Consider the following statements and choose the correct option
I. If the distance between the earth and the sun was half its present value, the number of days in a year would be 129.
II. According to Kepler's law of period `T^3 prop r^3`

Options

A. Statement I is true
B. Statement II is true
C. Both statements I and II are true
D. None is true

Answer

To solve the problem, we need to analyze the two statements given and determine their validity based on Kepler's laws of planetary motion.

### Step 1: Analyze Statement I
Statement I claims that if the distance between the Earth and the Sun was half its present value, the number of days in a year would be 129.

1. **Understanding Kepler's Third Law**: According to Kepler's Third Law, the square of the orbital period (T) of a planet is directly proportional to the cube of the semi-major axis (R) of its orbit:
\[ T^2 \propto R^3 \]

2. **Setting Up the Relationship**: Let’s denote the current radius (distance from the Earth to the Sun) as \( R_1 \) and the current period (one year) as \( T_1 \). We know:
\[ T_1^2 = k R_1^3 \]
where \( k \) is a constant.

3. **Changing the Radius**: If the distance is halved, then the new radius \( R_2 \) is:
\[ R_2 = \frac{R_1}{2} \]

4. **Applying Kepler's Law for the New Radius**: For the new period \( T_2 \):
\[ T_2^2 = k R_2^3 = k \left(\frac{R_1}{2}\right)^3 = k \frac{R_1^3}{8} \]

5. **Relating the Two Periods**: Now, we can relate \( T_1 \) and \( T_2 \):
\[ \frac{T_1^2}{T_2^2} = \frac{R_1^3}{\frac{R_1^3}{8}} = 8 \]
Thus, we have:
\[ T_2^2 = \frac{T_1^2}{8} \]

6. **Calculating \( T_2 \)**: Since \( T_1 = 365 \) days:
\[ T_2 = \sqrt{\frac{365^2}{8}} = \frac{365}{\sqrt{8}} = \frac{365}{2\sqrt{2}} \approx 129.0 \text{ days} \]
Therefore, Statement I is **correct**.

### Step 2: Analyze Statement II
Statement II states that according to Kepler's law of period, \( T^3 \propto R^3 \).

1. **Identifying the Correct Relationship**: Kepler's Third Law states:
\[ T^2 \propto R^3 \]
This means that the square of the period is proportional to the cube of the radius, not the cube of the period.

2. **Conclusion for Statement II**: Since Statement II incorrectly states the relationship as \( T^3 \propto R^3 \), it is **false**.

### Final Conclusion
- **Statement I**: True
- **Statement II**: False

Thus, the correct option is that only Statement I is true.