Kepler's law starts that square of the time period of any planet moving around the sun in an elliptical orbit of semi-major axis (R) is directly proportional to

Options

A. R
B. `(1)/(R)`
C. `R^(3)`
D. `(1)/(R_(3))`

Explanation

To solve the question regarding Kepler's law, we will analyze the relationship between the time period of a planet and the semi-major axis of its elliptical orbit.

### Step-by-Step Solution:

1. **Understanding 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 (R) of its orbit. This can be mathematically expressed as:
\[ T^2 \propto R^3 \]

2. **Expressing the Proportionality as an Equation**:
To convert the proportionality into an equation, we introduce a constant of proportionality (k):
\[ T^2 = k \cdot R^3 \]
where \( k \) is a constant that depends on the gravitational parameter of the central body (in this case, the Sun).

3. **Interpreting the Relationship**:
This equation indicates that if you know the semi-major axis (R) of a planet's orbit, you can calculate the square of its orbital period (T) by multiplying \( R^3 \) by the constant \( k \).

4. **Conclusion**:
Therefore, the answer to the question is that the square of the time period of any planet moving around the Sun in an elliptical orbit is directly proportional to the cube of the semi-major axis of its orbit:
\[ T^2 \propto R^3 \]

### Final Answer:
The square of the time period (T²) of any planet is directly proportional to the cube of the semi-major axis (R³) of its orbit.

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