# Double star system; relative orbit in the case of elliptical orbits

In my physics book they say that the relative distance between two stars (that both have elliptical orbits) in a double-star system equals $4.0 AU$ in the pericenter, and $16.0AU$ in the apocenter. Apparently, their relative orbit is an ellipse. However, I don't really understand the situation here. In the case of circular orbits, I understand that both stars share the centre of mass as the center of their relative orbit. We then have the equation $\frac{m_1}{m_2}=\frac{r_2}{r_1}$. But I don't understand the scenario with elliptical orbits. Could someone explain the picture here? Especially how this relative distance translates to an ellipse? Do they still have a stationary center of mass? Is one of the focal points the center of mass?

• Minor comment: I guess $AE =AU$. Feb 13, 2017 at 10:08
• If there is no external force on the system, the center of mass does not accelerate. "Do they have a stationary center of mass?" cannot be answered because it depends on the frame of reference. But one thing that is guaranteed is that the center of mass' velocity (if any) won't change when you are watching from an inertial reference frame. Feb 13, 2017 at 12:31
• What do you mean by "Apparently, their relative orbit is an ellipse."? Feb 13, 2017 at 12:33
• Possible duplicate of Semi-major axis and ellipticity of a binary system? Feb 14, 2017 at 2:11
• Possible duplicate of Meaning of the focus of an elliptical orbit Jul 19, 2018 at 17:05