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

Question

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

TestHub · Accepted Answer

Here gravitational force is equal to centripetal force required for motion of planet.
Hence,
$\frac{G M m}{r^{2}} = \frac{m v^{2}}{r}$
$v^{2} = \frac{G M}{r}$
Time period is given by,
$T = \frac{2 π r}{v}$
$T^{2} = \frac{4 π^{2} r^{2}}{v^{2}}$
$T^{2} = \frac{4 π^{2} r^{2}}{(\frac{G m}{r})} = \frac{4 π^{2} r^{3}}{G M}$

We have given,
$T^{2} = k r^{3}$
On comparing,
$k = \frac{4 π^{2}}{G M} \Rightarrow G M k = 4 π^{2}$

Physics - Gravitation Question with Solution | TestHub

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