The ratio of mean distances of three planets from the sun are `0.5 : 1: 1:5`, then the square of time periods are in the ratio of

Options

A. `1:4:9`
B. `1:9:4`
C. `1:8:27`
D. `2:1:3`

Explanation

To solve the problem, we will 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 mean distance (R) from the sun. Mathematically, this can be expressed as:

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

Given the ratio of the mean distances of the three planets from the sun as \( 0.5 : 1 : 1.5 \), we will denote these distances as:

- \( R_1 = 0.5 \)
- \( R_2 = 1 \)
- \( R_3 = 1.5 \)

### Step 1: Calculate the cube of the distances
We need to calculate the cube of each distance:

1. \( R_1^3 = (0.5)^3 = 0.125 \)
2. \( R_2^3 = (1)^3 = 1 \)
3. \( R_3^3 = (1.5)^3 = 3.375 \)

### Step 2: Write the ratios of the cubes
Now we can express the ratios of the cubes of the distances:

- The ratio of \( R_1^3 : R_2^3 : R_3^3 = 0.125 : 1 : 3.375 \)

### Step 3: Simplify the ratio
To simplify this ratio, we can multiply each term by 8 (to eliminate the decimal):

- \( 0.125 \times 8 = 1 \)
- \( 1 \times 8 = 8 \)
- \( 3.375 \times 8 = 27 \)

Thus, the ratio becomes:

\[ 1 : 8 : 27 \]

### Step 4: Relate this back to the time periods
According to Kepler's law, since \( T^2 \propto R^3 \), we can write:

\[ T_1^2 : T_2^2 : T_3^2 = R_1^3 : R_2^3 : R_3^3 \]

Therefore, the ratio of the squares of the time periods is:

\[ T_1^2 : T_2^2 : T_3^2 = 1 : 8 : 27 \]

### Final Answer
The square of the time periods are in the ratio \( 1 : 8 : 27 \).

---