Two planets are at distance `R_1 and R_2` from the Sun. Their periods are `T_1 and T_2` then `(T_1/T_2)^2` is equal to

Options

A. `R_1/R_2`
B. `(R_1/R_2)^2`
C. `(R_1/R_2)^3`
D. `(R_2/R_1)^3`

Answer

To solve the problem, we will use Kepler's Third Law of Planetary Motion, which states that the square of the period of orbit (T) of a planet is directly proportional to the cube of the semi-major axis (R) of its orbit around the Sun.

### Step-by-Step Solution:

1. **Understanding Kepler's Third Law**:
According to Kepler's Third Law, we have the relationship:
\[ T^2 \propto R^3 \]
This means that for any two planets orbiting the Sun, we can express this relationship as:
\[ \frac{T_1^2}{T_2^2} = \frac{R_1^3}{R_2^3} \]

2. **Rearranging the Equation**:
From the above relationship, we can rearrange it to find the ratio of the periods:
\[ \left(\frac{T_1}{T_2}\right)^2 = \frac{R_1^3}{R_2^3} \]

3. **Final Expression**:
Therefore, we can express the square of the ratio of the periods in terms of the distances:
\[ \left(\frac{T_1}{T_2}\right)^2 = \left(\frac{R_1}{R_2}\right)^3 \]

4. **Conclusion**:
The final answer to the question is:
\[ \left(\frac{T_1}{T_2}\right)^2 = \frac{R_1^3}{R_2^3} \]