Calculate the period of revolution of the neptune around the sun, given that diameter of its orbit is 30 times the diameter of the earth's orbit around the sun. Assume both the orbits to he circular.

Options

A. 165 yr
B. 163.4 yr
C. 154.3 yr
D. 164.3 yr

Answer

To calculate the period of revolution of Neptune around the Sun, we can use Kepler's Third Law of planetary motion, which states that the square of the period of revolution (T) of a planet is directly proportional to the cube of the semi-major axis (r) of its orbit. This can be expressed mathematically as:

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

### Step-by-Step Solution:

1. **Identify the Diameter of Earth's Orbit:**
The diameter of Earth's orbit around the Sun is considered as \( D_E \). The radius of Earth's orbit \( R_E \) is half of the diameter:
\[ R_E = \frac{D_E}{2} \]

2. **Determine the Radius of Neptune's Orbit:**
According to the problem, the diameter of Neptune's orbit is 30 times the diameter of Earth's orbit. Therefore, the radius of Neptune's orbit \( R_N \) is:
\[ R_N = \frac{30 \times D_E}{2} = 15 \times D_E \]

3. **Apply Kepler's Third Law:**
Using Kepler's Third Law, we can write the relationship for the periods of Earth and Neptune:
\[ \frac{T_N^2}{T_E^2} = \frac{R_N^3}{R_E^3} \]
Since the period of revolution of Earth \( T_E \) is 1 year, we have:
\[ T_E = 1 \text{ year} \]
Thus, we can simplify the equation:
\[ T_N^2 = T_E^2 \cdot \left(\frac{R_N^3}{R_E^3}\right) \]

4. **Substitute the Values:**
Substitute \( R_N = 15 \times D_E \) and \( R_E = \frac{D_E}{2} \):
\[ T_N^2 = 1^2 \cdot \left(\frac{(15 \times D_E)^3}{\left(\frac{D_E}{2}\right)^3}\right) \]
Simplifying the right side:
\[ T_N^2 = \left(\frac{15^3 \times D_E^3}{\frac{D_E^3}{8}}\right) = 15^3 \times 8 \]
\[ T_N^2 = 3375 \times 8 = 27000 \]

5. **Calculate the Period of Neptune:**
Taking the square root to find \( T_N \):
\[ T_N = \sqrt{27000} = 164.32 \text{ years} \]

Thus, the period of revolution of Neptune around the Sun is approximately **164.3 years**.

### Final Answer:
The period of revolution of Neptune around the Sun is **164.3 years**.