If I were to have a system of 2 masses that rotate around their barycenter such that $m_1$ has a radius of $r_1$ and $m_2$ has a radius of $r_2$, what would the ratio of the periods of the planets be?
My thinking: Following Kepler's 3rd law that says $T^2/R^3=4π^2/GM$, and since both would have a common $4π^2/GM$ ratio, this would result in the periods having a ratio of $T_1/T_2 = (r_1/r_2)$3/2
Answer: The period for both will be the same as they have the same angular velocity.
My question: So does this idea of the periods being the same for 2 planets orbiting their barycenter always hold? If so why? I can't seem to see the connection as to how if they are orbiting the barycenter, their angular velocities are equal.
Best Answer
You’re asking about a two-body system. Suppose the periods were slightly different: perhaps for every 1000 orbits of the more massive body, the less-massive one does 1001. If that were the case, the faster object would have to eventually “lap” the slower one, like a runner in a race.
But at some point during that overtaking, the two masses would have to be on the same side of the barycenter. That’s unphysical. The barycenter is the system’s center of mass.