From 678b0438dba299633de1d67fba5f0e0b4b619b4e Mon Sep 17 00:00:00 2001 From: Connor Date: Sun, 2 Jan 2022 17:05:32 -0700 Subject: [PATCH] Finished at ozo --- LaTeX/thesis.tex | 22 +++++++++++++++++++++- 1 file changed, 21 insertions(+), 1 deletion(-) diff --git a/LaTeX/thesis.tex b/LaTeX/thesis.tex index a9893fe..c8194df 100644 --- a/LaTeX/thesis.tex +++ b/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