Two Body Gravitational Motion
Newton’s law of gravitation tells us that two particles of mass m1 and m2 attract each other by a mutual force of magnitude,
where r is the distance between the two particles (at some time t). According to Newton’s IIIrd law of motion, the gravitational force F12 on 2 by 1, i.e.
where r is the position vector of particle 1 relative to 2.
Under the influence of gravitational force, the two particles individually move, with respect to an inertial frame in space. If we consider both the particles as an isolated system, then their center of mass C, remains at rest since there is no external force on the system; the mutual force of gravitation becomes the internal force of action-reaction. Hence, we can choose the frame attached to CM of the two particles as an inertial frame.
If r1 and r2 denote the position vectors of particles 1 and 2 with respect to CM frame, then Newton’s IInd law of motion gives,
where, r = r1 – r2.
The above equations exhibit the nature of a two-body problem, where motion of one particle is intrinsically coupled to the motion of other particle and vice-versa. That is, in order to solve for r1 we must know r2, while r2 in turn depends upon r1.
Fortunately, in Newtonian mechanics, a two-body problem can be solved by reducing it to a one body problem. To do that, we write
The above equation is interpreted as follows: the motion of particle 1 relative to 2 is the same as if particle 2 is fixed and the particle 2 is fixed and the particle 1 has a reduced mass 
Hence,
where F ( r ) is the magnitude of the force on particle 1 by 2; particle 2 being the fixed center of force. In fact,
Thus, we get an equation involving only one position vector r i.e.
That’s a one body problem. In case, m2 >> m1, we can take m1 + m2 ≃ m2, so that
The approximation is practically justified if we are considering motion of artificial satellite/planet around earth/Sun, or of electron around nucleus in case of Coulomb force.
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