we can immediately write the rectangular equations as

Here is where some work elliptic curve is needed. For an elliptic curve which takes the form of the give polar equation, the value of , termed eccentricity, describes the shape of the ellipse.

Incidentally, this value can also be written as

where c and a corresponding the those constants in the rectangular equation above. Graphically, the curve is shown as

Now, some geometric observation gives us

and so

Time for some astronomy knowledge. The semimajor axis a of an elliptical orbit is also called the mean distance because it is one-half the sum of the least and greatest values of r. These values of r correspond to  and . By this definition, we substitute these values of  into the polar equation to get the corresponding values of r to calculate a.

On rearranging, we get

We now define T as the period of m, that is the time require for one complete revolution in its orbit. We also take the area of the ellipse to be  and recalling Kepler’s second law as , we have

And so

on eliminating b and h. We previously defined , which depends on the gravitational constant and mass of the Sun, which are both constants. Hence for any planet rotation around our solar system they obey Kepler’s third law: The squares of the period of revolution of the planet is proportional to the cubes of their mean distances.

I hope you have enjoyed these 5 lessons on deriving Kepler’s three laws. As a conclusion I would like to say my opinion on the value of mathematics. What I have done is perhaps one of the more mathematical rigorous approaches in deriving the laws. Whether this proof surpasses that of using principles in physics, it is left for the student to decide. However, what I can safely say is that fully integrating math in the proof does somewhat imply a greater deal of precision. Under given assumptions, the equations don’t lie in math and once we can establish a law in terms of variables and numbers, the law will hold forever.