# [Physics] Orbital mechanics problem

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Here is a problem from the 2009 $F=ma$ contest on orbital mechanics:

Two stars, one of mass $M$, the other of mass $3M$, orbit their common center of mass. When the stars are collinear with the center of mass, the distance between the two stars is $d$. Find the period of orbit for the star of mass $3M$.

First of all, I am a bit clueless ab out how to solve this problem, partially because I don't understand the statement of the problem:

1. What does it actually mean for two bodies to "orbit their center of mass". Does that mean that the two bodies move in ellipses, and the center of mass is a foci of each ellipse?

2.Suppose we are given two masses with non-collinear initial velocity vectors. Assuming the only relevant force is the gravitational attraction between the two masses, does it follow that the bodies orbit their center of mass? In other words, under what conditions do two bodies orbit their center of mass?

I would appreciate the following two three things:

1. You answer my above questions.

2. You provide a solution to the problem.

3. You mention any interesting generalizations, related problems, or things I should study to solve problems like this.

Note: I am preparing for the F=ma exam, so any help is truly appreciated.

Edit: See the diagram here: http://www.aapt.org/physicsteam/2010/upload/2009_F-ma.pdf

What does it actually mean for two bodies to "orbit their center of mass". Does that mean that the two bodies move in ellipses, and the center of mass is a foci of each ellipse?

It is also called a barycenter. Any two (or more) objects in orbit around each other all orbit the barycenter. When working with 2 objects, the center of mass is the barycenter. I think you are confusing the center of mass for each object with the center of mass for the system.

Suppose we are given two masses with non-collinear initial velocity vectors. Assuming the only relevant force is the gravitational attraction between the two masses, does it follow that the bodies orbit their center of mass? In other words, under what conditions do two bodies orbit their center of mass?

The 2 object must always orbit their center of mass of the system.

As for solving the question, it appears there is not enough information provided to determine the period.

The period can be found from: $T=2\pi \sqrt {\frac{r^3}{G(M_1+M_2)}}$
The masses are M and 3M and the radii of the orbits is $\frac{1}{4}d + \frac{3}{4}d$.
$T=2\pi \sqrt {\frac{d^3}{4GM}}$
$T=\pi \sqrt {\frac{d^3}{GM}}$