**Double Star System**
A double star system consists of two stars, A and B, that orbit around a common center of mass due to their gravitational attraction. Each star has its own time period, radius, and mass. Let's analyze the given statements:

**a) If T1 > T2, then R1 > R2:**

- Time period (T) is the time taken by a star to complete one orbit around the center of mass of the system.
- The time period is inversely proportional to the radius of the orbit. As the time period increases, the radius of the orbit decreases.

If T1 > T2, it means star A takes more time to complete one orbit than star B. Hence, the orbit of star A is larger than the orbit of star B. This implies:

- Radius (R) is directly proportional to the time period (T).
- Therefore, if T1 > T2, then R1 > R2.

**b) If T1 > T2, then M1 > M2:**

- Mass (M) of a star determines its gravitational force, which in turn affects the time period and radius of the orbit.
- The more massive the star, the stronger its gravitational force, and the shorter its time period.

If T1 > T2, it means star A takes more time to complete one orbit than star B. This suggests that star B has a shorter time period and, therefore, a smaller mass compared to star A. Hence:

- Mass (M) is inversely proportional to the time period (T).
- Therefore, if T1 > T2, then M1 > M2.

**c) T1 = T2 and How?**

If T1 = T2, it means both stars A and B take the same time to complete one orbit around the center of mass. In this case:

- The time period is the same for both stars, indicating that they have the same orbital speed.
- The radii of their orbits can be different, but the ratio of their radii will be equal to the ratio of their masses.

This is known as Kepler's Third Law of Planetary Motion, which states that the square of the time period is proportional to the cube of the average distance between the stars:

(T1/T2)^2 = (R1/R2)^3

If T1 = T2, then the equation simplifies to:

1 = (R1/R2)^3

This implies:

- R1/R2 = 1
- R1 = R2

Therefore, if T1 = T2, then R1 = R2, indicating that both stars have the same radius.

In summary, for a double star system:
- If T1 > T2, then R1 > R2 and M1 > M2.
- If T1 = T2, then R1 = R2 and M1 = M2.