The time period of a satellite in a circular orbit around the earth is T . The kinetic energy of the satellite is proportional to `T^(-n)` . Then, n is equal to :

Options

A. `(1)/(2)`
B. `(2)/(3)`
C. `(4)/(3)`
D. `(3)/(2)`

Explanation

To solve the problem, we need to find the relationship between the kinetic energy of a satellite in a circular orbit and its time period \( T \). We will derive the relationship step by step.

### Step 1: Understand the Kinetic Energy Formula
The kinetic energy \( KE \) of an object is given by the formula:
\[ KE = \frac{1}{2} m v^2 \]
where \( m \) is the mass of the satellite and \( v \) is its velocity.

### Step 2: Find the Velocity of the Satellite
For a satellite in a circular orbit, the velocity \( v \) can be expressed in terms of the radius \( r \) of the orbit and the time period \( T \):
\[ v = \frac{2\pi r}{T} \]
This is because the satellite travels a distance of \( 2\pi r \) in one complete orbit, which takes time \( T \).

### Step 3: Substitute Velocity into the Kinetic Energy Formula
Now, substituting the expression for \( v \) into the kinetic energy formula:
\[ KE = \frac{1}{2} m \left(\frac{2\pi r}{T}\right)^2 \]
This simplifies to:
\[ KE = \frac{1}{2} m \cdot \frac{4\pi^2 r^2}{T^2} = \frac{2\pi^2 m r^2}{T^2} \]

### Step 4: Relate Radius \( r \) to Time Period \( T \)
According to Kepler's Third Law, the time period \( T \) is related to the radius \( r \) of the orbit:
\[ T^2 \propto r^3 \quad \Rightarrow \quad r \propto T^{2/3} \]
This means we can express \( r \) in terms of \( T \):
\[ r = k T^{2/3} \]
where \( k \) is a proportionality constant.

### Step 5: Substitute \( r \) Back into the Kinetic Energy Expression
Now, substituting \( r = k T^{2/3} \) into the kinetic energy equation:
\[ KE = \frac{2\pi^2 m (k T^{2/3})^2}{T^2} \]
This simplifies to:
\[ KE = \frac{2\pi^2 m k^2 T^{4/3}}{T^2} = 2\pi^2 m k^2 T^{4/3 - 2} = 2\pi^2 m k^2 T^{-2/3} \]

### Step 6: Determine the Value of \( n \)
From the expression \( KE \propto T^{-2/3} \), we can see that the kinetic energy is proportional to \( T^{-n} \) where \( n = \frac{2}{3} \).

### Final Answer
Thus, the value of \( n \) is:
\[ \boxed{\frac{2}{3}} \]