Using Newton’s Universal Law of Gravitation and Newton’s 3rd Law of Motion, we know that, the mutual gravitational interaction between two bodies can be expressed as the two following equations of motion (EoM):

\begin{align} \ddot{\mathbf{r}}_1 = -\frac{Gm_1m_2}{|\mathbf{r}_2-\mathbf{r}_1|^3}(\mathbf{r}_2-\mathbf{r}_1)~,\;\;\text{and} \;\;\; \ddot{\mathbf{r}}_2 = \frac{Gm_1m_2}{|\mathbf{r}_2-\mathbf{r}_1|^3}(\mathbf{r}_2-\mathbf{r}_1) \end{align}

Now you must define two new variables \(({\bf r}_1,{\bf r}_2)\rightarrow({\bf r},{\bf R})\) where

\begin{align} {\bf r}\equiv{\bf r}_2-{\bf r}_1,\;\;\text{and} \;\;\; {\bf R}\equiv \frac{m_1{\bf r}_1+m_2{\bf r}_2}{m_1+m_2} \end{align}

  1. Derive the EoM for \({\bf r}\) and \({\bf R}\) (i.e., obtain expressions for \(\ddot{\bf r}\) and \(\ddot{\bf R}\))

  2. Show that, in barycentric coordinates (i.e., \({\bf R}=0\)), the total angular momentum \(m_1{\bf r}_1\times\dot{\bf r}_1+m_2{\bf r}_2\times\dot{\bf r}_2\) is equal to \(\mu{\bf r}\times\dot{\bf r}\) where \(\mu=m_1m_2/(m_1+m_2)\) is the system’s reduced mass.

  3. Show that the following two vectors are constants of motion \begin{align} {\bf h}\equiv {\bf r}\times\dot{\bf r} \;\;\;\text{and} \;\;\; {\bf e}\equiv \frac{1}{G(m_1+m_2)}\dot{\bf r}\times({\bf r}\times\dot{\bf r})-\frac{\bf r}{|\bf r|} \end{align}

  4. Assuming the orbit is an ellipse (it is!), write down explicit expressions for \({\bf r}\) and \(\dot{\bf r}\). Now draw the orbit/ellipse in a convenient system of coordinate axes. In this coordinate system, draw a few vectors corresponding to \({\bf r}\) and \(\dot{\bf r}\) at different points along the orbit. Draw also \({\bf e}\). Comment on the direction in which \({\bf e}\) is pointing.

  5. What happens if there is an additional external force \({\bf f}_{\rm ext}\) acting on the system, namely \begin{align} \ddot{\bf r}={\bf f}_0 + {\bf f} _ {\rm ext} \end{align}

where \({\bf f}_0\) is the force (per unit mass) acting on the system when only mutual gravity is accounted for. Derive expressions for \(\dot{\bf h}\) and \(\dot{\bf e}\).