If the distance of earth form the sun were half the present value, how many days will make one year?

Explanation

To solve the problem of how many days would make one year if the distance of the Earth from the Sun were half the present value, we can use Kepler's Third Law of Planetary Motion. This law states that the square of the time period of a planet's orbit is directly proportional to the cube of the semi-major axis of its orbit.

### Step-by-Step Solution:

1. **Identify the Known Values:**
- Current time period of Earth (T1) = 365 days
- Current distance from the Sun (R1) = 1 astronomical unit (AU)
- New distance from the Sun (R2) = R1 / 2 = 1/2 AU

2. **Apply Kepler's Third Law:**
According to Kepler's Third Law:
\[ \frac{T1^2}{T2^2} = \frac{R1^3}{R2^3} \]
Where:
- T1 = 365 days (current time period)
- T2 = new time period (what we want to find)
- R1 = 1 AU (current distance)
- R2 = 1/2 AU (new distance)

3. **Substitute the Known Values:**
Substitute R1 and R2 into the equation:
\[ \frac{365^2}{T2^2} = \frac{1^3}{(1/2)^3} \]
Simplifying the right side:
\[ \frac{1}{(1/2)^3} = \frac{1}{1/8} = 8 \]
So the equation becomes:
\[ \frac{365^2}{T2^2} = 8 \]

4. **Rearranging the Equation:**
Rearranging gives us:
\[ T2^2 = \frac{365^2}{8} \]

5. **Calculating T2:**
Now calculate \( T2 \):
\[ T2 = \sqrt{\frac{365^2}{8}} = \frac{365}{\sqrt{8}} = \frac{365}{2\sqrt{2}} = \frac{365 \cdot \sqrt{2}}{4} \]
Approximating \( \sqrt{2} \approx 1.414 \):
\[ T2 \approx \frac{365 \cdot 1.414}{4} \approx \frac{517.1}{4} \approx 129.3 \text{ days} \]

6. **Final Answer:**
Thus, if the distance of the Earth from the Sun were half the present value, one year would be completed in approximately **129 days**.