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Newton's Gravitational Equations derived from Kepler's?
In his delightfully easy read book, "Five Equations that Changed the World", Michael Guillen, Ph.D., shows how Newton used Kepler's simple orbital equation T^2 = constant x d^3 to derive equality between Moon's orbital centrifugal force and gravitational attraction, by substituting T^2 in the equation for Centrifugal Force = (constant x m x d)/ T^2. This turns into:
Moon's Centrifugal Force = (constant x m x d)/ (constant x d^3), where T^2 in centrifugal equation was replaced with (constant x d^3), so you are left with the centrifugal-gravitational equivalence of F = new constant x m / d^2.
Of course, the "new constant" was Newton's G, which is 6.67E-11 m^3 kg^-1 s^-2, and the final result is F = GMm/d^2, where G is Newton's gravitational constant, M major mass, m is minor mass, and d is distance. Of course, everybody knows this, but it was nice to see it derived thus from Kepler's orbital equation. :)
I think what intrigues me about this is how a purely geometrical relationship, as discovered by Kepler, can turn into a usable equation for gravity, the equation that gets our space probes out into space.
-Ivan Alexander, Mar. 25, 2006
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