain the overall system momentum, but this amount
+			is negligible compared to the total momentum of the planet. The overall effect is that
+			the spacecraft arrives at the planet from one direction and, because of the influence of
+			the planet, leaves in a different direction.
+
+			This effect can be used strategically. The ``bend'' due to the flyby is actually
+			tunable via the exact placement of the fly-by in the b-frame, or the frame centered at
+			the planet, from the perspective of the spacecraft at $v_\infty$. By modifying the
+			turning angle of this bend. In doing so, one can effectively achieve a (restricted) free
+			impulsive thrust event.
 
 		\subsection{Multiple Gravity Assist Techniques}
-			Discuss the advantages of chaining together multiple gravity assists and highlight the
-			difficulties in choosing these assists. Here I can mention porkchop plots, Lambert's problem,
-			etc. Here I can also talk about Hybrid Optimal Control Problems.
 
-		\subsection{Ephemeris Considerations}
-			I can quickly mention SPICE here and talk a bit about validation.
+			Naturally, therefore, one would want to utilize these gravity flybys to reduce the fuel
+			cost to arrive at their destination target state. However, these flyby maneuvers are
+			quite restricted. The incoming hyperbolic velocity must be equal in magnitude to the
+			outgoing hyperbolic velocity. Also, the turning angle $\delta$, in the following
+			equation, correlates with the radius of periapsis of the hyperbolic trajectory crossing
+			the planet:
+
+			\begin{equation}
+				r_p = \frac{\mu}{v_\infty^2} \left[ \frac{1}{\sin\left(\frac{\delta}{2}\right)} - 1 \right]
+			\end{equation}
+
+			Where $v_\infty$ is the magnitude of hyperbolic velocity. Naturally, the radius of
+			periapsis must not fall below some safe value, in order to avoid the risk of the
+			spacecraft crashing into the planet or its atmosphere.
+
+			In order to visualize which trajectories are possible within these constraints, porkchop
+			plots are often employed, such as the plot in Figure~\ref{porkchop}. These plots outline
+			various incoming and outgoing qualities of the trajectory arc between two planetary
+			bodies. For instance, during an arc from launch at Earth to a flyby one might plot the
+			launch C3 against the Mars arrival $v_\infty$ for a variety of launch and arrival dates.
+
+			\begin{figure}
+				\centering
+				\includegraphics[width=\textwidth]{LaTeX/fig/porkchop}
+				\caption{A sample porkchop plot of an Earth-Mars transfer}
+				\label{porkchop}
+			\end{figure}
+
+			This is made possible by solving Lambert's problem for the planetary ephemeris at the
+			epochs plotted. Lambert's problem is concerned with determining the orbit between two
+			positions at two different times in space. There are a number of different Lambert's
+			problem algorithms that allow a mission designer to determine the velocity needed (and
+			thus the $\Delta V$) required to achieve a position at a later time. From this, the
+			designer can algorithmically determine trajectory properties in the porkchop plot for
+			easy visualization.
+
+			However, this is an impulsive thrust-centered approach. The solution to Lambert's
+			problem assumes a natural trajectory. However, to the low-thrust designer, this is
+			needlessly limiting. A natural trajectory is unnecessary when the trajectory can be
+			modified by a continuous thrust profile along the arc. Therefore, for the hybrid problem
+			of optimizing both flyby selection and thrust profiles, porkchop plots are less helpful,
+			and an algorithmic approach is preferred.
 
 	% \section{Genetic Algorithms}
 	%		I will probably give only a brief overview of genetic algorithms here. I don't personally know