For a planet revolving around sun, if `a` and `b` are the respective semi-major and semi-minor axes, then the square of its time period is proportional to

Options

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

Answer

To solve the question, we need to apply Kepler's Third Law of planetary motion, which relates the time period of a planet's orbit to the semi-major axis of its elliptical orbit.

### Step-by-Step Solution:

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

2. **Identifying the Semi-Major and Semi-Minor Axes**:
In the context of the question, we are given two axes: \( a \) (the semi-major axis) and \( b \) (the semi-minor axis). According to Kepler's law, only the semi-major axis is relevant for determining the time period.

3. **Applying the Law**:
Since \( a \) is the semi-major axis, we can directly apply Kepler's Third Law:
\[ T^2 \propto a^3 \]
This means that the square of the time period is proportional to the cube of the semi-major axis.

4. **Conclusion**:
Therefore, the square of the time period \( T^2 \) is proportional to \( a^3 \).

### Final Answer:
The square of the time period is proportional to \( a^3 \).

---