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Kepler's Third Law of Planetary Motion and Newton's Law of Gravitation
The elliptical orbits of the planets are very nearly circular.  
Here we will assume circular obits for simplicity.   
     
   
Applying Newton's law of gravitation to the situation shown here, we have   
   
   
   
this force gives the planet its centripetal acceleration so we can also write    
   
These two equations lead us to   
   
Remembering that   
   
gives the following equation   
   
which is a statement of Kepler's third law as everything on the right hand side is constant.   
   
This can be considered as further evidence to support the inverse square law of gravitation.   
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