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. (a)if `T_A gt T_B` , Then `R_A gt R_B`
B. (b)If `T_A gt T_B` , then `M_A gt M_B`
C. (c)`(T_A/T_B)^2=(R_A/R_B)^3`
D. (d)`T_A=T_B`

Explanation

To solve the problem of a binary star system consisting of two stars A and B with given parameters, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the System**:
- We have two stars, A and B, with masses \( M_A \) and \( M_B \), radii \( R_A \) and \( R_B \), and time periods \( T_A \) and \( T_B \).
- The stars revolve around their common center of mass.

2. **Defining Distances**:
- Let \( r_A \) be the distance from star A to the center of mass, and \( r_B \) be the distance from star B to the center of mass.
- According to the center of mass formula, we have:
\[ M_A \cdot r_A = M_B \cdot r_B \]

3. **Gravitational Force and Centripetal Force**:
- The gravitational force between the two stars provides the necessary centripetal force for their circular motion.
- The gravitational force \( F \) is given by:
\[ F = \frac{G M_A M_B}{(r_A + r_B)^2} \]
- The centripetal force acting on star A is:
\[ F_{cA} = \frac{M_A v_A^2}{r_A} \]
- And for star B, it is:
\[ F_{cB} = \frac{M_B v_B^2}{r_B} \]

4. **Relating Velocity and Time Period**:
- The velocities can be expressed in terms of the time periods:
\[ v_A = \frac{2\pi r_A}{T_A} \quad \text{and} \quad v_B = \frac{2\pi r_B}{T_B} \]

5. **Substituting Velocities into Centripetal Force**:
- Substitute \( v_A \) and \( v_B \) into the centripetal force equations:
\[ F_{cA} = \frac{M_A \left(\frac{2\pi r_A}{T_A}\right)^2}{r_A} = \frac{4\pi^2 M_A r_A}{T_A^2} \]
\[ F_{cB} = \frac{M_B \left(\frac{2\pi r_B}{T_B}\right)^2}{r_B} = \frac{4\pi^2 M_B r_B}{T_B^2} \]

6. **Setting Gravitational Force Equal to Centripetal Force**:
- For star A:
\[ \frac{G M_A M_B}{(r_A + r_B)^2} = \frac{4\pi^2 M_A r_A}{T_A^2} \]
- For star B:
\[ \frac{G M_A M_B}{(r_A + r_B)^2} = \frac{4\pi^2 M_B r_B}{T_B^2} \]

7. **Equating Time Periods**:
- From the equations, we can derive that:
\[ \frac{T_A^2}{r_A} = \frac{T_B^2}{r_B} \]
- This implies:
\[ T_A^2 \cdot r_B = T_B^2 \cdot r_A \]
- Therefore, we conclude that:
\[ T_A = T_B \]

8. **Final Conclusion**:
- The time period of both stars A and B is equal, hence the correct option is:
\[ \text{Option D: } T_A = T_B \]