PX282 - I5 - two body problem

for the one body problem (where mass of one object dominates), considering the sun in the origin, and planet at a position given by $\vec{r}$
the equation of motion, ie. the acceleration due to gravity:

\frac{d^{2} \vec{r}}{d t^{2}} = - \frac{G M}{r^{2}} \hat{r}

now, for the two body problem (where both masses matter), placing the centre of mass at the origin:

\frac{m_{1} \vec{r_{1}} + m_{2} \vec{r_{2}}}{m_{1} + m_{2}} = 0

also:

\vec{r} = {\vec{r}}_{2} - {\vec{r}}_{1}

\begin{aligned} ⟹ {\vec{r}}_{1} & = - \frac{m_{2}}{m_{1} + m_{2}} \vec{r} \\ {\vec{r}}_{2} & = \frac{m_{1}}{m_{1} + m_{2}} \vec{r} \end{aligned}

the two bodies have orbits with the same shape, both proportional to $\vec{r}$ , but of different amplitudes

\begin{aligned} \frac{d^{2} \vec{r}}{d t^{2}} & = \frac{d^{2} \vec{r_{2}}}{d t^{2}} - \frac{d^{2} \vec{r_{1}}}{d t^{2}} \\ = - \frac{G m_{1}}{r^{2}} \hat{r} - \frac{G m_{2}}{r^{2}} \hat{r} \\ = - \frac{G (m_{1} + m_{2})}{r^{2}} \end{aligned}

this is mathematically identical to the one-body problem
kepler's laws apply and both objects follow elliptical orbits with the centre of mass at a focus of the ellipse

generalized kepler's third law

$P^{2} = \frac{4 π^{2}}{G (m_{1} + m_{2})} (a_{1} + a_{2})^{2}$