I think I've finished my first revision

LaTeX/thesis.tex

@@ -1155,9 +1155,9 @@
 				uniform distribution within those bounds.
 
 				A unit launch direction is then also chosen as a 3-length vector of uniform random
-				numbers, then normalized. This unit vector is then multiplied by a uniform random
-				number between 0 and the square root of the maximum launch $C_3$ specified by the
-				user to generate an initial $\vec{v_\infty}$ vector at launch.
+				numbers and normalized. This vector is then multiplied by a uniform random number
+				between 0 and the root of the maximum launch $C_3$ specified by the user to generate
+				an initial $\vec{v}_\infty$ vector at launch.
 
 				Next, the times of flight of each phase of the mission is then decided. Since launch
 				date has already been selected, the maximum time of flight can be calculated by
@@ -1288,17 +1288,17 @@
 				a tuning parameter to determine the size of the tails and width of the distribution
 				set to $1.01$, but easily tunable.
 
-				The perturbation function, then steps through each parameter of the mission,
-				generating a new mission guess with the parameters modified by the above Pareto
-				distribution. After this perturbation, the NLP solver is then called again to find
-				a valid solution in the vicinity of this new guess. If the solution is better than
-				the current basin solution, it replaces that value and the drill counter is reset to
+				The perturbation function then steps through each parameter of the mission,
+				generating a new guess with the parameters modified by the Pareto distribution.
+				After this perturbation, the NLP solver is then called again to find a valid
+				solution in the vicinity of this new guess. If the solution is better than the
+				current basin solution, it replaces that value and the drill counter is reset to
 				zero. If it is better than the current total best, it replaces that value as well.
 				Otherwise, the drill counter increments and the process is repeated. Therefore, the
 				drill patience allows the mission designer to determine a maximum number of
-				iterations to perform without any improvements in a row before ending a given drill
-				loop. This process can be repeated essentially ''search patience`` number of times
-				in order to fully traverse all basins.
+				iterations to perform without improvement in a row before ending the drill loop.
+				This process can be repeated essentially ''search patience`` number of times in
+				order to fully traverse all basins.
 
 	\chapter{Results Analysis} \label{results}
 
@@ -1436,8 +1436,11 @@
 			with Mars after three and one half years to rendezvous in mid-December 2027.
 			Unfortunately, the launch energy required to effectively used the gravity assist with
 			Mars at this time is quite high. The $C_3$ value was found to be $60.4102$ kilometers
-			per second squared. However, for this phase, the thrusters are almost entirely turned
-			off, allowing for a nearly-natural trajectory to Mars rendezvous.
+			per second squared. 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 direction was neither
+			zero nor $\pi$, implying that these thrusts were steering thrusts rather than
+			momentum-increasing thrusts.
 
 			\begin{figure}[H]
 				\centering
@@ -1518,11 +1521,11 @@
 			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
-			at about the apoapsis point of its orbit in order to raise the periapsis enough to
-			rendezvous at roughly the same incidence angle in Mars' orbit, but one revolution ahead.
-			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.
+			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
@@ -1565,9 +1568,11 @@
 
 			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. 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.
+			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