In a certain binary star system, each star has the same mass as our sun. They revolve about their centre of mass. The distance between them is the same as the distance between earth and the sun. What is their period of revolution in years ?

Explanation

To solve the problem of finding the period of revolution of two stars in a binary star system, we can use Kepler's Third Law of planetary motion, which relates the period of revolution to the distance between the two stars.

### Step-by-Step Solution:

1. **Understanding the System**:
- We have two stars, each with a mass equal to that of the Sun (M = M_sun).
- The distance between the two stars (d) is equal to the distance between the Earth and the Sun (1 Astronomical Unit, AU).

2. **Identifying the Relevant Formula**:
- According to Kepler's Third Law, the square of the period (T) of revolution is proportional to the cube of the semi-major axis (R) of the orbit:
\[ T^2 \propto R^3 \]
- For two stars revolving around their common center of mass, the effective radius (R) for each star is half the distance between them:
\[ R = \frac{d}{2} = \frac{1 \text{ AU}}{2} \]

3. **Using the Gravitational Force**:
- The gravitational force provides the necessary centripetal force for the stars' circular motion. The gravitational force (F) between the two stars is given by:
\[ F = \frac{G M^2}{d^2} \]
- The centripetal force required for circular motion is:
\[ F = \frac{M v^2}{R} \]
- Setting these equal gives us:
\[ \frac{G M^2}{d^2} = \frac{M v^2}{R} \]

4. **Relating Velocity to Period**:
- The velocity (v) of the stars can be expressed in terms of the period (T):
\[ v = \frac{2 \pi R}{T} \]
- Substituting this into the centripetal force equation gives:
\[ \frac{G M^2}{d^2} = \frac{M \left(\frac{2 \pi R}{T}\right)^2}{R} \]

5. **Solving for the Period (T)**:
- Rearranging the above equation and solving for T yields:
\[ T^2 = \frac{4 \pi^2 R^3}{G M} \]
- Since R = d/2, we substitute d = 1 AU and M = M_sun:
\[ T^2 = \frac{4 \pi^2 \left(\frac{1 \text{ AU}}{2}\right)^3}{G M_{sun}} \]
- Simplifying gives:
\[ T = 2 \pi \sqrt{\frac{(1 \text{ AU})^3}{8 G M_{sun}}} \]

6. **Calculating the Period in Years**:
- Using the fact that 1 year corresponds to the period of the Earth around the Sun, we can express the result in years:
\[ T = 0.707 \text{ years} \approx 0.71 \text{ years} \]

### Final Answer:
The period of revolution of the two stars in the binary star system is approximately **0.71 years**.