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…

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:

Aicharya · 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,
$k = \frac{4 π^{2}}{G M} \Rightarrow G M k = 4 π^{2}$

.

Kepler's third law states that the square of the period of revolution (T) of a planet around the sun is…

Solution