\chapter{Application: Designing a Trajectory To Saturn} \label{results}

	To consider a relatively simple but representative mission design objective, a sample mission to
	Saturn was investigated.

	\section{Mission Scenario}

		The sample mission is defined to represent a general case for a near-future low-thrust
		trajectory to Saturn. No constraints are placed on the flyby planets, but a number of
		constraints were placed on the algorithm to represent a realistic mission scenario. 

		The first choice required by the application is one not necessarily designable to the
		initial mission designer (though not necessarily fixed in the design either) and is that of
		the spacecraft parameters. The application accepts as input a spacecraft object containing:
		the dry mass of the spacecraft, the fuel mass at launch, the number of onboard thrusters,
		and the specific impulse, maximum thrust and duty cycle of each thruster.

		For this mission, the spacecraft was chosen to have a dry mass of only 200 kilograms for a
		fuel mass of 3300 kilograms. This was chosen in order to have an overall mass roughly in the
		same zone as that of the Cassini spacecraft, which launched with 5712 kilograms of total
		mass, with the fuel accounting for 2978 of those kilograms\cite{cassini}. The dry mass of
		the spacecraft was chosen to be extremely low in order to allow for a variety of
		``successful'' missions in which the spacecraft didn't run out of fuel. That way, the
		delivered dry mass to Saturn could be thought of as a metric of success, without discounting
		mission that may have delivered just under whatever more realistic dry mass one might set,
		in case those missions are in the vicinity of actually valid missions.

		The thruster was chosen to have a specific impulse of 3200 seconds, a maximum thrust of
		250 millinewtons, and a 100\% duty cycle. This puts the thruster roughly in line with
		having an array of three NSTAR ion thrusters, which were used on the Dawn and Deep Space
		1 missions\cite{polk2001performance}.

		Also of relevance to the mission were the maximum $C_3$ at launch and $v_\infty$ at
		arrival values. In order to not exclude the possibility of a non-capture flyby mission,
		it was decided to not include the arrival $v_\infty$ term in the cost function and,
		because of this, the maximum value was set to be extremely high at 500 kilometers per
		second, in order to fully explore the space. In practice, though, the algorithm only
		looks at flybys below 10 kilometers per second in magnitude. The maximum launch $C_3$
		energy was set conservatively to 200 kilometers per second squared. This is upper limit
		is only possible, for the given start mass, using a heavy launch system such as the
		SLS\cite{stough2021nasa} or the comparable SpaceX Starship, though at values below about
		half of this maximum, it begins to become possible to use existing launch solutions.

		Finally, the mission is meant to represent a near future mission. Therefore the launch
		window was set to allow for a launch in any day in 2023 or 2024 and a maximum total time
		of flight of 20 years. This is longer than most typical Saturn missions, but allows for
		some creative trajectories for higher efficiency.

		It should be noted that each of these trajectories was found using an $n$ value of 20 as
		mentioned previously, but in post-processing, the trajectory was refined to utilize a
		slightly higher fidelity model that uses 60 sub-trajectories per orbit. This serves to
		provide better plots for display, higher fidelity analyses, as well as to highlight the
		efficacy of the lower fidelity method. Orbits can be found quickly in the lower fidelity
		model and easily refined later by re-running the NLP solver at a higher $n$ value.

		\subsection{Cost Function}

			Each mission optimization also allows for the definition of a cost function. This
			cost function accepts as inputs all parameters of the mission, the maximum $C_3$ at
			launch and the maximum excess hyperbolic velocity at arrival.

			The cost function used for this mission first generated normalized values for fuel
			usage and launch energy. The fuel usage number is determined by dividing the fuel
			used by the mass at launch and the $C_3$ number is determined by dividing the $C_3$
			at launch by the maximum allowed. These two numbers are then weighted, with the fuel
			usage value getting a weight of three and the launch energy value getting a weight
			of one. The values are summed and returned as the cost value, represented as the value
			$J$ below:

			\begin{equation}
				J(\vec{x}, m_{dry}, C_{3,max}) = 3 \left| \frac{h(\vec{x})}{m_{dry}} \right| +
				\left| \frac{k(\vec{x})}{C_{3,max}} \right|
			\end{equation}

			\noindent
			Where $h(\vec{x})$ represents the total fuel mass used during the trajectory and
			$k(\vec{x})$ represents the launch $C_3$ of the initial phase.

		\subsection{Flybys Analyzed}

			Since the algorithm itself makes no decisions on the actual choice of flybys, that
			leaves the mission designer to determine which flyby planets would make good
			potential candidates. A mission designer can then re-run the algorithm for each of
			these flyby plans and determine which optimized trajectories best fit the needs of
			the mission.

			For this particular mission scenario, the following flyby profiles were
			investigated (E: Earth, M: Mars, V: Venus, J: Jupiter, S: Saturn). These flyby choices
			were initially sampled randomly, but as patterns were noticed during the previous runs,
			certain trajectories were chosen to investigate phases that seemed promising.

			\begin{itemize}
				\setlength\itemsep{-0.5em}
				\item EJS
				\item EMJS
				\item EMMJS
				\item EMS
				\item ES
				\item EVMJS
				\item EVMS
				\item EVVJS
			\end{itemize}

			For each of these trajectories, the optimization algorithm was run. During the MBH phase
			of the optimization algorithm, anytime a new ``basin best'' mission was discovered, it
			was recorded. The resultant cost function values of each of those discovered missions
			can be found in the table below:

			\begin{longtable}{
					|
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					>{\centering}p{1.25in}
					>{\centering}p{1.1in}
					>{\centering}p{1in}
					>{\centering}p{0.8in}
					>{\centering\arraybackslash}p{0.8in}
					|
				}
				\hline
				\bfseries Flyby Selection &
				\bfseries Cost Function Value &
				\bfseries Mass Delivered (kg) &
				\bfseries Time of Flight (years) &
				\bfseries Launch $C_3$ ($\frac{\text{km}^2}{\text{s}^2}$) &
				\bfseries Arrival $V_\infty$ Norm \\
				\hline
				\endhead
				ES & 0.555 & 3423.49 & 5.945 & 97.89 & 0.009 \\
				ES & 0.5551 & 3422.41 & 5.945 & 97.73 & 0.0 \\
				ES & 0.5553 & 3425.31 & 5.897 & 98.26 & 0.0 \\
				ES & 0.561 & 3403.47 & 5.945 & 95.65 & 0.0 \\
				ES & 0.5612 & 3406.47 & 5.894 & 96.22 & 0.002 \\
				EMS & 0.648 & 3130.77 & 8.451 & 66.3 & 6.391 \\
				EMJS & 0.6601 & 2962.65 & 14.107 & 39.91 & 3.636 \\
				EMS & 0.6883 & 3004.49 & 9.229 & 52.71 & 4.245 \\
				EMS & 0.697 & 3037.04 & 7.984 & 60.04 & 6.021 \\
				EMJS & 0.7458 & 2837.9 & 14.036 & 35.65 & 4.816 \\
				EMJS & 0.7975 & 1905.95 & 12.99 & 16.92 & 2.686 \\
				EMJS & 0.8037 & 2652.62 & 13.793 & 15.48 & 3.209 \\
				EMJS & 0.8251 & 2760.39 & 13.857 & 38.23 & 3.818 \\
				EMMJS & 0.9115 & 2528.08 & 15.853 & 15.68 & 3.189 \\
				EMJS & 0.9415 & 2484.97 & 16.33 & 14.29 & 2.021 \\
				EMJS & 0.9614 & 2511.65 & 15.756 & 22.85 & 3.393 \\
				EMS & 1.0297 & 1655.14 & 10.412 & 3.18 & 4.529 \\
				EJS & 1.1285 & 1734.51 & 15.725 & 41.98 & 2.595 \\
				ES & 1.2317 & 1639.1 & 9.248 & 39.72 & 5.785 \\
				EVMS & 1.326 & 2241.72 & 8.87 & 49.49 & 4.977 \\
				EMS & 1.3288 & 1400.47 & 7.843 & 1.87 & 5.634 \\
				EMS & 1.3378 & 2705.69 & 15.848 & 131.39 & 5.151 \\
				EMJS & 1.3953 & 1904.96 & 13.813 & 5.62 & 5.146 \\
				EVMS & 1.4152 & 1963.98 & 11.315 & 19.72 & 9.117 \\
				EVMS & 1.4596 & 1963.09 & 11.885 & 28.45 & 6.7 \\
				EVMS & 1.4665 & 1915.47 & 11.691 & 21.67 & 8.919 \\
				EVMS & 1.5221 & 1966.29 & 12.002 & 41.49 & 7.085 \\
				EVMS & 1.5923 & 1811.71 & 7.612 & 29.05 & 8.892 \\
				EVMJS & 1.6694 & 2324.68 & 14.203 & 132.4 & 5.346 \\
				EMS & 1.7029 & 1652.8 & 12.064 & 23.93 & 8.898 \\
				EMJS & 1.7044 & 1687.48 & 17.45 & 30.16 & 6.148 \\
				EMS & 1.7811 & 1504.33 & 18.067 & 14.1 & 3.195 \\
				EVVJS & 2.0106 & 1362.61 & 14.71 & 35.72 & 6.315 \\
				EMJS & 2.1595 & 1026.4 & 17.548 & 7.86 & 7.977 \\
				EJS & 2.1622 & 1543.52 & 12.96 & 97.03 & 9.42 \\
				EMJS & 2.2217 & 2055.3 & 21.583 & 196.67 & 2.074 \\
				EMMJS & 2.3843 & 730.68 & 14.754 & 2.12 & 6.915 \\
				EMJS & 2.4246 & 1470.46 & 24.186 & 136.99 & 1.749 \\
				EMMJS & 2.4645 & 971.97 & 16.935 & 59.53 & 7.34 \\
				EVVJS & 2.4926 & 908.6 & 15.287 & 54.28 & 3.942 \\
				EMJS & 2.4948 & 872.56 & 12.943 & 48.55 & 8.548 \\
				EVMS & 2.7112 & 726.4 & 12.263 & 66.76 & 9.478 \\
				EMJS & 2.7681 & 496.32 & 16.0 & 38.71 & 10.348 \\
				\hline
				\caption{Table of resultant cost function values for every discovered mission}\label{cost_fn_table}\\
			\end{longtable}

	\section{Faster, Less Efficient Trajectory}

		In order to showcase the flexibility of the optimization algorithm (and the chosen cost
		function), two different missions were chosen to highlight. One of these missions is a
		slower, more efficient trajectory more typical of common low-thrust trajectories. The
		other is a faster trajectory, quite close to a natural trajectory, but utilizing more
		launch energy to arrive at the planet.

		It is the faster trajectory that we'll analyze first. Most interesting about this
		particular trajectory is that it's actually quite efficient despite its speed, in
		contrast to the usual dichotomy of low-thrust travel. The cost function used for this
		analysis did not include the time of flight as a component of the overall cost, and yet
		this trajectory still managed to be the lowest cost trajectory of all trajectories found
		by the algorithm, meaning that it has merit for both a flyby mission as well as a capture
		mission.

		The mission begins in late June of 2024 and proceeds first to an initial gravity assist with
		Mars after three and one half years to rendezvous in mid-December 2027. Unfortunately, the
		launch energy required to effectively use the gravity assist with Mars at this time is
		quite high. The $C_3$ value was found to be $60.4102 \frac{\text{km}^2}{\text{s}^2}$. While
		not as low as some of the other missions found to be very optimal, it should be noted that
		missions with this $C_3$ and launch mass are still quite feasible.

		However, for this phase, the thrust magnitudes are quite low, raising slowly only as the
		spacecraft approaches Mars, allowing for a nearly-natural trajectory to Mars rendezvous.
		Note also that the in-plane thrust angle was neither zero nor $\pi$, implying that these
		thrusts were steering thrusts rather than momentum-increasing thrusts.

		\begin{figure}[H]
			\centering
			\includegraphics[width=0.9\textwidth]{LaTeX/fig/EMS_plot}
			\caption{Depictions of the faster Earth-Mars-Saturn trajectory found by the
			algorithm to be most efficient; planetary ephemeris arcs are shown during the phase
			in which the spacecraft approached them}
			\label{ems}
		\end{figure}

		\begin{figure}[H]
			\centering
			\includegraphics[width=0.9\textwidth]{LaTeX/fig/EMS_plot_noplanets}
			\caption{Another depiction of the EMS trajectory, without the planetary ephemeris
			arcs}
			\label{ems_nop}
		\end{figure}

		The second and final leg of this trip exits the Mars flyby and, initially burns quite
		heavily along the velocity vector in order to increase its semi-major axis. After an initial
		period of thrusting, though, the spacecraft effectively coasts with minor adjustments until
		its rendezvous with Saturn just four and a half years later in June of 2032. The arrival
		$v_\infty$ is not particularly small, at $5.816058 \frac{\text{km}}{\text{s}}$, but this is
		to be expected as the arrival excess velocity was not considered as a part of the cost
		function. If capture was not the final intention of the mission, this may be of little
		concern. Otherwise, the low fuel usage of $446.92$ kilograms for a $3500$ kilogram launch
		mass leaves much margin for a large impulsive thrust to enter into a capture orbit at
		Saturn.

		\begin{figure}[H]
			\centering
			\includegraphics[width=0.9\textwidth]{LaTeX/fig/EMS_thrust_mag}
			\caption{The magnitude of the unit thrust vector over time for the EMS trajectory}
			\label{ems_mag}
		\end{figure}

		\begin{figure}[H]
			\centering
			\includegraphics[width=0.9\textwidth]{LaTeX/fig/EMS_thrust_components}
			\caption{The inertial x, y, and z components of the unit thrust vector over time for
			the EMS trajectory}
			\label{ems_components}
		\end{figure}

		In this case the algorithm effectively discovered that a higher-powered launch from
		the Earth, then a natural coasting arc to Mars flyby would provide the spacecraft with
		enough velocity that a short but efficient powered-arc to Saturn was possible with
		effective thrusting. It also determined that the most effective way to achieve this
		flyby was to increase orbital energy in the beginning of the arc, when increasing
		the semi-major axis value is most efficient. All of these concepts are known to skilled
		mission designers, but finding a trajectory that combined all of these concepts would
		have required much time-consuming analysis of porkchop plots and combinations of
		mission-design techniques. This approach is far more automatic than the traditional
		approach.

		The final quality to note with this trajectory is that it shows a tangible benefit of
		the addition of the Lambert's solver in the monotonic basin hopping algorithm. Since the
		initial arc is almost entirely natural, with very little thrust, it is extremely likely
		that the trajectory was found in the Lambert's Solution half of the MBH algorithm
		procedure.

	\section{Slower, More Efficient Trajectory}

		Next we'll analyze the nominally second-best trajectory. While the cost function provided to
		the algorithm can be a useful tool for narrowing down the field of search results, it can
		also be very useful to explore options that may or may not have quite as small of a cost
		function value, but beneficial for other reasons. By outputting many different optimal
		trajectories, the MBH algorithm can allow for this type of mission design flexibility.

		To highlight the flexibility, a second trajectory has been selected, which has nearly
		equal value by the cost function, coming in slightly lower. However, this trajectory
		appears to offer some benefits to the mission designer who would like to capture into
		the gravitational field of Saturn or minimize launch energy requirements, perhaps for a
		smaller mission, at the expense of increased speed.

		The first leg of this three-leg trajectory is quite similar to the first leg of the
		previous trajectory. However, this time the launch energy is considerably lower, with a
		$C_3$ value of only $40.4386$ kilometer per second squared. Rather than employ an almost
		entirely natural coasting arc to Mars, however, this trajectory performs some thrusting
		almost entirely in the velocity direction, increasing its orbital energy in order to
		achieve the same Mars rendezvous. In this case, the launch was a bit earlier, occurring
		in November of 2023, with the Mars flyby occurring in mid-April of 2026. This will prove
		to be helpful in comparison with the other result, as this mission profile is much
		longer.

		\begin{figure}[H]
			\centering
			\includegraphics[width=0.9\textwidth]{LaTeX/fig/EMJS_plot}
			\caption{Depictions of the slower Earth-Mars-Jupiter-Saturn trajectory found by the
			algorithm to be the second most efficient; planetary ephemeris arcs are shown during
			the phase in which the spacecraft approached them}
			\label{emjs}
		\end{figure}

		\begin{figure}[H]
			\centering
			\includegraphics[width=0.9\textwidth]{LaTeX/fig/EMJS_plot_noplanets}
			\caption{Another depiction of the EMJS trajectory, without the planetary ephemeris
			arcs}
			\label{emjs_nop}
		\end{figure}

		The second phase of this trajectory also functions quite similarly to the second phase
		of the previous trajectory. In this case, there is a little bit more thrusting necessary
		simply for steering to the Jupiter flyby than was necessary for Saturn rendezvous in the
		previous trajectory. However, most of this thrusting is for orbit raising in the
		beginning of the phase, very similarly to the previous result. In this trajectory, the
		Jupiter flyby occurs late July of 2029.

		\begin{figure}[H]
			\centering
			\includegraphics[width=0.9\textwidth]{LaTeX/fig/EMJS_thrust_mag}
			\caption{The magnitude of the unit thrust vector over time for the EMJS trajectory}
			\label{emjs_mag}
		\end{figure}

		\begin{figure}[H]
			\centering
			\includegraphics[width=0.9\textwidth]{LaTeX/fig/EMJS_thrust_components}
			\caption{The inertial x, y, and z components of the unit thrust vector over time for
			the EMJS trajectory}
			\label{emjs_components}
		\end{figure}

		Finally, this mission also has a third phase. The Jupiter flyby provides quite a strong
		$\Delta V$ for the spacecraft, allowing the following phase to largely be a coasting arc
		to Saturn almost one revolution later. During the most efficient part of the journey,
		some thrust in the velocity direction accounts for a little bit of orbit-raising, but
		the phase is largely natural. Because of this long coasting period, the mission length
		increases considerably during this leg, arriving at Saturn in December of 2037, over 8
		years after the Jupiter flyby.

		However, there are many advantages to this approach relative to the other trajectory.
		While the fuel use is also slightly higher at $530.668$ kilograms, plenty of payload
		mass is still capable of delivery into the vicinity of Saturn. Also, it should be noted
		that the incoming excess hyperbolic velocity at arrival to Saturn is significantly
		lower, at only $3.4774\frac{\text{km}}{\text{s}}$, meaning that less of the delivered
		payload mass would need to be taken up by impulsive thrusters and fuel for Saturn orbit
		capture, should the mission designer desire this.

	\section{Final Trajectory Analysis}

		Ultimately, two optimized trajectories were selected to be excellent candidates for further
		consideration. The resultant flyby selection, launch and arrival dates, and relevant cost
		function input of those trajectories can be found in Table~\ref{results_table} below:

		\begin{table}[h!]
		\begin{small}
		\centering
		\begin{tabular}{ | c c c c c c | }
			\hline
			\bfseries Flyby Selection &
			\bfseries Launch Date &
			\bfseries Mission Length &
			\bfseries Launch $C_3$ &
			\bfseries Arrival $V_\infty$ &
			\bfseries Fuel Usage \\
			&  & (years) & $\left( \frac{km}{s} \right)^2$ & ($\frac{km}{s}$) & (kg) \\
			\hline
			EMS & 2024-06-27 & 7.9844 & 60.41025 & 5.816058 & 446.9227 \\
			EMJS & 2023-11-08 & 14.1072 & 40.43862 & 3.477395 & 530.6683 \\
			\hline
		\end{tabular}
		\end{small}
		\caption{Comparison of the two most optimal trajectories}
		\label{results_table}
		\end{table}

		As mentioned before, the launch energy requirements of the second trajectory are quite a bit
		lower. Having a second mission trajectory capable of launching on a smaller vehicle could be
		valuable to a mission designer presenting possibilities. According to an analysis of the
		Delta IV and Atlas V launch configurations\cite{c3capabilities} in Figure~\ref{c3}, this
		reduction of $C_3$ from around 60 to around 40 brings the sample mission to just within
		range of both the Delta IV Heavy and the Atlas V in its largest configuration, neither of
		which are possible for the other result, meaning that either different launch vehicles must
		be found or mission specifications must change.

		\begin{figure}[H]
			\centering
			\includegraphics[width=\textwidth]{LaTeX/fig/c3}
			\caption{Plot of Delta IV and Atlas V launch vehicle capabilities as they relate to
			payload mass \cite{c3capabilities} from Vardaxis, et al, 2007 }
			\label{c3}
		\end{figure}
