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

Explanation

To find the relation between the constant \( K \) from Kepler's third law and the gravitational constant \( G \), we can follow these steps:

### Step 1: Understand Kepler's Third Law
Kepler's third law states that the square of the period \( T \) of revolution of a planet around the sun is proportional to the cube of the average distance \( r \) between the sun and the planet:
\[ T^2 = K r^3 \]
where \( K \) is a constant.

### Step 2: Write the Gravitational Force Equation
According to Newton's law of gravitation, the gravitational force \( F \) between the sun (mass \( M \)) and the planet (mass \( m \)) is given by:
\[ F = \frac{G M m}{r^2} \]
where \( G \) is the gravitational constant.

### Step 3: Relate Gravitational Force to Centripetal Force
For a planet in circular orbit, the gravitational force provides the necessary centripetal force to keep the planet in orbit. Thus, we can set the gravitational force equal to the centripetal force:
\[ \frac{M v^2}{r} = \frac{G M m}{r^2} \]
Here, \( v \) is the orbital velocity of the planet.

### Step 4: Solve for Orbital Velocity
From the equation above, we can simplify:
\[ M v^2 = \frac{G M m}{r} \]
Dividing both sides by \( M \) (assuming \( M \neq 0 \)):
\[ v^2 = \frac{G m}{r} \]

### Step 5: Find the Time Period \( T \)
The orbital velocity \( v \) can also be expressed in terms of the period \( T \):
\[ v = \frac{2 \pi r}{T} \]
Substituting this into the equation for \( v^2 \):
\[ \left(\frac{2 \pi r}{T}\right)^2 = \frac{G m}{r} \]
This simplifies to:
\[ \frac{4 \pi^2 r^2}{T^2} = \frac{G m}{r} \]

### Step 6: Rearrange to Find \( T^2 \)
Rearranging gives:
\[ T^2 = \frac{4 \pi^2 r^3}{G m} \]

### Step 7: Compare with Kepler's Third Law
Now we can compare this result with Kepler's third law \( T^2 = K r^3 \):
\[ K = \frac{4 \pi^2}{G m} \]

### Step 8: Establish the Relation between \( K \) and \( G \)
From the expression for \( K \):
\[ K G m = 4 \pi^2 \]
This shows the relationship between \( K \) and \( G \).

### Conclusion
The relation between \( K \) and \( G \) can be summarized as:
\[ K G m = 4 \pi^2 \]