A planet revolves in an elliptical orbit around the sun. The semi-major and minor axes are a and b , then the time period is given by :

Options

A. `T^(2) prop b^(3)`
B. `T^(2) prop ((a + b)/(2))^(3)`
C. `T^(2) prop a^(3)`
D. `T^(2) prop ((a - b)/(2))^(3)`

Explanation

To find the time period of a planet revolving in an elliptical orbit around the sun, we can use Kepler's Third Law of Planetary Motion. The law states that the square of the time period (T) of a planet is directly proportional to the cube of the semi-major axis (a) of its orbit.

### Step-by-Step Solution:

1. **Understanding Kepler's Third Law**:
Kepler's Third Law states that for any planet orbiting the sun, the square of its orbital period (T) is proportional to the cube of the semi-major axis (a) of its elliptical orbit. Mathematically, this can be expressed as:
\[ T^2 \propto a^3 \]

2. **Expressing the Proportionality**:
To convert the proportionality into an equation, we introduce a constant of proportionality (k):
\[ T^2 = k \cdot a^3 \]

3. **Identifying the Constant**:
The constant k depends on the mass of the sun and the gravitational constant (G). For our purposes, we can state that:
\[ k = \frac{4\pi^2}{G(M+m)} \]
where M is the mass of the sun and m is the mass of the planet. However, since the mass of the planet is negligible compared to the mass of the sun, we can simplify this to:
\[ k \approx \frac{4\pi^2}{GM} \]

4. **Final Expression for the Time Period**:
Substituting this back into our equation, we get:
\[ T^2 = \frac{4\pi^2}{GM} a^3 \]
Therefore, the time period T can be expressed as:
\[ T = 2\pi \sqrt{\frac{a^3}{GM}} \]

5. **Conclusion**:
Thus, the time period of a planet revolving in an elliptical orbit around the sun, with semi-major axis a, is given by:
\[ T^2 \propto a^3 \]