The distance of the two planets from the Sun are `10^(13)m` and `10^(12) m`, respectively. Find the ratio of time periods of the two planets.

Options

A. `(1)/(sqrt(10))`
B. 100
C. `10sqrt(10)`
D. `sqrt(10)`

Answer

To find the ratio of the time periods of two planets orbiting the Sun, we can use Kepler's Third Law of planetary motion, which 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. This can be expressed mathematically as:

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

From this, we can derive the relationship for two planets:

\[ \frac{T_1^2}{T_2^2} = \frac{R_1^3}{R_2^3} \]

Where:
- \( T_1 \) and \( T_2 \) are the time periods of the two planets.
- \( R_1 \) and \( R_2 \) are the distances of the two planets from the Sun.

### Step-by-Step Solution:

1. **Identify the distances of the planets from the Sun:**
- For Planet 1, \( R_1 = 10^{13} \, \text{m} \)
- For Planet 2, \( R_2 = 10^{12} \, \text{m} \)

2. **Calculate the ratio of the distances:**
\[ \frac{R_1}{R_2} = \frac{10^{13}}{10^{12}} = 10 \]

3. **Use Kepler’s Third Law to find the ratio of the time periods:**
\[ \frac{T_1^2}{T_2^2} = \frac{R_1^3}{R_2^3} \]

4. **Substituting the distances into the equation:**
\[ \frac{T_1^2}{T_2^2} = \frac{(10^{13})^3}{(10^{12})^3} = \frac{10^{39}}{10^{36}} = 10^{3} \]

5. **Taking the square root to find the ratio of the time periods:**
\[ \frac{T_1}{T_2} = \sqrt{10^{3}} = 10^{3/2} = 10 \sqrt{10} \]

6. **Conclusion:**
Thus, the ratio of the time periods of the two planets is:
\[ T_1 = 10 \sqrt{10} \, T_2 \]

### Final Answer:
The ratio of the time periods of the two planets is \( T_1 : T_2 = 10 \sqrt{10} : 1 \).

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