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

Options

A. (a)`65`
B. (b) `129`
C. (c)`183`
D. (d)`730`

Explanation

To solve the problem of how the number of days in one year would change if the distance between the Earth and the Sun were reduced to half its present value, we can use Kepler's Third Law of planetary motion. Here’s a step-by-step solution:

### Step 1: Understand Kepler's Third Law
Kepler's Third 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. Mathematically, this is expressed as:
\[ T^2 \propto r^3 \]
This means:
\[ \frac{T_2^2}{T_1^2} = \frac{r_2^3}{r_1^3} \]

### Step 2: Define the Variables
Let:
- \( T_1 \) = current orbital period of Earth (365 days)
- \( r_1 \) = current distance between Earth and Sun
- \( r_2 \) = new distance between Earth and Sun (which is half of \( r_1 \), so \( r_2 = \frac{r_1}{2} \))
- \( T_2 \) = new orbital period of Earth (which we need to find)

### Step 3: Substitute the Values
Substituting \( r_2 \) into Kepler's Third Law gives:
\[ \frac{T_2^2}{365^2} = \frac{\left(\frac{r_1}{2}\right)^3}{r_1^3} \]

### Step 4: Simplify the Equation
This simplifies to:
\[ \frac{T_2^2}{365^2} = \frac{\frac{r_1^3}{8}}{r_1^3} \]
\[ \frac{T_2^2}{365^2} = \frac{1}{8} \]

### Step 5: Solve for \( T_2^2 \)
Now, we can solve for \( T_2^2 \):
\[ T_2^2 = 365^2 \times \frac{1}{8} \]
\[ T_2^2 = \frac{365^2}{8} \]

### Step 6: Take the Square Root
Taking the square root of both sides gives:
\[ T_2 = \frac{365}{\sqrt{8}} \]
\[ T_2 = \frac{365}{2\sqrt{2}} \]

### Step 7: Calculate \( T_2 \)
Calculating \( T_2 \):
\[ T_2 \approx \frac{365}{2 \times 1.414} \]
\[ T_2 \approx \frac{365}{2.828} \]
\[ T_2 \approx 129 \text{ days} \]

### Conclusion
If the distance between the Earth and the Sun were reduced to half its present value, the number of days in one year would be approximately 129 days.