If a new planet is discovered rotating around Sun with the orbital radius double that of earth, then what will be its time period (in earth's days)

Answer

To find the time period of a new planet that is discovered rotating around the Sun with an orbital radius double that of Earth, we can use Kepler's Third Law of planetary motion. This law states that the square of the time 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 known values**:
- Let the orbital radius of Earth be \( R \).
- The time period of Earth \( T \) is 365 days.
- The new planet has an orbital radius \( R' = 2R \).

2. **Apply Kepler's Third Law**:
- According to Kepler's Third Law:
\[ T^2 \propto R^3 \]
- For Earth:
\[ T^2 = k \cdot R^3 \]
- For the new planet:
\[ T'^2 = k \cdot (R')^3 = k \cdot (2R)^3 \]

3. **Express \( T' \) in terms of \( T \)**:
- Substitute \( R' = 2R \) into the equation:
\[ T'^2 = k \cdot (2R)^3 = k \cdot 8R^3 \]
- Since \( T^2 = k \cdot R^3 \), we can relate \( T' \) and \( T \):
\[ T'^2 = 8 \cdot T^2 \]

4. **Take the square root**:
- Taking the square root of both sides gives:
\[ T' = \sqrt{8} \cdot T \]
- Since \( \sqrt{8} = 2\sqrt{2} \), we have:
\[ T' = 2\sqrt{2} \cdot T \]

5. **Substitute the value of \( T \)**:
- Substitute \( T = 365 \) days:
\[ T' = 2\sqrt{2} \cdot 365 \]

6. **Calculate \( T' \)**:
- Calculate \( 2\sqrt{2} \):
\[ 2\sqrt{2} \approx 2 \cdot 1.414 \approx 2.828 \]
- Therefore:
\[ T' \approx 2.828 \cdot 365 \approx 1032.42 \text{ days} \]
- Rounding gives approximately:
\[ T' \approx 1032 \text{ days} \]

### Final Answer:
The time period of the new planet is approximately **1032 days**.