Circular orbits in the two-body problem
Assume that the distance between two bodies is constant and equal to a. The gravitational force between the bodies accelerate them towards each other. It then seems possible that the bodies may circle a point P on the line connecting the bodies. Call the radii in the two orbits r1 og r2. Then r1 + r2 = a.
It is fairly obvious that the two bodies must have the same period T about P. Also the gravitational force on each of them is the same. Newton's gravitational law and second law and the formula for the centripetal acceleration gives
From this equation we see that
As r1 + r2 = a, we easily find
By comparing this with the theory for the center of mass, we see that P is the center of mass for the two bodies.
Inserting the expression for r1 in (1), we get
When the distance a between the bodies is given, we may then calculate the radii and periods of the circular orbits (the theory of elliptical orbits is more difficult). The velocities of the bodies may be calculated from