According to Kepler, the period of revolution of a planet (T) and its mean distance from the Sun (R) are related by the equation

Options

A. `T^(2)R^3` =constant
B. `T^(2)R^(-3)` =constant
C. `TR^3` =constant
D. `T^2R` =constant

Answer

To solve the problem regarding Kepler's third law of planetary motion, we need to establish the relationship between the period of revolution of a planet (T) and its mean distance from the Sun (R).

### Step-by-step Solution:

1. **Understanding Kepler's Third Law**:
Kepler's third law states that the square of the period of revolution (T) of a planet is directly proportional to the cube of the semi-major axis (R) of its orbit. This can be mathematically expressed as:
\[ T^2 \propto R^3 \]

2. **Introducing a Constant of Proportionality**:
To convert the proportionality into an equation, we introduce a constant \( k \):
\[ T^2 = k \cdot R^3 \]
Here, \( k \) is a constant that depends on the units used and the mass of the Sun.

3. **Rearranging the Equation**:
We can rearrange the equation to express the relationship in a different form. Dividing both sides by \( R^3 \):
\[ \frac{T^2}{R^3} = k \]
This shows that the ratio \( \frac{T^2}{R^3} \) is constant for all planets orbiting the Sun.

4. **Conclusion**:
Therefore, we conclude that the relationship between the period of revolution and the mean distance from the Sun can be expressed as:
\[ T^2 \cdot R^{-3} = \text{constant} \]
This means that \( T^2 R^3 = \text{constant} \).

### Final Answer:
The relationship according to Kepler's third law is:
\[ T^2 \cdot R^3 = \text{constant} \]