If the distance between the earth and the sun were half its present value, the number of days in a year would have been

Options

A. 64.5
B. 129
C. 182.5
D. 730

Explanation

To solve the problem of how many days would be in a year if the distance between the Earth and the Sun were halved, we can use Kepler's Third Law of Planetary Motion. This law states that 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.

### Step-by-Step Solution:

1. **Identify the Variables**:
- Let \( T_1 \) be the current orbital period of the Earth (which is 365 days).
- Let \( R_1 \) be the current distance between the Earth and the Sun.
- Let \( R_2 \) be the new distance, which is half of \( R_1 \): \( R_2 = \frac{R_1}{2} \).
- Let \( T_2 \) be the new orbital period we want to find.

2. **Apply Kepler's Third Law**:
According to Kepler's Third Law:
\[ \frac{T_2^2}{T_1^2} = \frac{R_2^3}{R_1^3} \]

3. **Substitute the Values**:
Substitute \( R_2 = \frac{R_1}{2} \) into the equation:
\[ \frac{T_2^2}{T_1^2} = \frac{\left(\frac{R_1}{2}\right)^3}{R_1^3} \]

4. **Simplify the Equation**:
Simplifying the right side gives:
\[ \frac{T_2^2}{T_1^2} = \frac{\frac{R_1^3}{8}}{R_1^3} = \frac{1}{8} \]

5. **Relate \( T_2 \) to \( T_1 \)**:
Now, we can express \( T_2 \) in terms of \( T_1 \):
\[ T_2^2 = \frac{1}{8} T_1^2 \]
Taking the square root of both sides:
\[ T_2 = T_1 \cdot \frac{1}{\sqrt{8}} = T_1 \cdot \frac{1}{2\sqrt{2}} \]

6. **Calculate \( T_2 \)**:
Substitute \( T_1 = 365 \) days:
\[ T_2 = 365 \cdot \frac{1}{2\sqrt{2}} \approx 365 \cdot 0.3536 \approx 129.04 \text{ days} \]

7. **Round the Result**:
Rounding \( T_2 \) gives approximately:
\[ T_2 \approx 129 \text{ days} \]

### Final Answer:
If the distance between the Earth and the Sun were half its present value, the number of days in a year would be approximately **129 days**.