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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 radiusthis is Kepler's third law of planetary motion.
Richard Fitzpatrick
20160331