The period T and mean distance R of a planet from the sun are related by

Options

A. `T^2 prop R^2`
B. `T prop R^3`
C. `T^3 prop R^2`
D. `T^(2//3) prop R`

Answer

To solve the problem regarding the relationship between the period \( T \) and the mean distance \( R \) of a planet from the sun, we can use Kepler's Third Law of planetary motion. Here’s a step-by-step solution:

### Step 1: Understand Kepler's Third Law
Kepler's Third Law states that the square of the orbital period \( T \) of a planet is directly proportional to the cube of the semi-major axis (mean distance) \( R \) of its orbit. Mathematically, this can be expressed as:
\[ T^2 \propto R^3 \]

### Step 2: Introduce a Constant of Proportionality
To express this relationship in an equation, we introduce a constant \( k \):
\[ T^2 = k R^3 \]

### Step 3: Rearranging the Equation
To analyze the relationship between \( T \) and \( R \), we can rearrange the equation. Taking the cube root of both sides gives:
\[ T^{2/3} = k' R \]
where \( k' = k^{1/3} \).

### Step 4: Interpret the Result
This shows that the period \( T \) raised to the power of \( \frac{2}{3} \) is directly proportional to the mean distance \( R \):
\[ T^{2/3} \propto R \]

### Conclusion
Thus, we conclude that the period \( T \) and the mean distance \( R \) of a planet from the sun are related by the equation:
\[ T^{2/3} = k R \]
where \( k \) is a constant.