Kepler's third law states that the square of period of revolution T of a planet around the sun is proportional to the third power of average distance, r between the sun and the planet i.e T 2 = K

Question

Kepler's third law states that the square of period of revolution $(T)$ of a planet around the sun is proportional to the third power of average distance, $r$ between the sun and the planet i.e $T^{2} = K r^{3}$ .Here, $K$ is constant.
If masses of the sun and the planet are $M$ and $m$ respectively, then as per Newton's law of gravitation force of attraction between them is $F = \frac{G M m}{r^{2}}$ , where $G$ is gravitational constant. The relation between $G$ and $K$ is described as

TestHub · Accepted Answer

As, G M m r 2 = m v 2 r = centripetal force ⇒ v 2 = G M r ⇒ T = 2 π r v ⇒ T 2 = 4 π 2 r 2 v 2 Out putting the value of v 2 , we get T 2 = 4 π 2 r 2 G M r ⇒ T 2 = 4 π 2 r 3 G M ⇒ T 2 = K r 3 Here, K is given by, 4 π 2 G M = K ⇒ G M K = 4 π 2

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