what are the conclusions of newton's law from kepler's law???????

Suppose a planet of mass m revolves around the sun of mass M in nearly circular orbit of radius r, with a constant angular velocity ω. Let T be the time period of the revolution of the planet around the sun.

The centripetal force acting on the planet,

According to Kepler’s third law which states that the square of the time period of revolution of a planet around the sun is directly proportional to the cube of semi major axis of its elliptical orbit i.e.

Where, K is proportionality constant.

According to Newton, the gravitational attraction between the sun and the planet is mutual. So if F depends upon the mass of the planet m then it should also be directly proportional to the mass of the sun, M. Hence,

This is Newton’s law of gravitation. Thus, Newton’s law of gravitation is derived from Kepler’s law of planetary motion.