Since external force is zero, the acceleration of center of mass is zero. This is the first constraint. For easy visualization of this constraint, we consider that center of mass of the system is at rest in a particular reference frame.

Now, since bodies are moving along two circular paths about "center of mass", their motions should be synchronized in a manner so that the length of line, joining their centers, is a constant . This is required; otherwise center of mass will not remain stationary in the chosen reference. Therefore, the linear distance between bodies is a constant and is given by :

Two body system - circular motion |
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Now this condition can be met even if two bodies move in different planes. However, there is no external torque on the system. It means that the angular momentum of the system is conserved. This has an important deduction : the plane of two circular trajectories should be same.

Mathematically, we can conclude this, using the concept of angular momentum. We know that torque is equal to time rate of change of angular momentum,

But, external torque is zero. Hence,

It means that “r” and “F” are always parallel. It is only possible if two planes of circles are same. We, therefore, conclude that motions of two bodies are coplanar. For coplanar circular motion, center of mass is given by definition as :

Two body system - circular motion |
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Taking first differentiation with respect to time, we have :

Now dividing second equation by first,

It means that two bodies move in such a manner that their angular velocities are equal.

Two body system - circular motion |
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**Gravitational force**

The gravitational force on each of the bodies is constant and is given by :

Since gravitational force provides for the requirement of centripetal force in each case, it is also same in two cases. Centripetal force is given by :