Newtonian Gravitation

 

The problem that arose with Kepler’s Laws is that they were created through the observations of the planets and by studying their properties and the relationships between each other; there was no mathematical formula that would prove beyond a doubt that these laws were accurate. For fifty years, mathematicians struggled to find a proof for Kepler’s Laws to no avail; science simply didn’t have the tools required to make such a calculation. It wasn’t until Newton developed calculus that a proof could be drawn up that would sufficiently account for the accuracy for the laws.

The key to proving Kepler’s Laws lay in Newton’s equation for gravitational force, which states that the gravitational force between two bodies is equal to the product of the gravitational constant G, the mass of the first body and the mass of the second body divided by the square of the distance between them.

http://csep10.phys.utk.edu/astr161/lect/history/newtongrav.html

The entire proof is rather lengthy and so is not covered here, however; it can be found in it’s entirety here: http://www.alcyone.com/max/physics/kepler/

 

Proving Kepler's third law, however; is a relatively simple process. We know that since planets move around the sun in a near circular orbit, there must be a centripetal acceleration, and hence, and centripetal force, acting inward toward the sun. We can calculate that the centripetal force acting toward the sun from the planet is m(4π^2R/T^2). The key to proving Kepler's Third Law is knowing that this centripetal force is equal to the force of gravity. So, by setting the two expressions equal to each other, we can derive this expression:

T^2 = (4π^2/Gm)/R^3

As we can see from this expression, T^2 and R^3 remain proportional, as required by Kepler's Third Law. What else should be noted is that the constant (4π^2/Gm) is included in the equation. thus creating an equation that not only proves Kepler's Third Law, but is also far reaching in its practical applications. For instance, if the peroid and distance from the sun of a planet were known, the mass of the planet could easily be calculated.