Solution To The Two  Body Problem
The center of mass ( barycenter ) will lay along a radial vector connecting masses and , where its position, , along the radial vector lies in exact proportion to the relative amount of mass at each end of the radial vector. That is, the ratios
determine the precise position of center of mass, , along radial vector .
Therefore,
which implies that
Now, Newton's force equations acting on and respectively becomes
Also, total kinetic energy for an isolated 2  body system becomes
Finally, whenever there is a relatively isolated 2  body system, there will by definition be no acceleration at barycenter ( center of mass ) since all of the forces balance  i.e.,
Notice, moreover, that this isolated 2  body system reduces down as follows:
If, on the other hand, is a small mass relative to such as an artificial satellite put up around a much larger planet or moon, the above equation for combined mass indicates that the barycenter of the 2  body system effectively becomes the center of mass of the much larger mass body!
§ References:
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