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

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 the sun and planet i.e. $T^2 = Kr^3$, here K is constant. If the masses of the sun and planet are M and m respectively, then as per Newton's law of gravitation, the force of attraction between them is $F = GMm/r^2$, here G is gravitational constant. The relation between G and K is described as:

Accepted Answer

The gravitational force provides the necessary centripetal force for the planet's circular motion around the Sun:
$F_g = F_c \Rightarrow \frac{GMm}{r^2} = \frac{mv^2}{r}$.
Solving for orbital velocity $v$, we get $v = \sqrt{\frac{GM}{r}}$.
The time period of revolution $T$ is given by $T = \frac{2\pi r}{v}$. Substituting $v$:
$T = \frac{2\pi r}{\sqrt{GM/r}} = 2\pi \sqrt{\frac{r^3}{GM}}$.
Squaring both sides:
$T^2 = \left(\frac{4\pi^2}{GM}\right) r^3$.
Comparing this with the given Kepler's law equation $T^2 = Kr^3$, we can see that the constant $K$ is:
$K = \frac{4\pi^2}{GM}$.
Rearranging the terms, we get:
$GMK = 4\pi^2$.

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