\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}{ | >{\centering}p{0.75in} >{\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}