Kepler's Third Law of Planetary Motion:

The square of the period of revolution around the sun is equal to the cube of the orbit’s semimajor axis.

          According to the first law, a planet’s orbit is elliptical in shape.  The length of the semimajor axis of the ellipse would be half the distance between the farthest points.  Using the equation for the ellipse derived in the first law, the two extremes of the curve are where  and , which occur when q  =0 and q  =p, respectively.  The length of the semimajor axis would therefore be

.

Integrating the area of the ellipse over a full period of revolution using the equation derived in the second law gives

.

For an ellipse, , where a and b are the semimajor axis and the semiminor axis, respectively.  By the definition of the eccentricity of an ellipse, b can be related to a and e  by

.

Plugging this into the area equation results in

,

which can be re-written as

.

Since k was originally defined as , this can be further reduced to the specific force of gravity by saying , where G is the Universal Gravitational Constant and M and m are defined as they were in the assumption starting the first law.  Replacing k with this expression gives

,

or, after squaring both sides,

,

which is often considered to be Newton’s correction of Kepler’s Third Law since it applies to the general case of any small body orbiting a large one, not just within our solar system.  It should be noted that in the case of the two bodies being of comparable mass, the reduced mass, , of the system should have been used before inserting the expression for k.  This would have given

.

After replacing k, the result would be

,

and squaring the expression would give

,

Which reduces down to the original form when M >> m, which is a good approximation when comparing stars and planets.  For the case of the Solar System, dividing the expression by itself with the values for Earth yields

.

If T is in units of years and a is in units of Astronomical Units, then  and , which reduces the above equation to

under these special conditions.  This is Kepler’s Third Law of Planetary Motion.  Although it is not the most general form, it is the simplest to use when dealing with the Solar System, which is what Kepler’s Laws were meant to describe.

source : http://astro.berkeley.edu/~converse/Lagrange/Kepler'sThirdLaw.htm