Finished at ozo

LaTeX/thesis.tex

@@ -219,8 +219,28 @@ Monotonic Basin Hopping}
         \ddot{\vec{r}} = - \frac{G \left( m_1 + m_2 \right)}{r^2} \frac{\vec{r}}{\left| r \right|}
       \end{equation}
 
+      Where $\vec{r}$ is the position of the spacecraft, $G$ is the universal gravitational
+      parameter, $m_1$ is the mass of the planetary body, and $m_2$ is the mass of the spacecraft.
+      Due to our assumption that the mass of the spacecraft is significantly smaller than the mass
+      of the primary body ($m_1 >> m_2$) we can reduce that formulation to simply:
+
+      \begin{equation}
+        \ddot{\vec{r}} = - \frac{\mu}{r^2} \hat{r}
+      \end{equation}
+
+      Where $\mu = G m_1$ is the specific gravitational parameter for our primary body of interest.
+
       \subsubsection{Kepler's Equations}
-        Detail Kepler's equations for astrodynamics.
+
+        Now that we've fully qualified the forces acting within the Two Body Problem, we can note
+        that the Problem is actually analytically solvable in the case when the position of the
+        spacecraft and the $\mu$ value of the primary body are known. This can be easily observed by
+        noting that there are three one-dimensional equations (one for each component of the
+        three-dimensional position) and three unknowns (the three components of the second
+        derivative of the position).
+
+        Therefore, we can use this analytically solvable force model to model the spacecraft's
+        motion in time, a more useful re-interpretation of the equations of motion.
 
     \subsection{Analytical Solutions to Kepler's Equations}
       Discuss how, since the 2BP is analytically solvable, there exists algorithms for solving