The distance of two planets from the sun are `10^(12)m and 10^(10)m` respectively. Then the ratio of their time periods is

Options

A. 10
B. 100
C. 1000
D. `10sqrt(10)`

Explanation

To solve the problem of finding the ratio of the time periods of two planets based on their distances from the sun, we can use Kepler's Third Law of planetary motion. According to this law, 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 around the sun.

### Step-by-Step Solution:

1. **Identify the Distances**:
- Let the distance of the first planet from the sun (R1) be \(10^{12} \, m\).
- Let the distance of the second planet from the sun (R2) be \(10^{10} \, m\).

2. **Apply Kepler's Third Law**:
- According to Kepler's Third Law, we have:
\[ \frac{T_1^2}{T_2^2} = \frac{R_1^3}{R_2^3} \]
- Here, \(T_1\) and \(T_2\) are the time periods of the first and second planets, respectively.

3. **Substitute the Values**:
- Substitute \(R_1 = 10^{12}\) and \(R_2 = 10^{10}\):
\[ \frac{T_1^2}{T_2^2} = \frac{(10^{12})^3}{(10^{10})^3} \]

4. **Simplify the Expression**:
- Calculate the cubes:
\[ \frac{T_1^2}{T_2^2} = \frac{10^{36}}{10^{30}} = 10^{6} \]

5. **Take the Square Root**:
- To find the ratio of the time periods, take the square root of both sides:
\[ \frac{T_1}{T_2} = \sqrt{10^{6}} = 10^{3} \]

6. **Final Result**:
- Therefore, the ratio of the time periods of the two planets is:
\[ T_1 : T_2 = 1000 : 1 \]

### Conclusion:
The ratio of their time periods is \(1000\).

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