According to Kepler's law of planetary motion, if T represents time period and `r` is orbital radius, then for two planets these are related as

Options

A. `((T_(1))/(T_(2)))^(3//2)=(r_(1))/(r_(2))`
B. `(T_(1))/(T_(2))=[(r_(1))/(r_(2))]`
C. `[(T_(1))/(T_(2))]^(2)=[(r_(1))/(r_(2))]^(3)`
D. `(T_(1)^(2))/(T_(2))=(r_(1)^(3))/(r_(1))`

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 semi-major axis (r) of its orbit.

### Step-by-Step Solution:

1. **Understanding Kepler's Third Law**: According to Kepler's Third Law, we can express the relationship between the time period (T) and the orbital radius (r) as:
\[ T^2 \propto r^3 \]
This means that for any two planets, the ratio of the squares of their time periods is equal to the ratio of the cubes of their orbital radii.

2. **Setting up the relationship for two planets**: Let’s denote the time periods and orbital radii of two planets as \( T_1 \), \( T_2 \) and \( r_1 \), \( r_2 \) respectively. According to Kepler's law, we can write:
\[ \frac{T_1^2}{T_2^2} = \frac{r_1^3}{r_2^3} \]

3. **Rearranging the equation**: We can rearrange this equation to express the relationship between the time periods and the orbital radii:
\[ \frac{T_1^2}{T_2^2} = \frac{r_1^3}{r_2^3} \]
This shows that the ratio of the squares of the time periods of the two planets is equal to the ratio of the cubes of their orbital radii.

4. **Final expression**: Hence, we can conclude that:
\[ \frac{T_1^2}{T_2^2} = \frac{r_1^3}{r_2^3} \]

### Conclusion:
The relationship between the time periods and the orbital radii of two planets can be summarized as:
\[ \frac{T_1^2}{T_2^2} = \frac{r_1^3}{r_2^3} \]