The areal velocity of a planet of mass m moving in elliptical orbit around the sun with an angular momentum of L units is equal to

Options

A. `(dA)/dt = L/m`
B. `(dA)/dt = (2m)/L`
C. `(dA)/dt = L/(2m)`
D. `(dA)/dt = m/L`

Answer

To solve the problem of finding the areal velocity of a planet of mass \( m \) moving in an elliptical orbit around the sun with an angular momentum \( L \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Areal Velocity**:
Areal velocity is defined as the rate at which area is swept out by the radius vector of the planet as it moves in its orbit. According to Kepler's second law, this rate is constant for a planet in orbit.

2. **Defining Angular Momentum**:
The angular momentum \( L \) of a planet in orbit can be expressed as:
\[ L = m \cdot r \cdot v_{\perp} \]
where \( r \) is the distance from the sun to the planet, and \( v_{\perp} \) is the component of the planet's velocity that is perpendicular to the radius vector \( r \).

3. **Calculating the Area Swept**:
When the planet moves a small distance \( dA \) in a time interval \( dt \), the area swept out can be approximated as:
\[ dA = \frac{1}{2} \cdot r \cdot v_{\perp} \cdot dt \]
This is the area of the triangle formed by the radius vector and the path of the planet.

4. **Finding Areal Velocity**:
The areal velocity \( \frac{dA}{dt} \) is given by:
\[ \frac{dA}{dt} = \frac{1}{2} \cdot r \cdot v_{\perp} \]

5. **Substituting Angular Momentum**:
From the expression for angular momentum, we can express \( v_{\perp} \) in terms of \( L \):
\[ v_{\perp} = \frac{L}{m \cdot r} \]
Substituting this into the areal velocity equation gives:
\[ \frac{dA}{dt} = \frac{1}{2} \cdot r \cdot \left(\frac{L}{m \cdot r}\right) \]

6. **Simplifying the Expression**:
Simplifying the equation:
\[ \frac{dA}{dt} = \frac{1}{2} \cdot \frac{L}{m} \]

7. **Final Result**:
Thus, the areal velocity of the planet is:
\[ \frac{dA}{dt} = \frac{L}{2m} \]

### Conclusion:
The areal velocity of a planet of mass \( m \) moving in an elliptical orbit around the sun with an angular momentum of \( L \) is given by:
\[ \frac{dA}{dt} = \frac{L}{2m} \]