A planet moves around sun in nearly circular orbit period of revolution 't', radius of orbit r mass of sun m.

Time period if direcyly proportional to mass of sun, distance between planet and sun and universal gravitational constant.

Prove T^{​​​​​​2} is directly proportional to r^{​​​​​​3}

YOUR ANSWER GOES IN THE WAY OF LAW STATED BY KEPLER.

3rd LAW OF KEPLER:

T² ∝R³

PROOF :

It is known as Law of periods..

Let us consider a planet P of mass m moving with a velocity v around the sun of mass M in a circular orbit of radius r.

The gravitational force of attraction of the sun on the planet is,

F=GMm/r².

The centripetal force is,F = mv²/r.

equating the two forces,

mv²/r=GMm/r².

v²=GM/r -----›(i)

If T be the period of revolution of the planet around the sun, then

v=2πr/T-------›(ii)

Substituting (ii) in (i)

4π²r²/T²=GM/r

r³/T²=GM/4π²

GM is a constant for any planet.

•°• T²∝R³.