The ratio of distance of two satellites from the centre of earth is `1:4`. The ratio of their time periods of rotation will be

Options

A. `1:4`
B. `4:1`
C. `1:8`
D. `8:1`

Explanation

To solve the problem, we need to find the ratio of the time periods of two satellites based on their distances from the center of the Earth. We will use Kepler's Third Law of planetary motion, which states that the square of the time period (T) of a satellite is directly proportional to the cube of the semi-major axis (r) of its orbit.

### Step-by-step Solution:

1. **Understanding the Given Ratio**:
- Let the distances of the two satellites from the center of the Earth be \( r_1 \) and \( r_2 \).
- According to the problem, the ratio of their distances is given as:
\[ \frac{r_1}{r_2} = \frac{1}{4} \]

2. **Applying Kepler's Third Law**:
- Kepler's Third Law states:
\[ T^2 \propto r^3 \]
- This can be expressed as:
\[ \frac{T_1^2}{T_2^2} = \frac{r_1^3}{r_2^3} \]

3. **Substituting the Ratio of Distances**:
- From the ratio of distances, we have:
\[ r_2 = 4r_1 \]
- Therefore, substituting \( r_2 \) in terms of \( r_1 \) into the equation gives:
\[ \frac{T_1^2}{T_2^2} = \frac{r_1^3}{(4r_1)^3} \]
- Simplifying the right side:
\[ \frac{T_1^2}{T_2^2} = \frac{r_1^3}{64r_1^3} = \frac{1}{64} \]

4. **Finding the Ratio of Time Periods**:
- Taking the square root of both sides to find the ratio of the time periods:
\[ \frac{T_1}{T_2} = \sqrt{\frac{1}{64}} = \frac{1}{8} \]

5. **Final Result**:
- Therefore, the ratio of the time periods of the two satellites is:
\[ \frac{T_1}{T_2} = \frac{1}{8} \]

### Conclusion:
The ratio of the time periods of the two satellites is \( 1:8 \).