If r is the distance between the Earth and the Sun. Then, angular momentum of the Earth around the sun is proportional to

Options

A. `sqrt(r)`
B. `r^(3//2)`
C. `r`
D. None of these

Explanation

To determine the angular momentum of the Earth around the Sun, we can follow these steps:

### Step 1: Understand the relationship between period (T) and distance (r)
According to Kepler's Third Law, the square of the orbital period (T) of a planet is directly proportional to the cube of the semi-major axis (r) of its orbit:
\[ T^2 \propto r^3 \]

### Step 2: Express T in terms of r
From the above relationship, we can express T as:
\[ T \propto r^{3/2} \]

### Step 3: Relate angular velocity (ω) to T
The angular velocity (ω) is given by:
\[ \omega = \frac{2\pi}{T} \]
Substituting the expression for T, we find:
\[ \omega \propto \frac{1}{r^{3/2}} \]

### Step 4: Write the expression for angular momentum (L)
The angular momentum (L) of the Earth around the Sun can be expressed as:
\[ L = m \cdot r^2 \cdot \omega \]
where m is the mass of the Earth.

### Step 5: Substitute ω into the angular momentum equation
Substituting the expression for ω into the angular momentum equation, we get:
\[ L \propto m \cdot r^2 \cdot \frac{1}{r^{3/2}} \]

### Step 6: Simplify the expression
Now, simplifying the expression:
\[ L \propto m \cdot r^{2 - 3/2} = m \cdot r^{1/2} \]

### Conclusion
Thus, the angular momentum of the Earth around the Sun is proportional to:
\[ L \propto r^{1/2} \]