 Laboratory paths of an equal-mass binary

For equal masses and relative position $\mathbf{r}$, the individual positions are ${\mathbf{r}}_A=\mathbf{R}-\mathbf{r}/2$ and ${\mathbf{r}}_B=\mathbf{R}+\mathbf{r}/2$, where the center of mass has uniform velocity in an isolated system. Bound inverse-square force motion makes $\mathbf{r}$ an ellipse; each mass follows a half-size ellipse in the center-of-mass frame. Adding a nonzero center-of-mass drift generally makes their laboratory trajectories nonclosed. The relative orbital period is not necessarily a period of either laboratory position.