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 the number of days in a year would change 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. The relationship can be expressed as:

\[ T^2 \propto R^3 \]

### Step-by-Step Solution:

1. **Identify the current situation**:
- The current average distance (R) between the Earth and the Sun is approximately 1 astronomical unit (AU).
- The current orbital period (T) of the Earth around the Sun is 1 year, which is approximately 365 days.

2. **Change in distance**:
- If the distance between the Earth and the Sun is halved, the new distance \( R' \) becomes:
\[ R' = \frac{R}{2} \]

3. **Apply Kepler's Third Law**:
- According to Kepler's Third Law:
\[ \frac{T'^2}{T^2} = \frac{R'^3}{R^3} \]
- Substituting \( R' \):
\[ \frac{T'^2}{T^2} = \frac{\left(\frac{R}{2}\right)^3}{R^3} \]
- Simplifying this gives:
\[ \frac{T'^2}{T^2} = \frac{R^3/8}{R^3} = \frac{1}{8} \]

4. **Solve for the new period \( T' \)**:
- Rearranging the equation:
\[ T'^2 = \frac{T^2}{8} \]
- Taking the square root of both sides:
\[ T' = \frac{T}{\sqrt{8}} = \frac{T}{2\sqrt{2}} \]

5. **Substituting the current period**:
- Since \( T \) is 365 days:
\[ T' = \frac{365}{2\sqrt{2}} \]
- Calculating \( 2\sqrt{2} \) gives approximately \( 2.828 \):
\[ T' \approx \frac{365}{2.828} \approx 129 \text{ days} \]

### Conclusion:
If the distance between the Earth and the Sun were halved, the number of days in a year would be approximately **129 days**.