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 orbitsr_{1}ogr_{2}. Thenr_{1}+r_{2}=a.It is fairly obvious that the two bodies must have the same period

Tabout 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(1)

From this equation we see that

As

r_{1}+r_{2}=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

r_{1}in (1), we getgiving

When the distance

abetween the bodies is given, we may then calculate the radii and periods of thecircular orbits(the theory of elliptical orbits is more difficult). The velocities of the bodies may be calculated from