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Kepler's third law states that square of period of revolution (T) of a planet around the Sun is proportional to third power of average distance r between Sun and planet, i.e. $T^2 = Kr^3$, 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 $F = \frac{GMm}{r^2}$, here G is gravitational constant. The relation between G and K is described as
A
$GK = 4\pi^2$
B
$GMK = 4\pi^2$
C
$K = G$
D
$K = \frac{1}{G}$
Detailed Solution
Orbital speed: $v = \sqrt{\frac{GM}{r}}$
$T = \frac{2\pi r}{v} = \frac{2\pi r^{3/2}}{\sqrt{GM}}$
$T^2 = \frac{4\pi^2}{GM}r^3$
Comparing with $T^2 = Kr^3$: $K = \frac{4\pi^2}{GM} \Rightarrow GMK = 4\pi^2$
$T = \frac{2\pi r}{v} = \frac{2\pi r^{3/2}}{\sqrt{GM}}$
$T^2 = \frac{4\pi^2}{GM}r^3$
Comparing with $T^2 = Kr^3$: $K = \frac{4\pi^2}{GM} \Rightarrow GMK = 4\pi^2$