Kepler's third law states that square of period revolution `(T)` of a planet around the sun is proportional to third power of average distance `i` between sun and planet i.e. `T^(2)=Kr^(3)`
here `K` is constant
if the mass of sun and planet are `M` and `m` respectively then as per Newton's law of gravitational the force of alteaction between them is `F=(GMm)/(r^(2))`, here `G` is gravitational constant. The relation between `G` and `K` is described as

Options

A. K = G
B. `K = 1/G`
C. `GK = 4pi^2`
D. `GMK = 4pi^2`

Answer

D

Explanation

Gravitational force of attraction between sun and planet provides centripetal force for the orbit of planet.
` therefore (GMm)/(r^2) = (mv^2)/(r ) rArr v^2 = (GM)/(r )` ....(i)
Time period of the planet is given by
` T = (2pi r)/( v) , T^2 = (4pi^2 r^2)/(v^2) = (4pi^2 r^2)/( (GM)/(r )) `
` T^2 = (4pi^2 r^3)/(GM)`
According to question, `T^2 = Kr^3` .....(iii)
Comparing equations (ii) and (iii), we get
`K = (4pi^2)/(GM) therefore GMK = 4pi^2`