In a double star system one of mass `m_(1)` and another of mass `m_(2)` with a separation d rotate about their common centre of mass. Then rate of sweeps of area of star of mass `m_(1)` to star of mass `m_(2)` about their common centre of mass is

Options

A. `(m_(1))/(m_(2))`
B. `(m_(2))/(m_(1))`
C. `(m_(1)^(2))/(m_(2)^(2))`
D. `(m_(2)^(2))/(m_(1)^(2))`

Answer

To solve the problem, we need to find the rate of area swept out by the two stars in a double star system rotating around their common center of mass. We will denote the masses of the stars as \( m_1 \) and \( m_2 \), and the separation between them as \( d \).

### Step-by-Step Solution:

1. **Identify the Center of Mass**:
The center of mass \( R_{cm} \) of the two-star system can be calculated using the formula:
\[ R_{cm} = \frac{m_1 \cdot r_1 + m_2 \cdot r_2}{m_1 + m_2} \]
where \( r_1 \) and \( r_2 \) are the distances of \( m_1 \) and \( m_2 \) from the center of mass, respectively.

2. **Calculate Distances from the Center of Mass**:
The distances \( r_1 \) and \( r_2 \) can be expressed as:
\[ r_1 = \frac{m_2 \cdot d}{m_1 + m_2} \]
\[ r_2 = \frac{m_1 \cdot d}{m_1 + m_2} \]

3. **Use the Area Swept Out Formula**:
According to Kepler's second law, the rate of area swept out by a body is given by:
\[ \frac{dA}{dt} = \frac{1}{2} r^2 \frac{d\theta}{dt} \]
Thus, for star \( m_1 \):
\[ \frac{dA_1}{dt} = \frac{1}{2} r_1^2 \frac{d\theta}{dt} \]
And for star \( m_2 \):
\[ \frac{dA_2}{dt} = \frac{1}{2} r_2^2 \frac{d\theta}{dt} \]

4. **Calculate the Ratio of Areas Swept Out**:
The ratio of the rates of area swept out by \( m_1 \) and \( m_2 \) is:
\[ \frac{dA_1/dt}{dA_2/dt} = \frac{r_1^2}{r_2^2} \]

5. **Substituting the Values of \( r_1 \) and \( r_2 \)**:
Substitute \( r_1 \) and \( r_2 \) into the ratio:
\[ \frac{dA_1/dt}{dA_2/dt} = \frac{\left(\frac{m_2 \cdot d}{m_1 + m_2}\right)^2}{\left(\frac{m_1 \cdot d}{m_1 + m_2}\right)^2} \]
Simplifying this gives:
\[ = \frac{m_2^2}{m_1^2} \]

6. **Final Result**:
Therefore, the rate of area swept out by star \( m_1 \) to star \( m_2 \) is:
\[ \frac{dA_1/dt}{dA_2/dt} = \frac{m_2^2}{m_1^2} \]

### Conclusion:
The final answer is:
\[ \frac{dA_1}{dA_2} = \left(\frac{m_2}{m_1}\right)^2 \]