The distance of two planets from the sun are `10^(13) and 10^(12)` m respectively. The ratio of the periods of the planet is

Options

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

Explanation

To find the ratio of the periods of two planets based on their distances from the sun, we can use Kepler's Third Law of Planetary Motion, which states that the square of the period of a planet (T) is directly proportional to the cube of the semi-major axis of its orbit (R). Mathematically, this can be expressed as:

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

From this, we can express the relationship between the periods and distances of two planets as:

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

Where:
- \( T_1 \) and \( T_2 \) are the periods of the first and second planets, respectively.
- \( R_1 \) and \( R_2 \) are the distances of the first and second planets from the sun, respectively.

Given:
- \( R_1 = 10^{13} \) m
- \( R_2 = 10^{12} \) m

We can now calculate the ratio of the periods:

1. **Calculate \( R_1^3 \) and \( R_2^3 \)**:
\[ R_1^3 = (10^{13})^3 = 10^{39} \]
\[ R_2^3 = (10^{12})^3 = 10^{36} \]

2. **Find the ratio \( \frac{R_1^3}{R_2^3} \)**:
\[ \frac{R_1^3}{R_2^3} = \frac{10^{39}}{10^{36}} = 10^{39 - 36} = 10^3 \]

3. **Taking the square root to find the ratio of the periods**:
Since \( \frac{T_1^2}{T_2^2} = \frac{R_1^3}{R_2^3} \), we have:
\[ \frac{T_1}{T_2} = \sqrt{\frac{R_1^3}{R_2^3}} = \sqrt{10^3} = 10^{3/2} = 10^{1.5} \]

4. **Expressing \( 10^{1.5} \)**:
\[ 10^{1.5} = 10 \times \sqrt{10} \approx 31.62 \]

Thus, the ratio of the periods of the two planets is approximately \( 10^{1.5} \) or \( 31.62 \).