Relativity Science Calculator - Solution To The Two - Body Problem

Solution To The Two - Body Problem

2-body-system.png

The center of mass ( barycenter ) will lay along a radial vector radial_vector.png connecting masses m_1.png and m_2.png , where its position, cm.png, 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

ratios.png

determine the precise position of center of mass, cm.png, along radial vector radial_vector.png.

Therefore,

vectors.png

which implies that

implies.png

Now, Newton's force equations acting on m_1.png and m_2.png  respectively becomes

force_vectors.png

Also, total kinetic energy for an isolated 2 - body system becomes

total_k.e.png

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.,

isolated_2_body_system.png

Notice, moreover, that this isolated 2 - body system reduces down as follows:

combined_mass.png

If, on the other hand, m_2.png  is a small mass relative to m_1.png 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 !

§ "The 55 Cancri Planetary System: Fully Self-Consistent N-body Constraints and a Dynamical Analysis ", by B. Nelson et al., pub. journal Monthly Notices of the Royal Astronomical Society, early online edition April 22, 2014, mathematical astronomy team led by Penn State University graduate student Benjamin Nelson in collaboration with faculty at the 'Center for Exoplanets and Habitable Worlds' at Penn State and five other astronomers from other institutions in the United States and Germany. Mysteries of a nearby ( only 39 light years away ) planetary system's dynamics in the Constellation Cancer are therefore now solved!