Kepler's third law
We have seen that the radius vector connecting our planet to the
origin sweeps out area at the constant rate . [See Equation (4.24).]
We have also seen that the planetary orbit is an ellipse. Now,
the major and minor radii of such an ellipse are
and
, respectively. [See Equations (A.107) and (A.108).] The area of the ellipse is
. We expect the
radius vector to sweep out the whole area of the ellipse in a single
orbital period, . Hence,

(4.32) 
It follows from Equation (4.31)
that

(4.33) 
In other words, the square of the orbital period of our planet is proportional to the cube
of its orbital major radius; this is Kepler's third law of planetary motion.