A binary star system consists of two stars A and B which have time period `T_A and T_B, radius R_Ba` and mass `M_a and M_B.` Then

Options

A. if `T_Agt T_B` then `R_A gt R_B`
B. if `T_A gt T_B` then `M_A gt M_B`
C. `((T_A)/(T_B))^2 = ((R_A)/(R_B))^3`
D. `T_A = T_B`

Explanation

To solve the problem regarding the binary star system consisting of two stars A and B, we need to analyze the relationship between their time periods, masses, and radii. Here’s a step-by-step solution:

### Step 1: Understanding the System
In a binary star system, two stars orbit around their common center of mass. The gravitational force between the two stars provides the necessary centripetal force for their circular motion.

### Step 2: Gravitational Force
The gravitational force \( F \) between the two stars can be expressed using Newton's law of gravitation:
\[ F = \frac{G M_A M_B}{D^2} \]
where \( G \) is the gravitational constant, \( M_A \) and \( M_B \) are the masses of stars A and B, respectively, and \( D \) is the distance between the two stars.

### Step 3: Centripetal Force
For a star to move in a circular path, the required centripetal force \( F_c \) is given by:
\[ F_c = m \omega^2 r \]
where \( m \) is the mass of the star, \( \omega \) is the angular velocity, and \( r \) is the radius of the circular path.

### Step 4: Equating Forces
Since the gravitational force provides the centripetal force for both stars, we can set the gravitational force equal to the centripetal force for each star:
\[ \frac{G M_A M_B}{D^2} = M_A \omega^2 R_A \quad \text{(for star A)} \]
\[ \frac{G M_A M_B}{D^2} = M_B \omega^2 R_B \quad \text{(for star B)} \]

### Step 5: Relationship Between Angular Velocity and Time Period
The angular velocity \( \omega \) is related to the time period \( T \) by the formula:
\[ \omega = \frac{2\pi}{T} \]
Thus, we can express the time periods for both stars:
\[ T_A = \frac{2\pi}{\omega_A} \quad \text{and} \quad T_B = \frac{2\pi}{\omega_B} \]

### Step 6: Conclusion
Since the angular velocities \( \omega_A \) and \( \omega_B \) are equal for both stars (as they orbit around the center of mass), we conclude that:
\[ T_A = T_B \]
This means the time periods of both stars are the same.

### Final Answer
Thus, the correct option is that the time periods of the two stars are equal:
\[ T_A = T_B \]